Probability Seminar Essen

Summer Semester 2023 (back to Contents)

Apr 04

Johannes Alt (University of Bonn)

Spectral Phases of Erdős–Rényi graphs
We consider the Erdős–Rényi graph on vertices with edge probability . It is well known that the structure of this graph changes drastically when is of order . Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of eigenvectors remains delocalized. In this talk, I will present the phase diagram depicting these localized and delocalized phases and our recent progress in establishing it rigorously.
This is based on joint works with Raphael Ducatez and Antti Knowles.

Apr 18

Arzu Ahmadova (University of Duisburg-Essen)

Convergence results for gradient systems in the artificial neural network training
The field of artificial neural network (ANN) training has garnered significant attention in recent years, with researchers exploring various mathematical techniques for optimizing the training process. Among these, deep neural networks, or deep learning, have gained popularity due to their ability to learn complex and abstract features from large datasets. In particular, this paper focuses on advancing the current understanding of gradient flow and gradient descent optimization methods. Our aim is to establish a solid mathematical convergence theory for continuous-time gradient flow equations and gradient descent processes, and show that bounded trajectories of gradient system dynamics actually converge to the set of extrema.
This talk is based on a joint work with Martin Hutzenthaler.

May 02

Dominik Schmid (University of Bonn)

Biased random walk on dynamical percolation
We consider a biased random walk on dynamical percolation and discuss the existence and the properties of the linear speed as a function of the bias. In particular, we establish a simple criterion to decide whether the speed is increasing or decreasing for large bias. This talk is based on joint work with Sebastian Andres, Nina Gantert, and Perla Sousi.

Jun 13

David Criens (University of Freiburg)

Propagation of Chaos for Controlled Mean Field SPDEs
The area of controlled McKean—Vlasov equations, also known as mean field control, has rapidly developed in the past years. Motivated for instance by financial applications of controlled delay equations, there is recently increasing interest in infinite dimensional mean field control. In a recent paper, Cosso et al (AAP, to appear) study controlled mean field SPDEs from a general point of view. They establish well-posedness, the DPP and a Bellman equation. Mean field dynamics are usually motivated by propagation of chaos, i.e., as limits of interacting particles. In this talk I discuss such a propagation of chaos motivation for controlled mean field SPDEs. In particular, I introduce a novel formulation of the corresponding propagation of chaos via Hausdorff metric convergence. The talk is based on work in progress. 

Jul 11

Lianne de Jonge (University of Osnabrück)

Coverage by an inhomogeneous Poisson cylinder model
We consider a random cylinder model that is constructed by placing points using a Poisson point process in and drawing a random line in through each of these points. The union of these lines defines a line set. The Minkowski sum of the line set with a -dimensional ball around the origin defines a cylinder set. In this talk, we will discuss a central limit theorem for the number of holes in the line set when the intensity of the point process goes to infinity. For the cylinder set, we discuss asymptotics of the minimum cylinder radius required to completely cover the -dimensional unit cube.
This talk is based on ongoing work with Hanna Döring and Xiaochuan Yang.


Winter Semester 2022/23 (back to Contents)

Feb 14

Sergey Tikhomirov (University of Duisburg-Essen)

Probabilistic aspects of shadowing for dynamical systems
Shadowing property is well-known topic in the context of hyperbolic theory of dynamical systems. The system has shadowing property if near any approximate trajectory (pseudotrajectory) there exists an exact one. Due to more recent works by Sakai, Abdenur, Diaz, Pilyugin, Tikhomirov shadowing is "almost" equivalent to hyperbolicity.  
At the same time numerical experiments by Hammel-Grebogi-Yorke for logistics and Henon maps shows that shadowing holds for relatively long pseudotrajectories. It poses a question which type of shadowing holds for systems, which are not necessarily hyperbolic. I consider probabilistic approach for the topic. I show that for infinite pseudotrajectories it does not change the notion. At the same time it shows that relatively long pseudotrajectories can be shadowed by exact trajectory with high probability. The main technique is a reduction to special form of gambler's ruin problems and mild form of large deviation principle for random walks. We show that our approach works for several examples -- skew product maps.

Knowledge of dynamical systems is not assumed. In the talk all necessarily notions such as shadowing, hyperbolicity, skew products will be introduced.
The talk is based on joint works with G. Monakov.

Jan 31

Jonas Arista (University of Bielefeld)

On some path transformations for random walks and Markov intertwinings
In this talk, I describe some interesting path transformations for random walks and establish (discrete-time) analogues of the geometric Pitman 2M-X theorem of Matsumoto and Yor and of the classical Dufresne identity, for a multiplicative random walk with Beta type II distributed increments. The proof of these results is based on certain Markov intertwinings between the associated transition kernels.

Jan 24

Sara Terveer (University of Bielefeld)

Central Limit Theorems for Hitting Times of Random Walks on Erdös-Rényi Random Graphs
In 2014, Löwe and Torres obtained a law of large numbers for the expected average starting and target hitting time of random walks on Erdös-Rényi random graphs. In this talk, we will analyze fluctuations around this law of large numbers and give insight into the main tools in the proof of asymptotic normality. Based on joint work with Matthias Löwe.

Jan 17 Osvaldo Angtuncio Hernandez (University of Duisburg-Essen)

Convergence of an Aldous-Broder-type algorithm on the discrete torus. 
The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with $N$ vertices. The Aldous-Broder Markov chain on a graph $G=(V,E)$ is a Markov chain with values in the space of rooted trees whose vertex set is a subset of $V$ with the uniform distribution on the space of rooted trees spanning $G$. In Evans, Pitman and Winter (2006) the so-called root growth with regrafting process (RGRG) was constructed. Further it was shown that the suitable rescaled Aldous-Broder Markov chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology. It was shown in Peres and Revelle (2005) that (up to a dimension depending constant factor) the continuum random tree is with respect to the Gromov-weak topology the scaling limit of the uniform spanning tree on the torus $\mathbb{Z}_N^d$, $d\ge 5$. In the present talk we discuss that a dynamic similar to the rescaled Aldous-Broder Markov chain on $\mathbb{Z}_N^d$ for $d\ge 5$, converges to the RGRG weakly with respect to the Gromov-Hausdorff topology when initially started in the trivial rooted tree.

Nov 15th

Thomas van Belle (University of Duisburg-Essen)

The consensus time of the voter model on Erdös-Rényi graphs
The voter model is a paradigmatic interacting particle system modelling the spread of opinions among agents on a social network. It is well known that the voter model is dual to a system of coalescing random walks, and therefore the consensus time of the voter model is almost surely finite on graphs where the simple random walk is recurrent. We develop a technique to determine the asymptotic consensus time of the voter model using spectral information about the transition matrix of a simple random walk on the graph. In particular, we will apply this technique to the voter model on connected Erdős–Rényi graphs.

Summer Semester 2022 (back to Contents)

Apr 12

Stefan Ankirchner (University of Jena)

Approximating stochastic gradient descent with diffusions: error expansions and impact of learning rate schedules

Applying a stochastic gradient descent method for minimizing an objective gives rise to a discrete-time process of estimated parameter values. In order to better understand the dynamics of the estimated values it can make sense to approximate the discrete-time process with a continuous-time diffusion. We refine some results on the weak error of diffusion approximations. In particular, we explicitly compute the leading term in the error expansion of an ODE approximation with respect to a parameter discretizing the learning rate schedule. The leading term changes if one extends the ODE with a Brownian diffusion component. Finally, we show that if the learning rate is time varying, then its rate of change needs to enter the drift coefficient in order to obtain an approximation of order 2.
The talk is based on joint work with Stefan Perko.

May 10
May 17

Steffen Dereich (University of Münster)

On the existence of optimal shallow networks

In this talk we discuss existence of global minima in optimisation problems over shallow neural networks. More explicitly, the function class over which we minimise is the family of all functions that can be expressed as artificial neural networks with one hidden layer featuring a specified number of neurons with ReLU (or Leaky ReLU) activation and one linear neuron (without activation function). We give existence results. Moreover, we provide counterexamples that illustrate the relevance of the assumptions imposed in the theorems.
The talk is based on joint work with Arnulf Jentzen (Münster) and Sebastian Kassing (Bielefeld).

May 24

Tobias Werner (University of Kassel)

Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations

Deep-learning based algorithms are employed in a wide field of real world applications and are nowadays the standard approach for most machine learning related problems. They are used extensively in face and speech recognition, fraud detection, function approximation, and solving of partial differential equations (PDEs). The latter are utilized to model numerous phenomena in nature, medicine, economics, and physics. Often, these PDEs are nonlinear and high-dimensional. For instance, in the famous BlackScholes model the PDE dimension $d \in \mathbb{N}$ corresponds to the number of stocks considered in the model. Relaxing the rather unrealistic assumptions made in the model results in a loss of linearity within the corresponding Black-Scholes PDE.
These PDEs, however, can not in general be solved explicitly and therefore need to be approximated numerically. Classical grid based approaches such as finite difference or finite element methods suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the PDE dimension $d \in \mathbb{N}$
and are therefore not appropriate. Among the most prominent and promising approaches to solve high-dimensional PDEs are deep-learning algorithms, which seem to handle highdimensional problems well.
However, compared to the vast field of applications these techniques are successfully applied to nowadays,
there exist only few theoretical results proving that deep neural networks do not suffer from the curse of
dimensionality in the numerical approximation of partial differential equations.
I will present a result, stating that deep neural networks are capable of approximating solutions of semilinear Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the required number of parameters in the networks grow at most polynomially in both dimension
$d \in \mathbb{N}$ and prescribed reciprocal accuracy ". Previously, this has only been proven in the case of semilinear heat equations.
The result is purely deterministic. However, the proof heavily relies on probabilistic tools, in particular
on full history recursive multilevel Picard approximations (MLP).
This talk is based on joint work with Martin Hutzenthaler and Petru A. Cioica-Licht

May 31

Josué Nussbaumer (Université Gustave Eiffel)

Algebraic two-level measure trees

Wolfgang Löhr and Anita Winter introduced algebraic trees which generalize the notion of graph-theoretic trees to potentially uncountable structures. They equipped the space of binary algebraic measure trees with a topology that relies on the Gromov-weak convergence of particular metric representations. They showed that this topology is compact and equivalent to the sample shape convergence on the subspace of binary algebraic measure trees, by encoding the latter with triangulations of the circle. We extended these results to a two-level setup, where algebraic trees are equipped with a probability measure on the set of probability measures. To do so, we encoded algebraic two-level measure trees with triangulations of the circle together with a two-level measure on the circle line. As an application, we constructed the algebraic nested Kingman coalescent.

Jun 14
Postponed to June 21st.
Jun 21

Katharina Pohl (University Duisburg-Essen)

On existence and uniqueness properties for solutions of stochastic fixed point equations with gradient-dependent nonlinearities

In this talk we discuss the connection between partial differential equations (PDEs) and stochastic fixed point equations (SFPEs) with the goal to find a stochastic representation of PDE solutions. To achieve this, we combine SFPEs arising from an application of the the Itô formula with the Bismut-Elworthy-Li formula.  Our main result shows the existence and uniqueness of solutions of SFPEs associated with suitable PDEs with Lipschitz continuous and gradient-dependent nonlinearities.
This talk is based on joint work with Martin Hutzenthaler.

Jul 05

Barbara Rüdiger-Mastandrea (University of Wuppertal)

The Boltzmann  –Enskog process

The theory of SDEs with Poisson noise  is used  here  to identify the „Boltzmann –Enskog –Process“. Its dynamic describes the space and velocity  evolution of a particle of a rarified gas which density evolves according to the Boltzmann – Enskog equation. This random dynamic is identified by a stochastic process solving a SDE, for  which the corresponding    Kolmogorov  equation is given by the „Boltzmann –Enskog equation“. It turns out that this is   the  solution of a Mc Kean –Vlasov type SDE  with Poisson noise: its compensator is determined by the  density solving the „Boltzmann –Enskog equation.
The talk is based on joint results with S. Albeverio and P. Sundar, as well as , M. Friesen and P. Sundar.


Winter Semester 2021/22 (back to Contents)

Oct 12

Tatiana Orlova (University of Duisburg-Essen)

Spectral Bootstrap confidence bands for Lévy-driven moving average processes

This report will consider the problem of constructing bootstrap confidence intervals for the Lévy density of the driving Lévy process based on highfrequency observations of a Lévy-driven moving average processes. Using a spectral estimator of the Lévy density, we propose a novel implementations of multiplier  bootstraps to construct confidence bands on a compact set away from the origin. We also provide conditions under which the confidence bands are asymptotically valid.


Oct 19

Philipp Petersen (University of Vienna)

Approximation and estimation of functions with structured singularities by deep neural networks

Functions with structured singularities appear in many application domains. For example, in image processing, jumps in intensity value are often associated to the boundaries of physical objects and appear along curves. In classification problems, decision boundaries are often assumed to possess a certain regularity. Finally, in the modelling of physical processes, such as fracture mechanics, singularities naturally appear. In the associated application areas, it is then necessary to finely resolve the singularities. We will analyse the extent to which deep neural networks are affected by discontinuous target functions if the discontinuity admits certain regularities. We will demonstrate that deep neural networks are surprisingly efficient at resolving discontinuities of potentially very high dimensional functions. 

Oct 26

Adrián González Casanova (UNAM)

[The talk takes place on site]

Lambda Selection and where to find it

There are many ways in which two populations can compete for resources and this leads to different forms of selection. A Lambda coalescent is a beautiful mathematical object that describes the genealogy of populations with skewed offspring distributions. We will discuss selection in the context populations in the Lambda universality class. Along the way, we will talk about a novel technique to relate coalescent processes with branching processes and about duality of Markov processes.

Most of the talk is based on joint work with Maria Emilia Caballero (UNAM, Mexico) and Jose Luis Perez (CIMAT, Mexico)


Nov 2
Yuri Kabanov (Lomonosov MSU and Université Bourgogne Franche-Comté)
Ruin probabilities for models with investments
We present several new results on asymptotics of ruin probabilities for models of insurance company investing its reserve in a risky asset including Sparre Andersen model with investment and models where the stock price has stochastic volatility.


Nov 9

Yan Dolinsky (Hebrew University of Jerusalem)

What if we knew what the future brings?

In this paper we study optimal investment when the investor can peek some time unites into the future, but cannot fully take advantage of this knowledge because of quadratic transaction costs. In the Bachelier setting with exponential utility, we give an explicit solution to this control problem with intrinsically infinite-dimensional memory. This is made possible by solving the dual problem where we make use of the theory of Gaussian Volterra integral equations. (joint work with P.Bank and M.Rasonyi)
Nov 23

Khoa N Le (TU Berlin)

[The talk takes place on site]

Taming singular SDEs numerically

We consider a generic and explicit  tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusion coefficient is uniformly elliptic, Hölder continuous and weakly differentiable in the spatial variables while the drift satisfies the Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and Röckner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation.  A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and be fine-tuned to achieve the standard 1/2-strong convergence rate with a logarithmic factor. Joint work with Chengcheng Ling, arXiv:2110.01343
Nov 30

Thomas Kruse (University of Giessen)

Inhomogeneous affine Volterra processes

We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel $K(t,s)$ and inhomogeneous drift and diffusion coefficients $b(s,X_s)$ and $\sigma(s,X_s)$. In the case of affine $b$ and $\sigma \sigma^T$ we show how the conditional Fourier-Laplace functional can be represented by a solution of an inhomogeneous Riccati-Volterra integral equation. 
For a kernel of convolution type $K(t,s)=\overline {K}(t-s)$ we establish existence of a solution to the stochastic inhomogeneous Volterra equation. 
If in addition $b$ and $\sigma \sigma^T$ are affine, we prove that the conditional Fourier-Laplace functional is exponential-affine in the past path. 

Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance. The talk is based on joint work with Julia Ackermann and Ludger Overbeck


Dec 7

Larisa Yaroslavtseva (University of Passau)

On strong approximation of SDEs with a discontinuous drift coefficient

In recent years, a number of results has been proven in the literature for strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space.

In many of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. In particular, for such SDEs, the classical L_p-error rate 1/2 has been recently proven in the literature for Euler-type schemes.

In this talk we discuss higher order methods for strong approximation of scalar SDEs of that type. We present a Milstein-type scheme that achieves an L_p-error rate 3/4 for approximation of the solution at the final time point in terms of the number of evaluations of the driving Brownian motion. We furthermore show that the L_p-error rate 3/4 can not be improved in general for such SDEs by no numerical method based on evaluations of the driving Brownian motion at fixed time points and, finally, we present a numerical method based on sequential evaluations of the driving Brownian motion, which achieves an L_p-error rate of at least 1 in terms of the average number of evaluations of the driving Brownian motion.

The talk is based on joint work with Thomas Mueller-Gronbach (University of Passau).

Dec 14

David Criens (University of Freiburg)

[The talk takes place on site]

A closer look at some properties of diffusions
In the talk I will discuss three questions related to multi and one dimensional diffusions. First I explain that the UI martingale property of a certain stochastic exponential (which is built from a multi dimensional diffusion) can be characterized by an explosion property of a drift and time changed test diffusion. Then I turn to a one dimensional setting and discuss the relation of the Feller—Dynkin and the martingale property of regular diffusions on natural scale. Finally I take a closer look at the connection of regular diffusions on natural scale and their speed measures. More precisely, I will explain a homeomorphic relation.
Dec 21
Simone Baldassarri(University of Florence)
Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions
In this talk we analyze metastability and nucleation in the context of the Kawasaki dynamics for the two-dimensional Ising lattice gas at very low temperature. Let $\Lambda\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Along each bond touching the boundary of $\Lambda$ from the outside to the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along each bond from the inside to the outside, particles are annihilated with rate $1$, where $\beta>0$ is the inverse temperature and $\Delta>0$ is an activity parameter. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $\Delta\in{(U_1,U_1+U_2)}$, so that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit as $\beta\rightarrow\infty$.  We investigate how the transition from empty to full takes place with particular attention to the critical configurations that asymptotically have to be crossed with probability 1. To this end, we provide a model-independent strategy to identify some unessential saddles (that are not in the union of minimal gates) for the transition from the metastable (or stable) to the stable states and we apply this method to our model. The derivation of some geometrical properties of the saddles allows us to identify the full geometry of the minimal gates and their boundaries for the nucleation in the strongly anisotropic case. We observe very different behaviors for this case with respect to the isotropic ($U_1=U_2$) and weakly anisotropic ($U_1<2U_2$) ones. Moreover, we derive sharp estimates for the asymptotic transition time for the strongly anisotropic case. This is based on a joint work with F. R. Nardi.
Jan 11

Julia Ackermann (University of Giessen)

Càdlàg semimartingale strategies for optimal trade execution in stochastic order book models
After a short introduction to optimal trade execution, we set up an order book model in continuous time where both order book depth and resilience may evolve randomly in time. We discuss the class of execution strategies and necessary modifications to the dynamics of the price deviation process and the cost functional. Next, we turn to a quadratic BSDE that under appropriate assumptions characterizes the solution of our control problem. We then find the minimal costs, identify conditions under which an optimal strategy exists, and, in that case, provide an expression for the optimal strategy. Furthermore, we illustrate the behavior of optimal strategies in several examples.

The talk is based on joint work with Thomas Kruse and Mikhail Urusov.

Jan 18

Kristin Kirchner (TU Delft)

A new class of spatiotemporal statistical models based on fractional stochastic PDEs

Most environmental data sets contain measurements collected over space and time. It is the purpose of spatiotemporal statistical models to adequately describe the underlying uncertain spatially explicit phenomena evolving over time. In this talk I will present a new class of spatiotemporal statistical models which is based on stochastic partial differential equations (SPDEs) involving fractional powers of parabolic operators. By means of semigroup theory the corresponding solution processes are rigorously defined, and their spatial and temporal regularity can be quantified. These regularity results provide a key motivation for employing this class of SPDEs in statistical applications: Namely, spatial and temporal smoothness are controlled via two positive parameters, which may be estimated from data in statistical inference. Besides this property, I will discuss further modeling advantages including the long-time behavior and marginal covariance structures.
This talk is based on joint work with Joshua Willems. 


Summer Semester 2021 (back to Contents)

Apr 20

Mark C. Veraar (TU Delft)
Maximal inequalities for stochastic convolutions

In this talk I will present a new approach to maximal inequalities for stochastic convolutions both in discrete and continuous time. The proofs are based on extensions of the vector-valued Burkholder-Rosenthal inequalities to the non-martingale setting. Our results are new in both Hilbert spaces and 2-smooth Banach spaces. Applications to stochastic evolution equations will be discussed as well. Here we prove the existence of continuous modifications and convergence of discretization schemes in time.

May 11

Jean Daniel Mukam (TU Chemnitz)
A Magnus-type integrator for semilinear parabolic non-autonomous SPDEs

Stochastic partial differential equations (SPDEs) are widely used to model many real world phenomena such as stock market prices and fluid flows. Since explicit solutions of many SPDEs are unknown, developing numerical schemes is a good alternative to provide their approximations, and is therefore a hot topic. Numerical methods for autonomous SPDEs are thoroughly investigated in the literature, while to the best of our knowledge the non-autonomous cases are not yet well understood. In this talk, we propose a Magnus-type integrator for time-dependent stochastic advection-reaction-diffusion equation, which is based on an approximation of the Magnus series [1]. We use finite element method for the approximation in space and provide the strong convergence error of the fully discrete scheme toward the mild solution.

Jun 1

Sarai Hernandez-Torres (Technion – Israel Institute of Technology)
Three-dimensional uniform spanning trees and loop-erased random walks

The uniform spanning tree (UST) on Z^3 is the infinite-volume limit of uniformly chosen spanning trees of large finite subgraphs of Z^3. The main theorem in this talk is the existence of subsequential scaling limits of the UST on Z^3. We get convergence over dyadic subsequences. An essential tool is Wilson’s algorithm, which samples uniform spanning trees by using loop-erased random walks (LERW).  This strategy imposes a restriction: results for the scaling limit of 3D LERW constrain the corresponding results for the 3D UST. We will comment on work in progress for the LERW that leads to the full convergence to the scaling limit of the UST.  This talk is based on joint work with Omer Angel, David Croydon, and Daisuke Shiraishi; and work in progress with Xinyi Li and Daisuke Shiraishi. 

Jun 8

Alexandra Neamtu (TU Konstanz)
Dynamical systems for stochastic evolution equations with fractional noise

We analyze stochastic partial di˙erential equations (SPDEs) driven by an infinite-dimensional fractional Brownian motion using rough paths techniques. Since the breakthrough in the rough paths theory there has been a huge interest in investigating SPDEs with rough noise. We con-tribute to this aspect and develop a solution theory, which can further be applied to dynamical systems generated by such SPDEs. This talk is based on joint works with Robert Hesse and Christian Kuehn.

Jun 15

Marcel Ortgiese (University of Bath)
Peturbations of preferential attachment networks

Preferential attachment networks form a popular class of evolving random graph models that share many features with real-life networks. The basic mechanism is that newly incoming nodes connect preferably to old vertices with high degree. We consider a perturbation of these networks, where the attractiveness of nodes is randomly perturbed. We can identify two different phases: if the perturbation is small, then the model behaves as if the perturbation is replaced by its mean, while if the perturbation is strong then the system is essentially driven by the extremes of the perturbation. In both cases, we have a detailed understanding of the behaviour of the degree of a typical vertex as well as the largest degree in the system. In particular, we show that for small perturbations `the old get richer' phenomenon is true, while in the other case younger nodes can compete. We will also compare these results to a class of evolving random graphs, where the preferential attachment mechanism is `switched off' that can be seen as a version of the well-known random recursive tree, but now in a random environment.
Joint work with Bas Lodewijks.

Jun 22 René Schilling (TU Dresden)
On the Liouville Property for Generators of Lévy Processes

We show necessary and sufficient conditions for the Liouville property and the strong Liouville property to hold for generators of Lévy processes. This extends the classical Liouville property known for Brownian motion, Random walks and the (discrete) Laplacian.

Jun 29

Leif Döring (University of Mannheim)
Results on stable SDEs (without drift) 

One-dimensional Brownian SDEs have been studied for decades, many techniques were developed to give a full picture of existence/uniqueness/properties. The situation is different for SDEs driven by stable processes, only for $\alpha>1$ the classical local time tricks can be extended to prove Engelbert-Schmidt type results. We discuss recent results for $\alpha<1$.

Jul 6

Ulrich Horst (Humboldt University of Berlin)
Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies

We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with with semi-martingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature as well as a microscopic foundation for the use of semi-martingale liquidation strategies. Our results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes. The talk is based on joint work with Evgueni Kivman.    

Jul 13

Laura Eslava (UNAM, México)
A branching process with deletions and mergers that matches the threshold for hypercube percolation

We define a graph process $G(p,q)$ based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$. We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on these graphs. However, it is left open whether the survival probability of $G(p,q)$ is monotone in $p$ or $q$.


Winter Semester 2020/21 (back to Contents)

Nov 10

Matthias Erbar (University of Bielefeld)
Optimal transport, stationary point processes, and log gases

In this talk I would like give an overview of the topics and aims of the project "Optimal transport for stationary point processes" within the SPP "Random geometric systems". I will give a brief introduction of the classical theory of optimal transport for finite measures and will outline how we envision to develop a counterpart to this theory in the setting of stationary point processes. I will also present first results in this direction from joint work with M. Huesmann and T. Lebl\'e which apply ideas from optimal transport in the study of log gases. The one-dimensional log gas in finite volume is a system of particles interacting via a repulsive logarithmic potential and confined by some external field. When the number of particles goes to infinity, their macroscopic empirical distribution approaches a deterministic limit shape. When zooming in one sees microscopic fluctuations around this limit which are described in the limit by a stationary point process, the Sine_\beta process constructed by Valko and Virag. We show that this process can be characterized as the unique minimizer of a renormalized free energy by leveraging strict convexity properties of this functional along interpolations built via optimal transport.

Nov 17 Sandra Palau (UNAM, Mexico)
Branching processes in varying environment


In this talk we are going to study branching processes and its extension when the offspring distribution is varying over time. We are going to analyze its extinction probability. By using a two-spine decomposition technique we are going to give the law of the process conditioned on non extinction.

Nov 24

Zakhar Kabluchko (University of Münster)
Random beta tessellations

We shall define and study several families of random tessellations of the Euclidean space, the sphere and the hyperbolic space that generalize the classical Poisson-Voronoi and Poisson-Delaunay tessellations. The talk is based on a joint work with Anna Gusakova and Christoph Thäle.

Dec 1

Lukas Gonon (LMU, Munich)
Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models

We study the expression rates of deep neural networks for option prices written on high-dimensional baskets of risky assets, whose log-​returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process that ensure that the size of the DNN required to achieve a given approximation accuracy grows only polynomially with respect to the dimension of the Lévy process and the reciprocal of the approximation accuracy, thereby overcoming the curse of dimensionality and justifying the use of DNNs in financial modelling of large baskets in markets with jumps.
In addition, we exploit parabolic smoothing of Kolmogorov partial integrodifferential equations for certain multivariate Lévy processes to present alternative architectures of ReLU DNNs that provide higher approximation rates, however, with constants potentially growing exponentially with respect to the dimension. Under stronger, dimension-​uniform non-​degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the present results.
The talk is based on joint work with Christoph Schwab.

Dec 8

Lisa Hartung (University of Mainz)
Branching Brownian motion among obstacles

In this informal talk, after introducing the class of models I will give an overview of the existent results in the literature on the behaviour of branching Brownian motion among obstacles. These results mainly concern the survival probability in the case of hard obstacles and the population size in the case of mild obstacles.
Then I will explain a couple of open questions which I would like to get a better understanding of.

Dec 15

Jiequn Han (Princeton University)
Deep BSDE Method and its Convergence Analysis for High-Dimensional PDEs & Games

Developing algorithms for solving high-dimensional partial differential equations, controls, and games has been an exceedingly difficult task for a long time, due to the notorious "curse of dimensionality". In the first part of the talk, I will introduce the Deep BSDE method for solving high-dimensional parabolic PDEs. The algorithm builds on the reformulation of backward stochastic differential equations and utilizes deep neural networks as efficient approximators to unknown high-dimensional components. Numerical results of various examples, including multi-agent games, demonstrate the efficiency and accuracy of the proposed algorithms in high-dimensions. In the second part of the talk, I will introduce some convergence analysis of the Deep BSDE method in terms of a posteriori error estimation and an upper bound for the minimized objective function.

Dec 22

Arno Siri Jégousse (UNAM, Mexico)
Site frequency spectrum of the Bolthausen-Sznitman coalescent

In this talk, I will study the concept of site frequency spectrum (SFS), which happens to be one of the most relevant statistics in population genetics. It is of particular use for genealogical model selection and estimation in evolution models. The SFS is closely related to the shape of the genealogical tree of the observed sample of a population.
In the particular case of the Bolthausen-Sznitman coalescent, which is now accepted to be the null model for rapidly evolving populations or populations under strong selection, the limit behaviour of the SFS can be studied thanks to approximations with random walks. More interestingly, there exists a construction of the coalescent by means of random recursive trees which yields very precise approximations of the moments of the SFS. This technique also gives new asymptotic results, leading to a complete picture of the statistic.
This is a joint work with Götz Kersting (Frankfurt) and Alejandro Wences (Mexico City).

Jan 12

Eva Kopfer (University of Bonn)
Random Riemannian Geometry

We study random perturbations of Riemannian manifolds by means of so-called Fractional Gaussian Field. The fields act on the manifolds via conformal transformation. Our focus will be on the regular case with Hurst parameter $H>0$, the celebrated Liouville geometry in two dimensions being borderline.

We want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap will change under the influence of the noise.

Jan 19

Gilles Bonnet (University of Bochum)
Concentration and Cumulants for Stabilizing Functionals on Point Processes

In this talk I will present aspects of the project "Concentration and Cumulants for Stabilizing Functionals on Point Processes". In particular, the notion of stabilizing functional will be introduced and several classical methods to obtain concentration bounds presented.

Jan 26

Christian Beck (WW University Münster)
Multilevel Picard approximations for semilinear elliptic PDEs

Multilevel Picard approximation methods have been successfully used to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic PDEs (see, e.g., Hutzenthaler et al. In this talk we extend multilevel Picard approximations for semilinear elliptic PDEs, state that they can overcome the curse of dimensionality in the numerical approximation of semilinear elliptic PDEs, point out differences between the parabolic and the elliptic situation, and comment on possible directions of future research.

Feb 2

Alexander Drewitz and Bingxiao Liu (University of Cologne)
Random polynomials and random Kähler geometry

We are presenting our SPP project which focuses on the interplay between complex geometry and probability theory. More precisely, we aim to combine methods from complex geometry and geometric analysis with probabilistic techniques in order to study several problems concerning local and global statistical properties of zeros of holomorphic sections of holomorphic line bundles over Kähler manifolds. A particularly important instance of this setting is given by the case of random polynomials. We are interested in the asymptotics of the covariance kernels of the polynomial / sections ensembles, universality of their distributions, central limit theorems and large deviation principles in this context.

Feb 9

Felix Lindner (University of Kassel)
On a class of stochastic differential-algebraic equations arising in industrial mathematics

The dynamics of inextensible fibers in turbulent airflows is of interest, e.g, in the context of mathematical models for spunbond production processes of non-woven textiles. In this talk, a model based on higher-index stochastic differential-algebraic equations is presented. It involves an implicitly given Lagrange multiplier process, the explicit representation of which leads to an underlying stochastic ordinary differential equation with non-globally monotone coefficients. Strong convergence is established for a half-explicit drift-truncated Euler scheme which fulfills the algebraic constraint exactly.


Summer Semester 2020 (back to Contents)

Apr 27

Thomas Kruse, University of Giessen
Multilevel Picard approximations for high-dimensional semilinear parabolic partial differential equations

We present new approximation methods for high-dimensional PDEs and BSDEs. A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of one of the proposed methods grows polynomially both in the dimension and in the reciprocal of the required accuracy. We illustrate the efficiency of the approximation methods by means of numerical simulations. The talk is based on joint works with Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Tuan Nguyen and Philippe Von Wurstemberger.

May 4

Cécile Mailler, University of Bath
Degree distribution in random simplicial complexes

In this joint work with Nikolaos Fountoulakis, Tejas Iyer and Henning Sulzbach (Birmingham), we study a random graph model introduced by Bianconi and Rahmede in 2015. This model is a simplicial complex model that generalises Apollonian networks and the random recursive trees, by, in particular, adding random weights to the nodes. For this general model, we prove limiting theorems for the degree distribution, and confirm the conjecture of Bianconi and Rahmede on the scale-free propoerties of this random graph.

May 11

Andreas Basse-O’Conner, Aarhus University
On infinite divisible laws on high dimensional spaces

Infinite divisible laws play a crucial role in many areas of probability theory. This class of laws corresponds exactly to all the possible weak limits of triangular arrays of random variables, due to the generalized central limit theorem, and hence infinite divisible laws are natural generalizations of the Gaussian laws. In this talk we will discuss properties and representations of infinite divisible probability measures on Banach spaces, with particular focus on their Lévy-Khintchine representations and their shot noise representations. We will see that these representations depend heavily on the Banach space under consideration. In particular, both the geometry of the Banach space and its “size” will play a big role. Many examples of high dimensional probability measures come from stochastic process theory, and we will illustrate the results within this framework.

May 18

Manuel Cabezas, Pontificia Universidad Católica de Chile
The totally asymmetric activated random walk model at criticality

Activated random walks is a system of particles which perform random walks and can spontaneously fall asleep, staying put. When an active particle falls on a sleeping one, the sleeping particle becomes active and continues moving. The system displays a phase transition in terms of the density of particles. If the density is small, all particles will eventually sleep forever, while, if the density is high, the system can sustain a positive proportion of active particles. In this talk we describe the critical behavior of the model in the totally asymmetric case. Joint work with Leo Rolla.

May 25 Airam Blancas Benítez, Stanford University

Coalescent models for trees within trees

Phylogenetic gene trees are contained within the branches of the species trees. In order to model genealogy backwards in time, of both, gene trees and species trees, simple exchangeable coalescent (snec) process are defined and characterized in talk. In particular, we study the coming down form infinity property for the so called nested Kingman. Finally, we present a model to include population structure in gene lineages.

June 8 Geronimo Uribe Bravo, UNAM

On the profile of trees with a given degree sequence

For a given (plane) tree $\tau$, let  $N_i$ be the quantity of individuals with $i$ descendants and define its degree sequence  as $s=(N_i)_{i\geq 1}$. We will be interested in the uniform distribution on trees whose degree sequence is $s$. We give conditions for the convergence of the profile (aka the sequence of generation sizes) as the size of the tree goes to infinity. This gives a more general formulation and a probabilistic proof of a conjecture due to Aldous for conditioned Galton-Watson trees.  Our formulation contains results in this direction obtained previously by Drmota-Gittenberger and Kersting. The technique, based on path transformations for exchangeable increment processes, also gives us a (partial) compactness criterion for the inhomogeneous continuum random tree.
Joint work with Osvaldo Angtuncio. 

June 15

Olivier Hénard, Université Paris-Sud
Parking  on critical GW trees

Lackner and Panholzer (2015) introduced the parking process on trees as a generalization of the classical parking process on the line. 
In this model, a random number of cars with mean m and variance σ2 arrive independently on the vertices of a critical Galton–Watson tree with finite variance Σ2 conditioned to be large. The cars go down the tree and try to park on empty vertices as soon as possible.  We show a phase transition depending on Θ:=(1−m)2 −Σ22 +m2 −m), confirming a conjecture by Goldschmidt and Pryzkucki (2016). Specifically, if Θ > 0, then most cars will manage to park, whereas if Θ < 0 then a positive fraction of the cars will not find a spot and exit the tree through the root. 
The proof relies on tools in percolation theory such as differential (in)equalities obtained through increasing couplings combined with the use of many-to-one lemmas and spinal decompositions of random trees.
This is joint work with Nicolas Curien (Paris Saclay).

June 22

Adrian Gonzalez Casanova, UNAM
From continuous state branching processes to coalescents

The relation between these two important families of processes has been investigated in some cases. In particular, in a a renowned paper by seven authors,  the $\beta$-coalescents are obtained as a functional of two independent $\alpha$-stable branching processes. 
Using Gillispie's sampling method, we find that an analogous relation holds for every lambda coalescent. Furthermore, functionals of independent CSBPs with different laws lead to frequency processes of coalescents with selection, mutation, efficiency and more.
This is a joint work in progress with Maria Emilia Caballero (UNAM) and Jose Luis Perez (CIMAT).

June 29

Antoine Lejay, Institut Élie Cartan de Lorraine (IECL)
Two Embeddable Markov Chain Schemes for simulating diffusions with irregular coefficients

We discuss the problem of the Monte Carlo simulation of sample paths. The Embeddable Markov Chain Schemes simulate the exact positions of a diffusion at some hitting times which are not known but only approximated. Such schemes aim at overcoming the situations in which the Euler-Maruyama scheme cannot be used, because of the presence of discontinuous coefficients, interfaces, sticky points, …
In this talk, we present two such schemes. The first one uses the explicit expressions of the resolvent kernel instead
of the density and deal with discontinuous coefficients. It leads to fast convergence. The second one generalises the Donsker approximation in presence of degenerate coefficients and appears to be fully flexible. Both schemes heavily rely on the underlying infinitesimal generator.
From joint work with Alexis Anagnostakis, Lionel Lenôtre, Géraldine Pichot and Denis Villemonnais.

July 6

Thomas Godland, University of Münster
Conical tessellations associated with Well chambers

In this talk, I consider d-dimensional random vectors Y1,,Yn that satisfy a mild general position assumption a.s. The hyperplanes


generate a conical tessellation of the Euclidean d-space, which is closely related to the Weyl chambers of type Bn. I will present a formulas for the number of cones which holds almost surely. For a random cone chosen uniformly at random from this random tessellation, I will address expectations for a general series of geometric functionals. These include the face numbers, as well as the conical intrinsic volumes and the conical quermassintegrals. All these expectations turn out to be distribution-free.
In a similar fashion, I will shortly discuss a conical tessellations which is closely related to the Weyl chambers of type An1. I will present analogous formulas the number of cones in this tessellation and the expectations of the same geometric functionals for the random cones obtained from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected by a certain linear subspace in general position.

July 13

Osvaldo Angtuncio, UNAM
On multitype random forests with a given degree sequence, the total population of branching forests and enumerations of multitype forests

In this talk, we introduce the model of uniform multitype forests with a given degree sequence (MFGDS). The construction is done using the results of Chaumont and Liu 2016, and a novel path transformation on multidimensional discrete exchangeable increment processes, which is a generalization of the Vervaat transform. By mixing the laws of MFGDS, one obtains multitype Galton-Watson (MGW) forests conditioned with the number of individuals of each type (CMGW). We also obtain the joint law of the number of individuals by types in a MGW forest, generalizing the Otter-Dwass formula. This allows us to get enumerations of multitype forests with a combinatorial structure (plane, labeled and binary forest), having a prescribed number of roots and individuals by types. Finally, under certain hypotheses, we give an easy algorithm to simulate CMGW forests, generalizing the unitype case given by Devroye in 2012. The previous results can be considered as the first step to obtain the profile of the multitype Lévy forest.


Winter Semester 2019/20 (back to Contents)

Oct 8 Máté Gerencsér (IST Austria)
Boundary renormalisation of stochastic PDEs

First, we discuss general methods for solving singular SPDEs endowed with boundary conditions. Then, starting from the Neumann problem for the KPZ equation, we discuss how and why boundary renormalisation effects arise in this context.

Oct 22 Josué Nussbaumer (University of Duisburg-Essen)

The alpha-Ford algebraic measure trees

We are interested in the infinite limit of the alpha-Ford model, which is a family of random cladograms, interpolating between the coalescent tree (or Yule tree) and the branching tree (or uniform tree). For this, we use the notion of algebraic measure trees, which are trees without edge length and equipped with a sampling measure. In the space of algebraic measure trees, the limit of the alpha-Ford model is well defined. We then describe some statistics on the limit trees, allowing for tests of hypotheses on real world phylogenies. Furthermore, the alpha-Ford algebraic measure trees appear as the invariant distributions of Markov processes describing the evolution of phylogenetic trees.

Oct 29

Emiel Lorist (TU Delft)
Singular stochastic integral operators

Joint work with Mark Veraar
Singular integral operators play a prominent role in harmonic analysis. By replacing the integration with respect to Lebesgue measure by integration with respect to Brownian motion, one obtains a stochastic singular integral of the form
\begin{equation*} S_K G(t) :=\int_{0}^\infty K(t,s) G(s) \,\mathrm{d} W_H(s), \qquad t\in \mathbb{R}_+, \end{equation*} which appears naturally in questions related to stochastic maximal regularity. Here $G$ is an adapted process, $W_H$ is a cylindrical Brownian motion and $K$ is allowed be singular.
In this talk I will study the $L^p$-boundedness for such singular stochastic integrals with operator-valued kernel $K$ using Calder\'on--Zygmund theory. The developed theory implies $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. This leads to mixed $L^p(L^q)$-theory for several stochastic partial differential equations, of which I will give a few examples.

Nov 12

Sascha Nolte (University of Duisburg-Essen)
Robust optimal stopping without time-consistency

In the context of robust optimal stopping one aim is to prove minimax identities, which play an important role in financial mathematics, especially in the characterization of arbitrage-free prices for American options. Normally, the proof relies on the assumption that the underlying set of probability measures (priors) satisfies the property of time-consistency which can be regarded as an extension of the tower property for conditional expectations. Unfortunately, time-consistency is very restrictive. In this talk we present a different kind of conditions that ensure the desired minimax result. The key is to impose a compactness assumption on the set of priors. The presented conditions reveal some unexpected connection between the minimax result and path properties of the corresponding process of densities. We exemplify our general results in the case of families of measures corresponding to diffusion exponential martingales.
Furthermore, we give a short outlook how to extend the minimax results to the model free situation where no reference probability measure is given in advance.

Nov 19

Nina Dörnemann (Ruhr University Bochum)
Likelihood ratio tests for many groups in high dimensions

In this work, we investigate the asymptotic distribution of likelihood ratio tests in models with several groups, when the number of groups converges with the dimension and sample size to infinity.

We derive central limit theorems for the logarithm of various test statistics and compare our results with the approximations obtained from a central limit theorem where the number of groups is fixed. In this talk, we will consider two testing problems, namely testing for a block diagonal covariance matrix and for equality of normal distributions.

Nov 21

Nicole Hufnagel (TU Dortmund)
Martingale estimators for the Bessel process

Martingale estimation functions are well studied by Bibby, Sørensen (1995) and Kessler, Sørensen (1999) in the case of discretely observed ergodic diffusion processes. In this talk we adapt the methodology of Kessler and Sørensen to achieve novel martingale estimation functions for a Bessel process which is a non-ergodic process. We can tackle this problem by considering a space-time transformation of the Bessel process.
We provide martingale estimation functions based on eigenfunctions of the diffusion generator for this transformed Bessel process. Following the approach of Kessler and Sørensen, consistency and asymptotic normality of these estimators can be derived. Furthermore, we compare the martingale estimation functions through a simulation study and discuss the emerging complications.

Nov 26

Kilian Hermann (TU Dortmund)
Limit theorems for Jacobi ensembles with large parameters

Consider Jacobi random matrix ensembles with the distributions $$c_{k_1,k_2,k_3}\prod_{1\leq i< j \leq N}\left(x_j-x_i\right)^{k_3}\prod_{i=1}^N \left(1-x_i\right)^{\frac{k_1+k_2}{2}-\frac{1}{2}}\left(1+x_i\right)^{\frac{k_2}{2}-\frac{1}{2}} dx$$ of the eigenvalues on the alcoves $A:=\{x\in\mathbb{R}^N| \> -1\leq x_1\leq ...\leq x_N\leq 1\}$. For $(k_1,k_2,k_3)=\kappa\cdot (a,b,1)$ with $a,b > 0$ fixed, we derive a central limit theorem for the distributions above for $\kappa\to\infty$. The drift and the inverse of the limit covariance matrix are expressed in terms of the zeros of classical Jacobi polynomials. We also rewrite the CLT in trigonometric form and determine the eigenvalues and eigenvectors of the limit covariance matrices. These results are related to corresponding limits for $\beta$-Hermite and $\beta$-Laguerre ensembles for $\beta\to\infty$ by Dumitriu and Edelman and by Voit.

Dec 3

Fabian Gerle (University of Duisburg-Essen)
Towards an invariance principle for symmetric Feller processes

Let $X$ be the speed-$\nu$ motion on a metric measure tree $(T,d,\nu)$. Athreya Löhr and Winter (2017) showed that such a sequence of symmetric Feller processes converges pathwise whenever the underlying sequence of metric measure trees converges in a suitable sense. Croydon (2018) extended this result to symmetric Feller processes associated with a resistance metric. 

Both approaches are tailored to discrete or (basically) linear state spaces. They fail in higher dimensions, where the resistance metric is not well-defined.

In this talk we lay out a path towards a general invariance principle. We consider a symmetric Feller process $X$ on a Lusin
topological space $S$ equipped with a measure $\nu$. We introduce a class of occupation time functionals and a notion of convergence of the state spaces based on these functionals. Given a sequence of symmetric Feller processes $X^{(n)}$ with state spaces $S^{(n)}$ we present an idea how pathwise convergence of the processes can be obtained from the convergence of the state spaces.

This is work in progress.

Dec 10

Tobias Hübner (University of Duisburg-Essen)
Solving optimal stopping problems for convex risk measures via empirical dual optimization

In the standard optimal stopping, model uncertainty is usually handled by considering as an objective the expected return. In this talk, we pursue a more versatile approach towards uncertainty and consider optimal stopping problems with conditional convex risk measures including  average value-at-risk and other risk measures.
Based on a generalization of  the additive dual representation of [Rogers 2002] to the case of optimal stopping under uncertainty, we develop a novel Monte Carlo algorithm for the approximation of the corresponding value function. The algorithm involves optimization of a genuinely penalized dual objective functional over a class of adapted martingales. This formulation allows to construct upper bounds for the optimal value with a reduced complexity. Further we discuss the convergence analysis of the proposed algorithm.

Jan 14

Benjamin Gess (MPI MiS Leipzig and Bielefeld University)
Stochastic thin film equations

In this talk we consider the stochastic thin-film equation. The stochastic thin-film equation is a fourth-order, degenerate stochastic PDE with nonlinear, conservative noise. This makes the existence of solutions a challenging problem. Due to the fourth order nature of the equation, comparison arguments do not apply and the analysis has to solely rely on integral estimates. The stochastic thin film equation can be, informally, derived via the lubrication/thin film approximation of the fluctuating Navier-Stokes equations and has been suggested in the physics literature to be an improved mesoscopic model, leading to better predictions for film rupture and propagation. In this talk we will prove the existence of weak solutions in the case of quadratic mobility. The construction of a solution will be based on an operator splitting technique, which at the same time gives rise to an easy to implement numerical method.

Jan 28

Gerónimo Rojas Barragán (University of Duisburg-Essen)
On the rate of convergence of diffusion approximation on compact trees

In this talk we introduce a new distance for stochastic processes taking values on compact metric measure trees based on hitting times. We show how this new metric can be bounded in terms of the Gromov-Prokhorov distance and argue how this yields rates of convergence in the f.d.d. sense. 

We complete the metric by adding a functional that captures tightness. We discuss how this complete metric can be used to derive rates for weak convergence in path-space.

As an application, we use this distance to derive a rate for weak convergence in path-space for SRW to BM on compact intervals.

This is work in progress.

Feb 4

George Andriopoulos, Shanghai (China)
Discrete scheme for Sinai's process in random media on Z.

We present Sinai's random walk in random environment, and focus on its functional limit theorem. We begin by recalling SInai's model and its long time behaviour. We illustrate that the proof of the known functional limit theorem can be shortened, when verifying the convergence of the resistance metric measure spaces rather than that of the random walks associated with these.

Summer Semester 2019 (back to Contents)

Apr 16

Viktor Schulmann, TU Dortmund
Life span estimation for randomly moving particles based on their places of death

Consider the following problem from physics: A radiation source is placed at the center of a screen. At certain time intervals the source releases particles. These move around the screen following a path of some known random process $(Y_t)_{t\geq 0}$ without interacting with each other and without us being able to observe their movement until they die after some random time $T$. During its death a particle leaves a mark such that we can measure the distance $X=||Y_T||_2$ it traveled from the source during its lifetime. Based on these observed distances we wish to infer the life span $T$ of a particle or, in particular, the density $f_T$ of $T$.
We assume $(Y_t)_{t\geq 0}$ from our physics experiment to be a multi-dimensional L\'{e}vy processes with spherical symmetry. Norms of such processes exhibit structural similarities to one-dimensional L{\'e}vy processes. For that case an estimator was given by Belomestny and Schoenmakers (2016) using the Mellin and Laplace transforms. Applying their techniques we construct a non-parametrical estimator for $f_T$, calculate its convergence rate and show its optimality in the minimax sense.

Apr 23

Sascha Kissel, University Bochum
Dynamical Gibbs-non-Gibbs transitions in Curie-Weiss Widom-Rowlinson models

In this talk, we consider the Curie-Weiss Widom-Rowlinson model for particles with spins and holes, with a repulsion strength $\beta>0$ between particles of opposite spins. A closed solution of the symmetric model will be provided. After this, we talk about the dynamical Gibbs-non-Gibbs transitions for the time-evolved model under independent stochastic symmetric spin-flip dynamics. It will be shown that, for sufficiently large $\beta$ after a transition time, continuously many bad empirical measures appear.

May 7

Gabriel Hernán Berzunza Ojeda, University Uppsala
Fluctuations of the giant cluster for supercritical percolation on split trees

A split tree of cardinality $n$ is constructed by distributing $n$ "balls" (which often represent "key numbers") in a subset of vertices of an infinite tree. In this talk, we will discuss Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. The main goal is to show for appropriate percolation regimes that there exists a unique giant cluster that is of a size comparable of that of the entire tree (where size is defined as either the number of vertices or the number of balls). Furthermore, it will be shown that in such percolation regimes, (also known as supercritical regimes) the fluctuations of the size of the giant cluster are non-Gaussian as $n$ grows. Instead, they are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work is a generalization of previous results for random $m$-ary recursive trees which is one specific case of split trees. Other important examples of split trees include $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. The approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which allows us to apply a classical limit theorem for the convergence of triangular arrays to infinitely divisible distributions.

May 14

Jan Nagel, University Dortmund
Random walk on barely supercritical branching random walk

The motivating question behind this project is how a random walk behaves on a barely supercritical percolation cluster, that is, an infinite percolation cluster when the percolation probability is close to the critical value.

As a more tractable model, we approximate the percolation cluster by the embedding of a Galton-Watson tree into the lattice. When the random walk runs on the tree, the embedded process is a random walk on a branching random walk. Now we can consider a barely supercritical branching process conditioned on survival, with survival probability approaching zero. In this setting the tree structure allows a fine analysis of the random walk and we can prove a scaling limit for the embedded process under a nonstandard scaling. The talk is based on a joint work with Remco van der Hofstad and Tim Hulshof.

May 21

José Manuel Pedraza, London School of Economics and Political Science
Predicting in a $L_p$ sense the last zero of an spectrally negative Lévy process

Given a spectrally negative Lévy process drifting to infinity, we are interested in the last time g in which the process is below zero. At any time t, the value of g is unknown and it is only with the realisation of the whole process when we can know when the last zero of the process occurred. However, this is often too late, we usually are interested in know how close is the process to g at time t and take some actions based on this information.

We are interested on finding a stopping time which is as close as possible to $g$ (on a $L_p$ distance). We prove that solving this optimal prediction problem is equivalent to solve an optimal stopping problem in terms of a two dimensional Markov process which involves the time of the current excursion away from the negative half line and the L\'evy process. We state some basic properties of the last zero process and prove the existence of the solution of the optimal stopping problem.

Then we show the solution of the optimal stopping problem (and therefore the optimal prediction problem) is given as the first time that the process crosses above a non-increasing and non-negative curve dependent on the time of the last excursion away from the negative half line.

May 28

Carina Betken, University Bochum     
Stein's method and preferential attachment random graphs

We consider a general preferential attachment model, where the probability that a newly arriving vertex  connects to an older vertex is proportional to a (sub-)linear function  of the indegree of the older vertex at that time.
We develop Stein's method for the asymptotic indegree distribution of a vertex, chosen uniformly at random, and deduce rates of convergence as the number of vertices tends to $ \infty $. Using Stein's method for Poisson and  Normal approximation we also show limit theorems for the outdegree distribution as well as for the number of isolated vertices.

June 4 Jiří Cerny, University Basel  

Gaussian free field on regular graphs

We study the behaviour of level sets of 0-mean Gaussian free field on regular expanding graphs, and (at least partially) prove that they exhibit a similar phase transition as Bernoulli percolation and the vacant set of random walk on such graphs.

Jun 25

David Belius, University Basel  
The TAP approach to mean field spin glasses

The topic of this talk is mean-field spin glasses, in particular the Sherrington-Kirkpatrick (SK) model. I will revisit the Thouless-Andersson-Palmer approach to the these models from the physics literature, and report on our efforts to use it as a basis for an alternative mathematically rigorous treatment.

Jul 9

Max Grieshammer, University Erlangen
Measure representation of evolving genealogies

We present a method of describing evolving genealogies, i.e. a special kind of processes that take values in the space of
(marked) ultra-metric measure spaces and satisfy some sort of consistency condition. We relate measure-valued processes
to certain genealogical quantities and use this connection to prove a tightness result for evolving genealogies.
Finally, we sketch how this result applies to Moran models.

Aug 13

Sungsoo Byun, Seoul National University
On the interface between Hermitian and normal eigenvalue ensembles

In 1998, Fyodorov, Khoruzhenko and Sommers introduced the weakly non-Hermitian regime, which provides a natural bridge between Hermitian and normal random matrix theories. In this talk, I will discuss ensembles of a similar appearance and their universality, in particular from the viewpoint of Ward equations. Such universality classes we discovered include not only weakly non-Hermitian ensembles with certain external potentials but also general weakly circular ensembles under the soft/hard edge constraint. This is based on two different joint works: one with Yacin Ameur, and the other with Yacin Ameur and Seong-Mi Seo.

Winter Semester 2018/19
(back to Contents)

Oct 8

Abel Klein, Irvine (USA)
Manifestations of dynamical localization in the random XXZ quantum spin chain

Oct 8 Tobias Müller, Groningen (Netherlands)
The critical probability for confetti percolation equals 1/2
Oct 15

Vitalii Konarovskyi, Leipzig
A particle model for Wasserstein type diffusion

Oct 22 PI-meeting
Oct 29

Piotr Graczyk, Angers (France)
Squared Bessel particle systems and Wishart processes

Nov 5

Vlada Limic, Strasbourg (France)
Large random graphs and excursions of Levy-type processes

Nov 12

Wioletta Ruszel, Delft (Netherlands)
A zoo of scaling limits of odometers in divisible sandpile models

Nov 19

Noemi Kurt, Berlin
Modelling the Lenski experiment

Nov 26

Tran Viet Chi, Lille (France)
Exploration of a social network by a respondent driven sampling survey

Nov 27

Diyora Salimova, (ETH Zürich)
Deep artificial neural networks in the numerical approximation of Kolmogorov PDEs:

In recent years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a numerous of computational problems including, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations (PDEs). Such numerical simulations indicate that DNNs seem to admit the fundamental flexibility to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the dimension of the function which the DNN aims to approximate in such computational problems.  In this talk we show that DNNs do overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. We prove that the number of parameters used to describe the employed DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy  and the PDE dimension.

Dec 3

Martin Slowik, Berlin
Green kernel asymptotics for two-dimensional random walks among random conductances

Dec 17

Michael Hinz, Bielefeld
Hydrodynamic limits of weakly asymmetric exclusion processes on fractals

Jan 7

Sebastian Riedel, Berlin
A random dynamical system for stochastic delay differential equations

Jan 14

Mike Keane, Leiden (Netherlands)
The World of Fibonacci, Thue-Morse and Toeplitz Substitutions

Jan 15 Georgiy Shevchenko, Taras Shevchenko National University of Kyiv

Stratonovich stochastic differential equation with power nonlinearity: Girsanov's example revisited

Jan 21

Thomas Mikosch, Copenhagen (Denmark) and Bochum
Regular variation and heavy-tail large deviations for time series

Jan 28

Johannes Heiny, Bochum (RTG)
Assessing the dependence of high-dimensional time series via autocovariances and autocorrelations

Mar 19

Boris Baeumer, University of Otago, Neuseeland
Boundary conditions for Levy processes on bounded domains and their governing PDEs.

Levy processes are jump processes governed by non-local operators on R^d and are used to model dispersive systems where the occasional large dispersal event (many standard deviations) is driving the system. In modelling, boundaries appear naturally and in 1D we answer the question of what type of boundary condition for the non-local operator corresponds to what type of boundary behaviour of the process by using numerical approximation schemes. 


Summer Semester 2018 (back to Contents)

Apr 17

Anton Klimovsky (Universität Duisburg-Essen)
Stochastic population models on evolving networks

Many models of complex systems can be seen as a system of many interacting (micro)variables evolving in time. We focus on the situation, where the network of interactions between the variables is complex and possibly itself evolves in time. We discuss a modeling framework for interacting particle systems on evolving networks based on such familiar ingredients as exchangeability and Markovianity. In some simple cases, we discuss the genealogies of such population models.
(Based on joint work in progress with Jiří Černý.)

Apr 24

Anselm Hudde (Universität Duisburg-Essen)
A perturbation Theory and applications to numerical approximation of SDEs

In this talk we will discuss a perturbation theory which can be applied to find strong $L^2$-convergence rate for approximations schemes of SDEs and SPDEs.

May 8

Fernando Cordero (Universität Bielefeld)
On the stationary distribution of the block-counting process in populations with mutation and selection

The $\Lambda$-Wright-Fisher model is a population model subject to selection, mutation and neutral reproduction (described by a finite measure $\Lambda$ on $[0,1]$). The block-counting process traces back the number of potential ancestors of a sample of the population at present. In absence of selection and mutation the latter coincides with the $\Lambda$-coalescent. Selection and mutation translate into additional branching and pruning. Under some conditions the block-counting process is positive-recurrent and its stationary distribution is described via a linear system of equations. In this talk, we first characterise the measures $\Lambda$ leading to a geometric stationary distribution, the Bolthausen-Sznitman model being the most prominent example having this feature. For a general measure $\Lambda$, we show that the probability generating function of the stationary distribution of the block-counting process satisfies an integro-differential equation. We solve the latter for the Kingman model and the star-shaped model.
(Based on joint work with M. Möhle).

May 15

Roland Meizis (Universität Duisburg-Essen)
Convergence of metric two-level measure spaces

We extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces): An m2m space $(X, r, \nu)$ is a Polish metric space $(X, r)$ equipped with a two-level measure $\nu \in \mathcal{M}_f(\mathcal{M}_f(X))$, i.e. a finite measure on the set of finite measures on $X$. We define the set $\mathbb{M}^{(2)}$ of (equivalence classes of) m2m spaces and provide a complete metric on this set. Furthermore, we introduce a convergence determining set of test functions on $\mathbb{M}^{(2)}$, which is suitable for defining generators of Markov processes on $\mathbb{M}^{(2)}$.
The framework introduced in this talk is motivated by applications in biology. It is well suited for modeling the random evolution of the genealogy of a population in a hierarchical system with two levels, for example, host-parasite systems or populations which are divided into colonies.

May 29

Daniel Pieper (Universität Duisburg-Essen)
Altruistic defense traits in structured populations: Many-demes limit in the sparse regime

We discuss spatially structured Wright-Fisher type diffusions modelling the frequency of an altruistic defense trait. These arise as the diffusion limit of spatial Lotka-Volterra type models with a host population and a parasite population, where one type of host individuals (the altruistic type) is more effective in defending against the parasite but has a weak reproductive disadvantage. For the many-demes limit (mean-field approximation) hereof, we prove a propagation of chaos result in the case where only a few diffusions start outside of an accessible trap. In this "sparse regime", the system converges in distribution to a forest of trees of excursions from the trap.

Jun 5

Richard Kraaij (Ruhr Universität Bochum)
Fluctuations for a dynamic Curie-Weiss model of self-organized criticality

The Curie-Weiss model of self-organized criticality was introduced by Cerf and Gorny(2014,2016) as a modification of the Curie-Weiss model of ferromagnetism that drives itself into a criticalstate. We consider a dynamic variant of this model, i.e. a system of interacting SDE's, and study its dynamical fluctuations.
Based on joint work with Francesca Collet(Delft) and Matthias Gorny(Paris-Sud).

Jun 12

Anita Winter (Universität Duisburg-Essen)
Aldous move on cladograms in the diffusion limit

In this talk we are interested in limit objects of graph-theoretic trees as the number of vertices goes to infinity. Depending on which notion of convergence we choose different objects are obtained. One notion of convergence with several applications in different areas is based on encoding trees as metric measure spaces and then using the Gromov-weak topology. Apparently this notion is problematic in the construction of scaling limits of tree-valued Markov chains whenever the metric and the measure have a different scaling regime. We therefore introduce the notion of algebraic measure trees which capture only the tree structure but not the metric distances. Convergence of algebraic measure trees will then rely on weak convergence of the random shape of a subtree spanned a sample of nite size. We will be particularly interested in binary algebraic measure trees which can be encoded by triangulations of the circle. We will show that in the subspace of binary algebraic measure trees sample shape convergence is equivalent to Gromov-weak convergence when we equip the algebraic measure tree with an intrinsic metric coming from the branch point distribution.
The main motivation for introducing algebraic measure trees is the study of a Markov chain arising in phylogeny whose mixing behavior was studied in detail by Aldous (2000) and Schweinsberg (2001). We give a rigorous construction of the diffusion limit as a solution of a martingale problem and show weak of the Markov chain to this diffusion as the number of leaves goes to infinity.

Jul 3

Lionel Lenôtre (CMAP - Ecole Polytechnique)
A simulation method called GEARED and its application to Skew Diffusions

The GEARED or GEneralized Algorithm based on REsolvent for Diffusion is a simulation method of Feller's processes whatever they admit a representation as a stochastic differential equation or not. The only requirements is to be able to sample a particular random variable whose density is given by the resolvent kernel of the stochastic process that one wants to simulate. This mainly means that an analytical form of the resolvent kernel is required.
In this talk, we present the GEARED method and provide and application of it on Skew Diffusions with piecewise constant coefficients. On this particular case of Feller's processes, we show through numerical experiments that the GEARED method has a quite fast convergence and that it conserves important properties in physics such a good repartition of mass.

Jul 17

Robert Link (Universität Duisburg-Essen)
Existence and uniqueness of solutions of infinite dimensional Kolmogorov equations

It is well known from the Feynman-Kac formula that a classical solution of the Kolmogorov backward equation can be written as the expectation of the solution of the corresponding SDE. In 2015 M. Hairer, M. Hutzenthaler, and A. Jentzen gave a finite dimensional example of a Kolmogorov backward equation with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique viscosity solution is not locally Hölder continuous. Moreover, they proved in the finite dimensional case that under suitable assumption the Kolmogorov backward equation has a unique viscosity solution which can be represented as the expectation of the solution of the corresponding SDE.
In the talk I will generalize this result to infinite dimensional Hilbert spaces and SPDEs. Therefore I will use a more general notation of viscosity solution introduced by H. Ishii and show that under suitable assumptions the expectation of the solution of an SPDE is the unique viscosity solution of the corresponding Kolmogorov backward equation.


Winter Semester 2017/18 (back to Contents)

Oct 17 Sara Mazzonetto (Universität Duisburg-Essen)

About some skewed Brownian diffusions: explicit representation of their transition densities and exact simulation

In this talk we first discuss an explicit representation of the transition density of Brownian dynamics undergoing their motion through semipermeable and semireflecting barriers, called skewed Brownian motions.
We use this result to present an exact simulation of these diffusions, and comment some (still) open problems. Eventually we consider the exact simulation of Brownian diffusions whose drift admits finitely many jumps.

Oct 24

Tuan Anh Nguyen (Universität Duisburg-Essen)
The random conductance model under degenerate conditions

The aim of the talk is to briefly introduce some ideas in my PhD thesis. Random motions in random media is an interesting topic that has been studied intensively since several decades. Although these models are relatively simple mathematical objects, they have a wide variety of interesting properties from the theoretical point of view. In the talk I would like to draw your attention to an important branch within this topic, namely reversible random walks moving among nearest neighbour random conductances on $\mathbb{Z}^d$ -- the random conductance model. Reversibility provides the model a variety of interesting connections with other fields in mathematics, for instance, percolation theory and especially stochastic homogenization. Many questions coming from this model have been answered by techniques from partial differential equations and harmonic analysis. As seen in the name of the talk, I would like to consider this model under ''degenerate conditions''. Here, ''degenerate'' has essentially two meanings. First, the conductances are not assumed to be bounded from above and below and stochastically independent. Second, we also consider the case of zero conductances, where the random walk can only move on a subgraph of $\mathbb{Z}^d$. Since there are percolation clusters, where the existence of the infinite cluster does not rely on stochastic independence, it is reasonable to accept the lack of stochastic independence. In the first part of the talk I introduce quenched invariance principles (joint work with Jean-Dominique Deuschel and Martin Slowik). We assume that the positive conductances have some certain moment bounds, however, not bounded from above and below, and give rise to a unique infinite cluster and prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infinite cluster. An essential ingredient of our proof is a new anchored relative isoperimetric inequality. In the second part I would like to talk about Liouville principles. As in the first part, I also assume some moment bounds and prove a first order Liouville property for this model. Using the corrector method introduced by Papanicolaou and Varadhan, the first and the second part are closely related to each other at the technical level. I also introduce a discrete analogue of the Dirichlet-to-Neumann estimate, which compares the tangential and normal derivatives of a harmonic function on the boundary of a domain. Although it is a purely deterministic classical result, it is used in the second part and perhaps useful for numerical analysis.

Nov 7

Clemens Printz (Universität Duisburg-Essen)
Stochastic averaging for multiscale Markov processes applied to a Wright-Fisher model with fluctuating

We present a new result on stochastic averaging for sequences of bivariate Markov processes $((X_t^n,Z_t^n)_{t\in[0,\infty)})_{n\in\mathbb{N}}$ whose components evolve on different time scales. Under suitable conditions, convergence of certain functionals of the fast variables $Z^n$ guarantees convergence of the (not necessarily Markovian on its own) slow variables $X^n$ to a limiting Markov process $(X_t)_{t\in[0,\infty)}$.
With this tool we can generalize the well-known diffusion limit of a Wright-Fisher model with randomly fluctuating selection. Whereas the classical result assumes the selection coefficients to be independent for different generations, we allow the environment to persist with a positive probability. The diffusion limit turns out to depend on this probability.
This talk is based on joint work with Martin Hutzenthaler and Peter Pfaffelhuber.

Nov 14 Simon Eberle (Universität Duisburg-Essen)

Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation

We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this talk we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy ${\mathcal{F}}$ on a time-constrained manifold. First, we prove existence of solutions by passing to the limit in an explicit Euler scheme obtained by minimizing $h {\mathcal{F}}(\varrho)+W_2^2(\varrho^0,\varrho)$ among all $\varrho$ satisfying the constraint for some $\varrho^0$ and time-step $h>0$. Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities.

Nov 21

Michael Weinig (Universität Duisburg-Essen)
Nested Monte-Carlo Simulation Estimates of Expected Value of Partial Perfect Information

The expected value of partial perfect information (EVPPI) expresses the value gaining by acquiring further information on certain unknowns in a decision-making process. In this talk, we will show and compare various approaches to estimate the EVPPI like nested Monte-Carlo simulations, Multi-Level Monte-Carlo simulations, and regression-based algorithms.
Further, we introduce an unbiased estimator based on a randomized version multi-level Monte-Carlo algorithm.
The EVPPI estimate is mostly used in medical decision making. We will show some numerical result in evaluating the benefit of further information when deciding between two different treatment options.

Nov 28

Sebastian Mentemeier (TU Dortmund)
Solutions to complex smoothing equations

In many models of Applied Probability and Statistical Physics quantities of interest satisfy distributional limit theorems (i.e. convergence in law) which are nonstandard in the sense that neither normal nor $\alpha$-stable laws appear as limiting distributions. Sometimes, there is even no convergence in law, but periodical fluctuations centered at some limit law occur. Common to all these models is that the limiting distributions satisfy so-called smoothing equations. These are equations of the form
\begin{equation}\label{SFPE} X \stackrel{\text{law}}{=} \sum_{j \ge 1} T_j X_j + C, \end{equation}
where $X_1, X_2, \dots$ are i.i.d.~copies of the (unknown)  random variable $X$ and independent of the (given) random variables $(C, T_1, T_2, \dots)$.
I will start my talk by presenting several examples where this phenomenon occurs. Then I will present a general theory for solving equations of type (1) in the case where $X$ as well as $C, T_1, T_2, \dots$ are complex-valued random variables. These results are then applied to the examples mentioned in the introduction.
This talk is based on joint work with Matthias Meiners, University Innsbruck.

Dec 5 Christoph Schumacher (TU Dortmund)

The asymptotic behavior of the ground state energy of the Anderson model on large regular trees

The Anderson model was invented 1985 by Anderson and describes the quantum mechanical motion of a particle in a random potential in $Z^d$. It is related to random walks in a random environment.  The Anderson model on regular tree was introduced 1973 by Abou-Chacra, Thouless and Anderson. We give a detailed description of the ground state energy on large finite symmetric subtrees. This is joint work with Francisco Hoecker-Escuti (TU Hamburg-Harburg).

Dec 12 

Larisa Yaroslavtseva (Universität Passau)
On sub-polynomial lower error bounds for strong approximation of SDEs

We consider the problem of strong approximation of the solution of a stochastic differential equation (SDE) at the final time based on finitely many evaluations of the driving Brownian motion $W$. While the majority of results for this problem deals with equations that have globally Lipschitz continuous coefficients, such assumptions are typically not met for real world applications. In recent years a number of positive results for this problem has been established under substantially weaker assumptions on the coefficients such as global monotonicity conditions: new types of algorithms have been constructed that are easy to implement and still achieve a polynomial rate of convergence under these weaker assumptions.
In our talk we present negative results for this problem. First   we show that there exist  SDEs  with bounded smooth coefficients such that their solutions can not be approximated by means of any kind of adaptive method with a polynomial rate of convergence. Even worse, we show that for any sequence $(a_n)_{n \in \mathbb N}\subset (0, \infty)$, which may converge to zero arbitrarily slowly, there exists an SDE  with bounded smooth coefficients such that no approximation method based on $n$ adaptively chosen evaluations of $W$ on average can achieve a smaller absolute mean error than the given number $a_n$. While the diffusion coefficients of these pathological SDEs are globally Lipschitz continuous, the first order partial derivatives of the drift coefficients are, essentially, of exponential growth. In the second part of the talk we show that sub-polynomial rates of convergence may happen even when the first order partial derivatives of the coefficients have at most polynomial growth, which is one of the typical assumptions in the literature on numerical approximation of SDEs with globally monotone coefficients.
The talk is based on joint work with Arnulf Jentzen (ETH Zürich) and Thomas Müller-Gronbach (University of Passau).

Jan 9

Sung-Soo Byun (Seoul National University/Universität Bielefeld)
CFT and SLE in a doubly connected domain

In this talk, I will present certain implementations of conformal field theory (CFT) in a doubly connected domain. The statistical fields in these implementations are generated by central charge modifications of the Gaussian free field with excursion reflected/Dirichlet boundary conditions. I will explain Ward’s equation in terms of a stress energy tensor, Lie derivative operators and the modular parameter. Combining Ward’s equation with the level 2 degeneracy equation for the boundary condition changing operator, I will outline the relation between CFT and Schramm-Loewner Evolution (SLE) theory. As applications, I will present a version of restriction property and Friedrich-Werner’s formula for annulus SLE and explain how to apply the method of screening to find explicit solutions to the partial differential equations for the annulus SLE partition functions introduced by Lawler and Zhan.
This is based on joint work with Nam-Gyu Kang and Hee-Joon Tak.

Jan 16

Sebastian Hummel (Universität Bielefeld)
Lines of descent in a deterministic model with mutation, selection and pairwise interaction

We consider a classical deterministic model for the evolution of a haploid population with two allelic types which is subject to mutation, selection, and a special form frequency-dependent selection. The deterministic model arises (also) as the large population limit of the Moran model, in which neither parameters nor time are rescaled. Despite the deterministic nature of this limiting process, the ancestry of single individuals in the population is still stochastic. In the case with mutation and selection, we describe it via a killed ancestral selection graph and connect it to the deterministic process via duality; this leads to a stochastic representation of the deterministic solution. In particular, the stationary state obtains a nice probabilistic interpretation. We generalise the construction to the case with frequency-dependent selection.

Jan 23

Thomas Kruse (Universität Duisburg-Essen)
Multilevel Picard approximations for high-dimensional nonlinear parabolic partial differential equations

In this talk we present a family of new approximation methods for high-dimensional PDEs and BSDEs. A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. Thereby we obtain a class of multilevel Picard approximations. Our error analysis proves that for semi-linear heat equations, the computational complexity of one of the proposed methods is bounded by $O(d\,\eps^{-(4+\delta)})$ for any $\delta > 0$, where $d$ is the dimensionality of the problem and $\eps\in(0,\infty)$ is the prescribed accuracy. We illustrate the efficiency of one of the proposed approximation methods by means of numerical simulations presenting approximation accuracy against runtime for several nonlinear PDEs from physics (such as the Allen-Cahn equation) and financial engineering (such as derivative pricing incorporating default risks) in the case of $d=100$ space dimensions.
The talk is based on joint work with Weinan E., Martin Hutzenthaler and Arnulf Jentzen.

Jan 29/30 GRK 2131 Workshop in Bochum (RUB)
Transformations and phase transitions
Feb 13

Lukas Knichel (RUB)
On the Rate of Convergence to a Gamma Distribution on Wiener Space

We consider a sequence $(F_n)$ of random variables living inside a fixed Wiener chaos of order $p$, converging in law to a centred $\chi^2$ distribution with $\nu$ d.f. It can be shown (using the Malliavin-Stein approach) that the rate of convergence is dominated by the differences of the third and fourth cumulants. More precisely, if $G(\nu)$ is a centred Gamma variable with parameter $\nu$, we have $$ d_{W}(F_n,G(\nu)) \leq_C \sqrt{\max ( \vert\kappa_3(F_n) - \kappa_3(G(\nu)) \vert, \vert \kappa_4(F_n) - \kappa_4(G(\nu))\vert ) }. $$ The same result (even in TV-distance) is available if we replace $G(\nu)$ by a standard Gaussian variable $N$. In 2015, I.Nourdin and G. Peccati showed that in this case, the \textit{exact} rate of convergence is given by the above-mentioned expression, if we omit the square root.
In this talk, we will discuss the question if we can also dispose of the square root in the case of a centred Gamma/$\chi^2$ random variable, and we will see by means of an explicit example, that this is not possible using the Malliavin-Stein approach.

Summer Semester 2017
(back to Contents)

April 25 Sandra Kliem (Universität Duisburg-Essen)

Travelling wave solutions to the KPP equation with branching noise

The one-dimensional KPP-equation driven by space-time white noise, $$ \partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 $$ is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. This SPDE arises for instance as the high density limit of particle systems which undergo branching random walks and allow for extra death due to overcrowding.

If $\theta$ is below a critical value $\theta_c$, solutions die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$. For initial conditions that are ‘’uniformly distributed in space’’, a complete convergence result holds, that is, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution. What can be said for solutions with finite initial mass if we condition on their survival?

In this talk I start with an overview of the main probabilistic ideas and arguments of the above mentioned concepts. In a next step, I explain how to obtain a travelling wave solution to this SPDE. Here, the choice of the wave front marker plays an important role. Finally, I outline why an understanding of the behaviour of travelling wave solutions can help to answer questions on convergence of solutions with arbitrary initial conditions.

May 3 Barbara Gentz (Universität Bielefeld)

Metastability in diffusion processes and synchronization

Non-standard time & location: 17:15 @ WSC-S-U-4.02

May 9 Anton Klimovsky (Universität Duisburg-Essen)

The phase diagram of the complex branching Brownian motion energy model

We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and $L^1$ martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy. This is joint work with Lisa Hartung (Courant Institute, NYU)

May 16 Tsiry Randrianasolo (Montan University Leoben, Austria)

Time-discretization of stochastic 2-D Navier-Stokes equations by a penalty-projection method

In this talk, I will present a time-discretization method of the stochastic incompressible Navier--Stokes problem using a penalty-projection method. Basically, the talk will consist of three parts. A brief introduction of the mathematical problem. Then an overview of the main computational issue that shares the stochastic and the deterministic form of Navier--Stokes. Different algorithm will be introduced: a main algorithm and in order to treat the nonlinear character of the equation two auxiliary algorithms. At the end of the day we will arrive at the convergence with rate in probability and a strong convergence of the main algorithm.

May 23 Zakhar Kabluchko (Westfälische Wilhelms-Universität Münster)

Convex Hulls of Random Walks: Expected Number of Faces

Consider a random walk $S_i= \xi_1+\dots+\xi_i$, $1\leq i\leq n$, starting at $S_0=0$, whose increments $\xi_1,\dots,\xi_n$ are random vectors in $\mathbb R^d$, $d\leq n$. We are interested in the properties of the convex hull $C_n:=\mathrm{Conv}(S_0,S_1,\dots,S_n)$. Assuming that the tuple $(\xi_1,\dots,\xi_n)$ is exchangeable and some general position condition holds, we derive an explicit formula for the expected number of $k$-dimensional faces of $C_n$ in terms of the Stirling numbers of the first and second kind. Generalizing the classical discrete arcsine law for the position of the maximum due to E. Sparre Andersen, we compute explicitly the probability that for given indices $0\leq i_1 < \dots < i_{k+1}\leq n$, the points $S_{i_1},\dots,S_{i_{k+1}}$ form a $k$-dimensional face of $\mathrm{Conv}(S_0,S_1,\dots,S_n)$. This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of a direct product of finitely many reflection groups of types $A_{n-1}$ and $B_n$. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. All formulae are distribution-free, that is do not depend on the distribution of the $\xi_k$'s. (Joint work with Vladislav Vysotsky and Dmitry Zaporozhets.)

May 30 Sander Dommers (Ruhr-Universität Bochum)

Continuous spin models on annealed generalized random graphs

We study spin models where the spins take values in a general compact Polish space and interact via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution. We prove a variational formula for the corresponding annealed pressure.

We furthermore study classes of models with second order phase transitions which include models on an interval and rotation-invariant models on spheres, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model, but we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.

This is joint work with Christof Külske and Philipp Schriever.

June 13 Airam Blancas Benítez (Goethe-Universität Frankfurt am Main)

Simple nested coalescents and two-level Fleming Viot processes

Simple nested coalescent has been introduced to model backwards in time the genealogy, of both, species trees and genes trees. In this setting, the Kingman case corresponds to binary coalescences in the species trees or the genes trees, but not simultaneously. On the other hand, two level Fleming Viot with two level selection arises in Dawson (2015) as the limit in distribution of multilevel multitype population undergoing mutation, selection, genetic drift and spatial migration. In this talk, I will establish a duality relation between Kingman nested coalescents and the two level Fleming Viot associated with a two-level multitype population with genetic drift. Using this relation we can read off the genealogy backwards in time of a Kingman nested.

June 20 Gabriel Hernán Berzunza Ojeda (Georg-August-Universität Göttingen)

Cutting-down random trees

Imagine that we destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. We then record the genealogy induced by this destruction process in a random rooted binary tree, the so-called cut-tree. The goal of this talk will be to show a general criterion for the convergence of the rescaled cut-tree in the Gromov-Prohorov topology to a real tree, when the underlying tree has a small height. In particular, we consider uniform random recursive trees, binary search trees, scale-free random trees. The approach relies in the introduction of a continuous version of the cutting-down procedure which we allow us to represent the destruction process up to a certain finite time as Bernoulli bond percolation.

June 27 Stein Andreas Bethuelsen (Technische Universität München)

Stochastic domination in space-time for the supercritical contact process

The contact process is a classical model for the spread of infections in a population. In this talk, we focus on the contact process in the supercritical regime for which infections may spread forever with positive probability. Our goal is to understand how this process behaves compared with a process having no spatial correlations. In particular, does the contact process stochastically dominate a non-trivial independent (in space) spin-flip process? Such questions were studied by Liggett and Steif (2006) who proved that, for the process on $\mathbb{Z}^d$, the upper invariant measure stochastically dominates a Bernoulli product measure. We present some space-time versions of their results for the contact process on general graphs. From our methods, we furthermore conclude strong (uniform) mixing properties for certain space-time projections of the contact process. Based on joint work with Rob van den Berg (CWI Amsterdam and VU Amsterdam).

July 4 Vladimir V. Ulyanov (Moscow State University)

Non-asymptotic analysis of non-linear forms in random elements

We review recent results on non-asymptotic analysis for the distributions of quadratic and almost quadratic forms in random elements with values in a Hilbert space. The study of almost quadratic forms is motivated by approximation problems in multidimensional mathematical statistics. A number of results are optimal - they can not be improved without additional assumptions. A unified approach will be proposed for constructing non-asymptotic approximations on the basis of the general result on approximation accuracy for symmetric functions of several variables.

Vladimir Panov (Higher School of Economics)

Low-frequency estimation for moving average Lévy processes

This talk is devoted to statistical inference for the stochastic integrals of the type $$ Z_{t}=\int_{R} K(t-s) dL_{s},$$ where $K$ is a deterministic function, and $L$ is a Lévy process. In particular, I will present several new ideas yielding the construction of a consistent estimator for the Lévy density of $L$. Moreover, I intend to discuss the mixing properties, which draw particular interest from the theoretical point of view.

July 5 Nina Gantert (Technische Universität München)

Biased random walk in random environments

We explain two models for biased random walks in random environment (biased random walk on percolation clusters, biased random walk among random conductances) which describe transport in an inhomogenous medium. We give an overview of typical questions and present several results about Einstein relation and monotonicity of the speed.

Time: 17:15 @ WSC-S-U-4.02 (Math Colloquium)
July 11 Manfred Opper (Technische Universität Berlin)

Approximation techniques for probabilistic inference --
a statistical physics perspective

Probabilistic, Bayesian methods provide important approaches to data modelling in the field of machine learning. Inference on unobserved variables or parameters typically requires the performance of high-dimensional integrals or sums. If the dimensionality of the problem is very large, one often has to apply approximate inference methods which approximate intractable multivariate probability distributions by tractable ones.

In this talk I will focus on the so-called “Expectation Propagation” message passing techniques for approximate inference in latent Gaussian variable models. These are related to (and were, in parts motivated by) the “Thouless-Anderson-Palmer” (TAP) mean field approach to spin-glass models in physics.

I will give an introduction to the basic idea behind this approach and then discuss a combination of ideas from random matrix theory and dynamical functional methods of statistical physics which could be used to improve the efficiency of such methods and to study the convergence of inference algorithms for large systems.

July 18 Oleg Butkovsky (Technische Universität Berlin)

Convergence of Markov processes to the invariant measure with applications to SPDEs and stochastic delay equations

While convergence of finite-dimensional Markov processes (e.g., SDEs) to the invariant measure is quite well understood by now, less is known about the convergence of infinite-dimensional Markov processes (e.g., SPDEs). In the first part of the talk we explain how the classical methods (which are based on the construction of a Lyapunov function) can be extended to study convergence of infinite-dimensional Markov processes in the Wasserstein metric. This generalizes recent results of M. Hairer, J. Mattingly, M. Scheutzow (2011). In the second part of the talk we provide some specific applications to SPDEs and stochastic delay equations and discuss the arising challenges. (Joint work with Alexey Kulik and Michael Scheutzow)

[1] O. Butkovsky (2014). Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Annals of Applied Probability, 24, 526-552.

[2] O. Butkovsky, M. Scheutzow (2017). Invariant measures for stochastic functional differential equations. arXiv:1703.05120.

July 25 Stephan Gufler (Goethe-Universität Frankfurt am Main)

Exchangeable ultrametrics and tree-valued Fleming-Viot processes

The Donnelly-Kurtz lookdown model contains an evolving genealogy. The genealogical tree of the population at each time can be described by an isomorphy class of a metric measure space to obtain a tree-valued Fleming-Viot process. The states of this process can be viewed as ergodic components of the genealogical distance matrices. In this context, we also discuss a general representation for exchangeable ultrametrics in terms of sampling from marked metric measure spaces.

Winter Semester 2016/17 (back to Contents)

October 18 Marvin Zorn (Hochschule Koblenz)

Stochastische Optimierungsstrategien fur Insider Trading auf einem zeitstetigen Finanzmarkt

Wir legen einen Wahrscheinlichkeitsraum $(\Omega, \mathcal{F},P)$ zugrunde, der die Entwicklung eines Finanzmarkts beschreibt. Auf diesem agiert neben den üblichen Teilnehmern ein Trader, der im Besitz nicht öff entlicher Information ist. Für diesen Insider wird seine optimale Investment Strategie bzgl. einer vorgegebenen Nutzenfunktion ermittelt. Anschließend wird untersucht wie sich der Verlust von Information auf die Strategie und den Nutzen auswirkt. Hierzu werden nicht injektive Funktionen auf die Zufallsvariable, die die Insider Information repräsentiert, angewandt. Es werden Identitäten entwickelt, die zur Gewinnung der neuen aus der alten Strategie dienen konnen.

October 25 Anita Winter (University of Duisburg-Essen)

BM on trees: a cover time bound

Simple symmetric random walk on a finite, connected graph is positive recurrent and the mean time to visit all vertices is finite. Recently, variable speed motions on metric measure trees have been constructed. In this talk we will raise the question what happens if we replace the assumption on the finiteness of the graph by completeness of the tree. (Joint with Omer Angel, Siva Athreya & Manjunath Krishnapur).

November 8 Daniel Pieper (Georg-August-Universität Göttingen)

Asymptotic growth of supercritical Galton-Watson processes

The Kesten-Stigum theorem describes the growth of supercritical Galton-Watson processes. Its classical proof relies on the analysis of probability generating functions. We look at a conceptual proof due to Lyons, Pemantle and Peres that uses size-biased Galton-Watson trees and basic measure theory. We also discuss its generalization to the case of stationary and ergodic environments.

November 9 Jakob Herrenbrück (Universität Bielefeld)

Nonlinear Schrödinger equations on spheres

We will discuss the Cauchy problem for Schrödinger equations with a nonlinearity of odd order posed on spheres of arbitrary dimension. The methods used are multilinear eigenfunction estimates and a spectral decomposition of the Laplacian, which allows the construction of suitable function spaces. In these spaces we can prove space-time estimates of Strichartz type. As the main result, we will see that the Cauchy problem is locally well-posed for initial data in an appropriate Sobolev space. This extends previous results of Herr and Burq-Gérard-Tzvetkov.

Non-standard weekday (Wednesday), time (16:15) & place (WSC-S-U-4.01)

November 22 Ekaterina Krymova (Universität Duisburg-Essen)

Structural adaptive dimension reduction for musical signals modelling

In the talk we discuss problem of simplification of musical signals which arise in cochlear implant music transmission. The main peculiarity of musical signals is highly varying temporal and spectral structure. Also the hearing loss and the hearing aid limitations should be used as an a-priori knowledge. The simplification of the music signals may be achieved by an adaptive dimension reduction of spectral data and by the extraction of important musical features. To this end, we develop a novel unsupervised segmentation procedure for music signals which relies on an explained variance criterion in the eigenspace of the constant-Q spectral domain.

November 29 Alexander Klug (University Cologne)

Speciation and recombination on fitness landscapes

Sungmin Hwang (University Cologne)

The number of fitness maxima in the random NK model

Visiting Session @ Rhein-Ruhr SPP Seminar
16:00 @ Ground floor lecture hall, Biozentrum, Universität zu Köln
(Building 304, Zuelpicher Strasse 47b, D-50674 Cologne)

December 8 Yan Dolinsky (Hebrew University)

Super-replication with fixed transaction costs

We study super-replication of contingent claims in markets with fixed transaction costs. The first result in this paper reveals that in reasonable continuous time financial market the super–replication price is prohibitively costly and leads to trivial buy–and–hold strategies. Our second result is derives non trivial scaling limits of super–replication prices in the binomial models with small fixed costs. (Joint work with P.Bank)

Non-standard weekday (Thursday), time (10:15), place (WSC-S-U-3.03)

December 13 Wolfgang Löhr  (Universität Duisburg-Essen)

State spaces of (continuum) trees: R-trees versus Algebraic Trees

In order to construct tree-valued stochastic processes, one needs a topological space of trees as state space. While this is not an issue for finite (or countable) graph-theoretic trees, we want to consider global limits as the number of vertices tends to infinity. We call the limiting objects also "tree" but have to make precise what we mean by this. The standard approach is to consider continuum trees to be metric (measure) spaces (so-called $\mathbb{R}$-trees) and equip the space of them with Gromov-Hausdorff or Gromov-weak topology.

We argue that sometimes it is more natural to consider a different type of structure, namely the tree-structure instead of the distance. First, because distances may behave very ``wild'' in certain cases and neglecting them makes proving some limit results more feasible. Second, because sometimes one might want to preserve structural properties such as being binary in the limit.

We present a framework for a space of such (continuum) trees possessing no metric- but only a tree-structure. We call them algebraic trees, because we formalise the tree-structure by a tertiary operation on the tree, namely the branch point map. We construct a natural, topology on spaces of sufficiently nice algebraic trees. In the binary case, the resulting space is compact and intimately related to the set of triangulations of the circle as introduced by Aldous, equipped with the Hausdorff metric.

Visiting Session @ Rhein-Ruhr SPP Seminar
16:30 @ Ground floor lecture hall, Biozentrum, Universität zu Köln
(Building 304, Zuelpicher Strasse 47b, D-50674 Cologne)

January 17 Johannes Wirtz, Alex Klassmann (Universität zu Köln)

Visiting Session @ Rhein-Ruhr SPP Seminar
16:00 @ Ground floor lecture hall, Biozentrum, Universität zu Köln
(Building 304, Zuelpicher Strasse 47b, D-50674 Cologne)

Januar 24 Vladimir Ulyanov (Lomonosov Moscow State University)

Bootstrap confidence sets for spectral projectors of sample covariance

Alexey Naumov (Skoltech), Vladimir Ulyanov (MSU)

Let $X_1, \ldots ,X_n$ be i.i.d. sample in $\mathbb{R}^p$ with zero mean and the covariance matrix $S$. The problem of recovering the projector onto the eigenspace of $S$ from these observations naturally arises in many applications. Recent technique from [Koltchinskii and Lounici, 2015b] helps to study the asymptotic distribution of the distance in the Frobenius norm between the true projector $P_r$ on the subspace of the $r$-th eigenvalue and its empirical counterpart $\hat{P}_r$ in terms of the effective trace of $S$. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector $P_r$ from the given data. This procedure does not rely on the asymptotic distribution of $\Vert P_r - \hat{P}_r \Vert_2$ and its moments, it applies for small or moderate sample size $n$ and large dimension $p$. The main result states the validity of the proposed procedure for finite samples with an explicit error bound on the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimension. Numeric results confirm a nice performance of the method in realistic examples. These are the joint results with V.Spokoiny.

Non-standard time & location: 12:00 @ WSC-O-4.43

Januar 24 Kohei Suzuki (Hausdorff Center for Mathematics)

Convergence of Brownian motions on metric measure spaces under the Riemannian Curvature-Dimension condition

The goal of this talk is to characterize the weak convergence of Brownian motions in terms of a geometric convergence of the underlying spaces. As main results, we show several equivalences between the weak convergence of Brownian motions and the pointed measured Gromov convergence of the underlying spaces satisfying the Riemannian Curvature-Dimension (RCD) condition. Here the RCD condition is a generalization of "Ricci curvature $\geq K$" and admits various singular spaces like the Gromov-Hausdorff limit of Riemannian manifolds with Ric $\geq K$, or several infinite-dimensional spaces.

February 7 Rebecca Neukirch (Rheinische Friedrich-Wilhelms-Universität Bonn)

The recovery of a recessive allele in a Mendelian diploid model

Summer Semester 2016 (back to Contents)

May 3 Dr. Alexander Kulikov (Moscow Institute of Physics and Technology)

Optimal hedging and risk contribution in energy markets via coherent risk measures

Let us consider an energy company that sells a fluctuating volume $V$ of energy units at random price $S$ per unit and obtains the income $X=VS$. Today's markets do not provide simple instruments that can be used for hedging volumetric risk, i.e. the risk that appears due to the randomness in $V$. In the current literature hedging of volumetric risk is settled via utility functions. Here we consider this task using coherent risk measures and obtain explicit hedging strategies in some cases. If we extract oil in one country and sell it in another one, then we additionally have currency risk. Motivated by the dependence between oil prices and currency exchange rates we settle a problem of hedging both currency and volumetric risks via multidimensional coherent risk measures, with put options on oil prices as hedging instruments, and present its solution in some situations.

June 14 Luis Enrique Osorio Puentes (University of Duisburg-Essen)

Distributional properties of the stochastic heat equation via Malliavin Calculus

Malliavin Calculus is an infinite dimensional calculus defined on the Wiener space. It was developed in 1976 by Paul Malliavin. It has been used to study regularity properties (conditions for being absolutely continuos and/or having smooth density) of functionals defined on the Wiener space as well as in some applications in finance (via the Clark-Ocone formula). In this talk we consider some properties of the Mallivin derivative and its adjoint operator, known as the Skorokhod integral. We then apply this theory to study distributional properties of the Stochastic Heat Equation. More specifically, we will see how this theory helps us find conditions under which the solution to this equation has a smooth density.

July 5 Geronimo Rojas Barragan (University of Duisburg-Essen)

Random processes conditioned to stay positive and occupation times

Random walks provide some of the most rich examples of stochastic processes. Starting with a random walk with real values (zero mean and finite variance), we are interested in finding out how it behaves if we restrict the state space. In particular, what happens when we are only interested in the positive real line. This version of the process is known as random walk conditioned to stay positive. Given the analogies between random walks with finte variance and Brownian motion, we also conditioned the latter to stay positive. The resulting process can be seen as 3-d Bessel process. Once these conditioned processes have been introduced we analyse the asymptotic behaviour of their occupation times; to be more precise, we seek for deterministic functions that describe the limiting behaviour of these processes.

Winter Semester 2015/16 (back to Contents)

October 13 Dr. Yan Dolinsky (Hebrew University of Jerusalem)

Limiting behavior of super-replication prices with small transaction costs: multi-asset version of Kusuoka results

We consider super-replication with small transaction costs in complete multi-asset multinomial markets. For this setup we prove that the limit of the corresponding super-replication prices equals to a multidimensional $G$-expectation which we find explicitly. These results can be interpreted as a multidimensional extension of Kusuoka (1995). This is a joint work with P.Bank and A.P.Perkkio.

October 20 Dr. Robert Fitzner (Eindhoven University of Technology)

High-dimensional percolation

Percolation is one of the simplest ways to define models in statistical physics and mathematics which displays a non-trivial critical behaviour. This model describes how a graph/lattice/network behaves under random removal of edges. In the last 40+ years, it has been an active field of research due to richness of the behaviour of the model and due to its numerous applications. We introduce the classical version of the model on the lattice $\mathbb{Z}^d$, where percolation has the remarkable feature that it undergoes a sharp phase transition: There exists a critical value $p_c = p_c(d) \in (0,1)$ such that if you randomly retain a fraction of less than $p_c$ edges (i.e., remove a fraction of more than $1-p_c$ of all edges in i.i.d. manner), then the resulting graph consists of finite connected components only. Conversely, if you retain a fraction of more than $p_c$ edges (i.e., randomly remove a smaller than $1-p_c$ fraction of edges), then the resulting graph will contain exactly one infinite connected component. What happens at criticality, i.e., if we randomly retain exactly the critical fraction of $p_c$ edges is only understood for $d = 2$ (where $p_c(2) = 0.5$) and for $d > 10$ (where $p_c(d) \approx 1/(2d-1)$). In the talk, we review known results about percolation and give an idea of how to obtain results for $d > 10$.

November 3 Prof. Leif Döring (University of Mannheim)

Perpetual integrals for Lévy processes

Due to time-change properties of stochastic differential equations, it is important to understand finiteness properties of time-integrals of Brownian motion (or Lévy Processes). We show how to use fluctuation theory and Jeulin’s lemma to prove a 0-1 law for time-integrals of Lévy processes with local times.

November 10 Dr. Michael Hinz (University of Bielefeld)

Time dependent perturbations of symmetric Markov processes and related evolution problems

A diffusion subject to a deterministic movement and and external supply of particles can be modelled by a Brownian motion with drift and killing. In probability this can be studied by Girsanov and Feynman-Kac transforms, in analysis by a heat equation with first order and potential terms. If the drift depends on time, we can no longer operate with one-parameter semigroups, and if a related Markov process exists, it will not be homogeneous. Combining evolution semigroups, time-dependent Dirichlet forms and space-time processes we deduce probabilistic representations for classical solutions of abstract evolution problems. The result generalizes a formula well known for Brownian motion and applies for instance to Lévy processes, Random walks or diffusions on fractals. The talk is based on joint work with Gerald Trutnau, SNU Seoul.

November 17 Dr. Daniel Zivkovic (LMU Munich)

The allelic spectrum and its applications on genomic data and to an epidemiological model of host-parasite coevolution

Advances in empirical population genetics have made apparent the need for models that simultaneously account for selection and demography. To address this need, I consider the Wright-Fisher diffusion under selection and piecewise constant population sizes. In the case of genic selection, I will sketch the derivation of the transition density. For general diploid selection, I will apply a moment-based approach to devise an efficient and fast algorithm for the computation of the allele frequency spectrum. I will discuss several applications that are of interest for the analysis of whole-genome sequences. In respect of a further application, I will introduce and discuss a system of coupled differential equations describing the coevolution of host and parasite alleles.

November 24 Stefan Häfner (University of Duisburg-Essen)

Higher order variance reduction for discretised diffusions via regression

A novel approach towards variance reduction for discretised diffusion processes is presented. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance of the terminal functionals. In this way the complexity order of the plain Monte Carlo algorithm ($\varepsilon^{-3}$ in the case of a first order scheme and $\varepsilon^{-2.5}$ in the case of a second order scheme) can be reduced down to $\varepsilon^{-2+\delta}$ for any $\delta\in [0,0.25)$ with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.

December 1 Dr. Alexey Muravlev (Steklov Mathematical Institute, Moscow)

Upper and lower estimates for boundaries in non-linear optimal stopping problems

We consider a sequential testing problem of two hypotheses concerning the drift value of a fractional Brownian motion. We show that it can be reduced to the optimal stopping problem for a standard Brownian motion with non-linear cost of observation. Using standard technique one may characterize optimal stopping boundaries as an unique solution of a system of non-linear integral equations, which can be used for numerical evaluation via backward induction technique. However, for this, one needs to have some end-point T, after which the values of optimal boundaries are known. The main point of the talk is how to obtain some upper and lower estimates for the boundaries which provide the choice of such a T. In contrast to known methods based on the study of corresponding free-boundary problem, our approach is more probabilistic.

December 8 Dr. Mikhail Zhitlukhin (Steklov Mathematical Institute, Moscow)

Bounds for the expected maximum of a fractional Brownian motion and related processes

We study expected maxima of Gaussian processes that are Holder continuous in L2-norm and/or satisfy the opposite inequality for the L2-norms of their increments. Examples of such processes include the fractional Brownian motion and some of its relatives. We establish upper and lower bounds for the expected maximum of such a process and investigate the rate of convergence to that quantity of its discrete approximation. Further properties of these two maxima are established in the special case of the fractional Brownian motion. This is a joint work with Konstantin Borovkov, Yulia Mishura and Alexander Novikov.

December 15 Dr. Alexander Cox (University of Bath)

Maximising functions of the average of a martingale with given terminal law

In this talk, we consider the problem of finding the martingale which maximises the expected value of some function of the average of the martingale, subject to a given terminal law of the process. This is a problem which arises in model-independent pricing of financial derivatives, and is commonly known as a Martingale Optimal Transport problem. Our methods differ from most approaches to these problems in that we consider the problem as a dynamic programming problem, where the controlled process is the conditional distribution of the martingale at the final time. By formulating the problem in this manner, we are able to determine the maximal price through a PDE formulation. Notably, this approach does not require specific constraints on the function (e.g. convexity), and would appear to be generalisable to many related problems. (Joint work with S. Källblad).

December 22 Dr. Sigurd Assing (University of Warwick)

On trading American put options with interactive volatility

We introduce a simple stochastic volatility model, which takes into account hitting times of the asset price, and study the optimal stopping problem corresponding to a put option whose time horizon (after the asset price hits a certain level) is exponentially distributed. We obtain explicit optimal stopping rules in various cases one of which is interestingly complex because of an unexpectedly disconnected continuation region. Finally, we discuss in detail how these stopping rules could be used for trading an American put when the trader expects a market drop in the near future.

January 12 Prof. Peter Tankov (University Paris-Diderot -- Paris 7)

Asymptotic lower bounds for optimal tracking: a linear programming approach

We consider the problem of tracking a target whose dynamics is modeled by a continuous Ito semimartingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control of Brownian motion, which is characterized as a deterministic linear programming problem. A comprehensive list of examples with explicit expressions for the lower bounds is provided.

January 19 Dr. Giorgio Ferrari (University of Bielefeld)

Nash equilibria of threshold type for two-player nonzero-sum games of stopping

In this talk I consider two-player nonzero-sum games of optimal stopping on a class of regular diffusions with singular boundary behaviour (in the sense of Itô and McKean, p. 108). I show that Nash equilibria are realised by stopping the diffusion at the first exit time from suitable intervals whose boundaries solve a system of algebraic equations. Under mild additional assumptions we also prove uniqueness of the equilibrium. The talk is based on a joint work with Tiziano De Angelis and John Moriarty.

January 26 Prof. Mihail Zervos (London School of Economics)

Dynamic contracting under moral hazard

We consider a contracting problem that a firm faces in the presence of managerial moral hazard and stochastic cashflows. We first develop a general contracting setting. We then restrict attention to contracts that admit appropriate state space representations. In the latter context, we establish the link between the optimal contract and the solution to a suitable stochastic control problem.

February 2 Prof. Georgiy Shevchenko (Taras Shevchenko University Kiev)

Fine integral representations through small deviations of quadratic variation

The talk is based on a joint paper with Taras Shalaiko (Mannheim University). It will be devoted to stochastic integral representations of the form $$\xi = \int_0^T \phi(s) dB^H(s),$$ where $\phi$ is an adapted process, and $B^H$ is a fractional Brownian motion with Hurst parameter $H>1/2$. Studying such representations is motived by applications in financial mathematics, where $\phi$ plays a role of the risky component of a self-financing portfolio. I will discuss how the estimates for probabilities of small deviations of realised quadratic variation of $B^H$ can be used to construct the above integral representations under some mild assumptions on $\xi$. To this end, we define a generalisation of the fractional integral introduced by Martina Zähle.

February 9 Prof. Cornelia Pokalyuk (University of Magdeburg)

Sweeps in the ancestral selection graph

We will see, why and how the ancestral selection graph can be used to study fixation of beneficial alleles in structured populations.


Summer Semester 2015 (back to Contents)

April 14 Prof. Stefan Tappe (Uni Hannover)

Invariance of closed convex cones for stochastic partial differential equations

In this talk, we provide necessary and sufficient conditions for stochastic invariance of closed convex cones in Hilbert spaces for semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures. Several examples accompany our results.

April 21 Dr. Nikolaus Schweizer (University of Duisburg-Essen)

Perturbation theory for Markov chains via Wasserstein distance

Perturbation theory for Markov chains addresses the question how small differences in the transitions of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the n-th step distributions of two Markov chains when one of them satisfies a Wasserstein contractivity condition.

Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak conditions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings and stochastic Langevin algorithms.

This is joint work with Daniel Rudolf (University of Jena)

May 5 PhD Alexander Kulikov (Moscow Institute of Physics and Technology)

Hedging price risk in commodity markets in presence of volumetric and currency risk

Let us consider an energy company that sells fluctuating volume $V$ of energy units at random price~$S$ per unit and obtains the income $X=VS$. Markets today do not provide simple instruments that can be used for hedging volumetric risk (See~\cite{OOD06}). Traditional measure of risk which is widely used in practice is V@R. We consider the problem of V@R minimization in incomplete markets using put options. We try to find the optimal strategy in $h\in {\mathbb R}$ that solves the following problem: $$ V@R\Bigl(V * S + h ((K-S)^+ - EP(K))\Bigr) \rightarrow \min_{h\in{\mathbb R},K\in {\mathbb R}_{+}}, $$ (due to incompleetness of commodity markets we could not make a perfect hedge). Here we introduce the solution of finding optimal number of put options bought with strike $K$ to minimize V@R in some cases (also the case of ''natural hedging'' is also introduced). To solve it we use the following function: $$q(K)(a,h)={\sf P}(VS+h((K-S)^{+}-P(K))\leq a)$$ and investigate its properties. So we try to find optimal hedge in presence of volumetric risk. However when you extract oil in one country and sell it in another you also have currency risk. So you need to solve the same task in presence of currency risk where usually only price risk is traded. Here we provide some fundamental facts which allow to model currency exchange rate and oil price together (correlation does not provide good results in this case). Using this fact we provide some examples of optimal hedging in presence of currency and volume risk using put options on oil prices.

May 12 Prof. Albert Shiryaev (Steklov Mathematical Institute, Moscow)

Optimal stopping problems for a Brownian motion with drift and disorder: Applications to financial mathematics.

We consider solutions of the several sequential problems for a Brownian motion. For example, we show how to solve Chernoff's problem of testing statistical hypotheses about the sign of drift. We show how to obtain a solution to the problem of sequential testing of three statistical hypotheses about the drift of a Brownian motion. We consider financial problems of the type "When to sell Apple?" and some other problems.

May 19 Prof. Martin Keller-Ressel (TU Dresden)

Implied Volatilities from Strict Local Martingales

Several authors have proposed to model price bubbles in stock markets by specifying a strict local martingale for the risk-neutral stock price process. Such models are consistent with absence of arbitrage (in the NFLVR sense) while allowing fundamental prices to diverge from actual prices and thus modeling investors’ exuberance during the appearance of a bubble. We show that the strict local martingale property as well as the “distance to a true martingale” can be detected from the asymptotic behavior of implied option volatilities for large strikes, thus providing a model-free asymptotic test for the strict local martingale property of the underlying. This talk is based on joint work with Antoine Jacquier.

May 26 Marco Noll (University of Frankfurt)

Regularity of Cox-Ingersoll-Ross processes in the case of an accessible boundary point.

Cox-Ingersoll-Ross (CIR) processes are widely used in financial modeling such as in Heston's stochastic volatility model as the variance process for the pricing of financial derivatives. In Heston's stochastic volatility model, it is commonly assumed that the parameters of the variance process satisfy Feller's boundary condition. As a consequence, the variance process is strictly positive. On the other side, if Feller's boundary condition is not satisfied, we can show for half of the parameter regime in which the boundary point 0 is accessible that the variance process spends $\mathbb{P}$-almost surely only Lebesgue time $0$ at the boundary point $0$. Since calibrations of the Heston model frequently result in parameters such that the boundary is accessible, we focus on this interesting case. In the literature, positive strong convergence rates for numerical approximations of CIR processes have been established in the case of an inaccessible boundary point. Our main result shows for every $p\in(0,\infty)$ that the drift-implicit square-root Euler approximations proposed in Alfonsi (2005) converge in the strong $ L^p $-distance with a positive rate for half of the parameter regime in which the boundary point is accessible. Moreover, we prove local Lipschitz continuity in the initial value for CIR processes in the case where the boundary point $0$ is accessible.

June 2 Dr. Hendrik Weber (University of Warwick)

SPDEs, criticality and renormalisation

In this colloquium I will report on recent progress in the theory of stochastic PDEs and their connections with statistical mechanics. My main examples will be the KPZ equation and the dynamic $\Phi^4$ model. Both of these equations arise as scaling limits of interacting particle systems: the KPZ equation as a scaling limit for surface growth models and the $\Phi^4$ model as a scaling limit of an Ising-type model of a ferromagnet. The mathematical treatment of these equations is challenging due to the low regularity of the random ``noise term'' present. In particular, solutions are too irregular to apply the usual ``deterministic'' solution techniques. Even worse sometimes ``infinite counterterms'' have to be subtracted to produce non-trivial solutions. I will discuss this renormalisation procedure in some detail for the $\Phi^4$ model. I will first explain, why this procedure is necessary and how it can be implemented mathematically. Then I will discuss the interpretation on the level of particle approximations.

June 9 Prof. Andrea Barth (University of Stuttgart)

Galerkin approximations for stochastic partial differential equations

In this talk I give an introduction to Finite Element approximations for parabolic and hyperbolic stochastic partial differential equations. I demonstrate that in the parabolic case, due to the nice properties of the parabolic operator, Galerkin approximations give the expected convergence results (expected as for deterministic equations). In the case of an hyperbolic equation, however, some artificial smoothing has to be employed, since the driving noise process and therefore the solution is too irregular for a (standard) Galerkin approximation. For an approximation of these problems, I introduce a (stochastic) version of a Streamline Diffusion method.

June 16 Prof. Stefan Ankirchner (Uni Jena)

Controlling the occupation time of a geometric martingale

We consider control problems in which the volatility of a geometric martingale can be chosen to take any value between a minimal and a maximal value. Special attention is paid to the strategy that consists in choosing the minimal volatility if the martingale is above a given threshold and to choose the maximal volatility else (play safe if ahead, take risks if behind).

June 23 PhD Vladimir Panov (HSE Moscow)

Semiparametric estimation in the normal variance-mean mixture model

In this talk, I intend to present some fresh ideas concerning the estimation in the normal variance-mean mixture models. These models are closely related to the class of time-changed L{\'e}vy processes, and naturally appear in some interesting problems like modelling of the sizes of diamonds in marine deposits of South West Afrika. The focus of our research is the simultaneous estimation of all finite-dimensional parameters of the model and the mixing distribution.

June 30 Ryan Kurniawan (ETH Zurich)

Numerical approximations of stochastic partial differential equations with superlinearly growing nonlinearities

In this talk, we study numerical approximations of stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities. We first give examples of some of such SPDEs, including stochastic Burgers equations, stochastic 2-D Navier-Stokes equations, Cahn-Hilliard-Cook type equations, and stochastic Kuramoto-Sivashinsky equations. We then show that the fully discrete exponential Euler method converges in the almost sure sense with some rate for all of the aforementioned examples of SPDEs. Finally, we will discuss about a work in progress on how to use this result to show that a newly proposed scheme, called the fully discrete nonlinearity-stopped exponential Euler method, converges strongly in the $L^p$-norm for all $p\in(0,2)$ for all of the aforementioned examples of SPDEs.

July 2 Dr. Pavel Yaskov (Moscow)

Random orthogonal projections, universality, and the smallest singular values of random matrices

We start with a particular problem that arises in econometrics and deals with random orthogonal projection matrices. We show that this problem is intimately related to the Marchenko-Pastur theorem and, in general, the universality phenomenon in the random matrix theory. We also discuss some lower bounds on the smallest singular values of random matrices that are needed to solve the above problem, but can be of independent interest in a much more general context.

July 14 Rebecca Neukirch (University of Bonn)

Genetic Variability in the Mendelian Diploid Model

Adaptive dynamics are well studied for the haploid reproduction model. In this talk we are interested in the genetic evolution of a diploid hermaphroditic population, which is modeled by a three-type nonlinear birth-and-death process with competition and Mendelian reproduction. In a recent paper, Collet, Méléard and Metz (2013) have shown that on the mutation time-scale the process converges to the Trait-Substitution Sequence of adaptive dynamics, stepping from one homozygotic state to another with higher fitness.

In the first part of the talk, we prove that, under the assumption that the dominant allele is also the fittest one, the recessive allele survives for a time of order at least $K^{1/6}$, where $K$ is the size of the population.

In the second part we give a sufficient condition on the competition such that the recessive a- allele can recover after the occurrence of a new fitter mutant. This leads to genetic variability in the populations.


Winter Semester 2014/15 (back to Contents)

Oct 14

Dr. Fabian Dickmann (University of Duisburg-Essen)

Pricing Bermudan options via multi-level approximation methods

This talk is about an approach to reduce the computational complexity of various approximation methods for pricing Bermudan options. Given a sequence of continuation value estimates corresponding to different levels of spatial approximation, we propose a multi-level low biased estimate of the price. It turns out that the resulting complexity gain can be of order $1 / \epsilon$ where $\epsilon$ denotes the desired precision measured in terms of the root-mean-squared error.

Oct 21

Dr. Thomas Kruse (Université d'Evry Val d'Essonne)

A generalized Donsker theorem and approximating SDEs with irregular coefficients

We provide a new method for approximating the law of a diffusion $M$ solving a stochastic differential equation with coefficients satisfying the Engelbert-Schmidt conditions. To this end we construct Markov chains whose law can be embedded into the diffusion M with a sequence of stopping times that have expectation $1/N$, where $N \in \mathbb{N}$ is a discretization parameter. The transition probabilities of the Markov chains are determined by a reference probability measure $\mu$, scaled with a factor depending on $N$ and the state. We show that the Markov chains converge in distribution to the diffusion $M$, as $N \to \infty$, thus refining the Donsker-Prokhorov invariance principle. Finally, we illustrate our results with several examples.

The talk is based on joint work with Stefan Ankirchner and Mikhail Urusov.

Oct 28

Tigran Nagapetyan (Wias Berlin)

Weak Multilevel Monte Carlo algorithm

We discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. This general approach can be used in a wide range of problems, while we exemplify this idea in the case of weak Euler scheme for Lévy driven stochastic differential . The numerical performance of the new "weak" MLMC method is illustrated by several examples.
The talk is based on joint work with Denis Belomestny.

Nov 4

Lisa Hartung (University of Bonn)

Extended convergence of the extremal process of branching Brownian motion

Branching Brownian motion is a popular example for a Gaussian process indexed by a tree. A property of key interest is the structure of the process near its maximum for large times. In this talk I recall the construction of the extremal process of branching Brownian motion obtained by Arguin, Bovier and Kistler. Then I explain how this convergence can be extended by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. (joint work with A. Bovier)

Nov 11

Paul Krühner (TU Dortmund)

Optimal density bounds for SDE's with discontinuous drift coefficients

In this talk we study the regularity of solutions to the SDE $dX(t) = b(t,X(t))dt + a(t,X(t))dW(t)$ in a finite dimensional space where $b$ is only assumed to be measurable and bounded. Malliavin invented a great method to study the density properties -- like boundedness of the density -- of $X(t)$ under smoothness assumptions. His approach has been generalised in various directions and to various different applications. We find sharp upper and lower bounds for the density without using any type of variational calculus. This talk is based on joint work with David Baños.

Nov 18

Anselm Hudde (University of Freiburg)

Malliavin Calculus and the Calculation of Greeks

In this talk we introduce the Malliavin Calculus for Isonormal Gaussian processes. This allows us to differentiate and integrate random variables with respect to the white noise that is generated by an Isonormal Gaussian process. We present some of its main properties, including an integration by parts formula. With this formula we can find stochastic weights such that option sensitivities in the Black Scholes model can be easily calculated without differentiating the payoff function and without knowledge of the density of the underlying asset.

Nov 25

Felix Jordan (LMU München)

Evolution of Altruistic Defense Traits in Structured Populations

An altruist is an individual that helps others at having more offspring while paying the cost of having less offspring itself. This would intuitively suggest that bearers of genes for altruism have a disadvantage compared to non-altruists and, therefore, become extinct. Nevertheless, it is known that in a structured population, altruists can prevail, if the benefit to other altruists outweighs the cost to the individual. In the case of altruism taking the form of defense against a parasite, there is a negative feedback for altruists. When altruists become abundant, they reduce the number of parasites, which in turn reduces the benefit that each altruist brings. It is not a priori clear how this mechanism affects the evolution of the defense trait. We analyze this scenario in an island model using diffusion theory. In the limit of infinitely large populations on infinitely many islands, we obtain a condition determining whether a gene for altruism becomes extinct or fixates in the host population.

Dec 2

Dr. Sonja Cox (University of Amsterdam)

Burkholder-Davis-Gundy inequalities for vector-valued stochastic integrals

Stochastic integrals of processes taking values in a Banach space are used to study the solutions to stochastic partial differential equations and obtain regularity results that are not accessible using Hilbert space theory only. A key ingredient for defining these stochastic integrals are the Burkholder-Davis-Gundy inequalities. These inequalities however do not hold in all Banach spaces. In my talk I will discuss different types of decoupling inequalities for random sums in a Banach space, by means of which one can characterize the spaces in which the BDG inequalities hold. Some characterizations are well-known, and some were established in recent work by Stefan Geiss and myself.

Dec 9

Dr. Martin Sauer (TU Berlin)

Analysis and Approximation of Stochastic Nerve Axon Equations

This talk concerns spatially extended conductance based neuronal models with noise described by certain stochastic reaction diffusion equations with only locally Lipschitz continuous nonlinearities. We study the wellposedness of such equations, the approximation in space using the finite difference method and derive explicit error estimates, in particular pathwise and strong convergence rates.

Jan 13, 2015

PD Dr. Volker Krätschmer (University of Duisburg-Essen)

Optimal stopping under model uncertainty

In the talk we consider optimal stopping problems with conditional convex risk measures of the form $ \rho^{\Phi}_t(X)=\sup_{\mathbb{Q}\in {\cal Q}_{t}}\left(\mathbb{E}_{\mathbb{Q}}[X|\mathcal{F}_t]- \mathbb{E}\left[\Phi\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)\big|{\cal F}_{t}\right]\right),$ where $\Phi: [0,\infty[\rightarrow [0,\infty]$ is a lower semicontinuous convex mapping and ${\cal Q}_{t}$ stands for the set of all probability measures ${\cal Q}$ which are absolutely continuous w.r.t. a given measure $\mathbb{P}$ and $\mathbb{Q}= \mathbb{P}$ on \({\cal F}_{t}.\) Here the model uncertainty risk depends on a (random) divergence $\mathbb{E}\left[\Phi\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)\big|{\cal F}_{t}\right]$ measuring the distance between a hypothetical probability measure we are uncertain about and a reference one at time $t.$ Let \((Y_t)_{t\in [0,T]}\) be an adapted nonnegative, right-continuous stochastic process fulfilling some proper integrability condition and let \({\cal T}\) be the set of stopping times on \([0,T]\), then without assuming any kind of time-consistency for the family \((\rho_t^{\Phi}),\) we derive a novel representation \begin{eqnarray*} \sup_{\tau\in {\cal T}}\rho^{\Phi}_0(Y_\tau)= \inf_{x\in\RRR}\left\{\sup\limits_{\tau\in{\cal T}}\mathbb{E}\bigl[\Phi^*(x + Y_{\tau}) - x\bigr]\right\}, \end{eqnarray*} which makes the application of the standard dynamic programming based approaches possible. In particular, we generalize the additive dual representation of Rogers (2002) to the case of optimal stopping under uncertainty. The talk is based on a joint work with Denis Belomestny.

Jan 20, 2015 Shota Gugushvili (University of Leiden)

Non-parametric Bayesian inference for multi-dimensional compound Poisson processes

Given a sample from a discretely observed multi-dimensional compound Poisson process, we study the problem of non-parametric estimation of its jump size density $r_0$ and intensity $\lambda_0.$ We take a non-parametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply existence of Bayesian point estimates that converge to the true parameter pair $(r_0,\lambda_0)$ at these rates.

Jan 27, 2015 Michael Fielder (University of Hannover)

Extensions of Transition Semigroups onto $L^p$-spaces and Applications to SDE's

Given both, a Markov semigroup $(P_t)_{t\geq0}$, a priori defined on the bounded and Borel measurable functions, $B_b(E)$, over a metric space $E$, with corresponding Markov kernels $(p_t(x,\cdot))_{t\geq0,x\in E}$, and a semigroup $(R_t)_{t\geq0}$ of linear Operators, defined on $B_b(E)$, of the shape $$R_t\varphi(x)=\int_E\varphi(y)r_t(x,dy)\quad x\in E,\varphi\in B_b(E), t\geq0,$$ with corresponding finite transition kernels $(r_t(x,\cdot))_{t\geq0,x\in E}$, satisfying $r_t(x,\Gamma)\leq e^{at}p_t(x,\Gamma)$ for all $t\geq0$, $x\in E$, $\Gamma\in\mathscr{B}(E)$ and some constant $a\in\mathbb{R}$, we ask under which conditions the semigroup $(R_t)_{t\geq0}$ can be extended from $B_b(E)\cap L^p(E,\mathscr E,\mu)$ onto $L^p(E,\mathscr E,\mu)$, $p\in[1,\infty)$, such that the extension is even strongly continuous. In the talk we give sufficient conditions, slightly generalizing existing results by Da Prato and Zabczyk, under which we can answer in the affirmative. To illustrate the results we will apply them to Lévy-semigroups and some stochastic differential equations.

Feb 10, 2015 Prof. Dr. Antonis Papapantoleon (TU Berlin)

An equilibrium model for commodity spot and forward prices

The aim of this project is to determine the forward price of a consumption commodity via the interaction of agents in the spot and forward commodity market. We consider a market model that consists of three agents: producers of the commodity, consumers and financial investors (sometimes also called speculators). Producers produce a fixed amount of the commodity at each time point, but can choose how much they offer in the spot market and store the rest for selling at the next time period. They also have a position in forward contracts in order to hedge the commodity price uncertainty. Consumers are setting the spot price of the commodity at each time point by their demand. Finally, investors are investing in the financial markets and, in order to diversify their portfolios, also in the forward commodity market. The equilibrium prices for the commodity are the ones that clear out the spot and forward markets. We assume that producers and investors are utility maximizers and have exponential preferences, while the consumers' demand function is linear. Moreover, the exogenously priced financial market and the demand function are driven by Lévy processes. We solve the maximization problem for each agent and prove the existence of an equilibrium. This setting allows to derive explicit solutions for the equilibrium prices and to analyze the dependence of prices on the model parameters and the agent's risk aversion. This is joint work with Michail Anthropelos and Michael Kupper.

Summer Semester 2014 (back to Contents)

Sept. 23

Dr. Mikhail Zhitlukhin (Steklov Mathematical Institute Moskow)

Testing complex hypotheses about the drift of a Brownian motion

We consider two problems of sequential testing of the hypotheses whether the drift of a Brownian motion is positive or negative. The sequential setting assumes that a decision rule consists of a stopping time when the observation is terminated and a decision function choosing the hypothesis to accept. An observer is penalized for the duration of observation and a wrong decision. The goal is to find the optimal trade-off between the duration and the chance of a wrong decision.

The first criterion of optimality we consider was proposed by H. Chernoff and assumes the unknown drift is a priori normally distributed and one needs to minimize the sum of the observation time and the penalty for a wrong decision proportional to the absolute value of the drift. The second criterion, proposed by J. Kiefer and L. Weiss, minimizes the maximum mean observation time for all possible values of the drift parameter among criteria that have fixed probability of a wrong decision.

In the both problems, asymptotically optimal decision rules were obtained long ago. The new result presented in this talk concerns exact optimal decision rules. We show that they can be found from solving some optimal stopping problems and obtain them in an explicit form.

Sept. 16

Dr. Alexey Muravlev (Steklov Mathematical Institute Moskow)

On the methods of sequential hypothesis testing for a fractional Brownian motion in the Bayesian setting

The problem of sequential hypothesis testing lies at the origins of sequential analysis and was considered by many authors. Among the most well-known setting are Wald's criteria, Bayesian testing of two simple hypothesis, H. Chernoff's (Bayesian) problem and Kiefer-Weiss criteria. The greatest number of results refers to the case when we observe i.i.d. sequence of random variables or a Brownian motion with unknown drift.

Recently, U.Çetin, A. Novikov and A. Shiryaev considered the Bayesian problem of sequential estimation of the drift of a fractional Brownian motion (that is not Markovian). In this talk we consider the sequential hypothesis testing problems for the same process, and present some general methods how to deal with them. In particular:

1) How to reduce the sequential testing problem for fractional Brownian motion to the similar problem for a standard one (with Hurst index equal to 1/2).

2) How to reduce it to the standard optimal stopping problem of a special form.

3) How to describe continuation and stopping regions.

August 06

Richard Bernstein (Otto von Guericke Universität Magdeburg)

Der Satz von Girsanow

Der Satz von Girsanow ist von großer Bedeutung für die Finanzmathematik. Zu seiner Formulierung ist es notwenig, einige Begriffe wie "Adaptiertheit", "Brownsche Bewegung" oder "quadratische Variation" zu erklären. Auch soll eine Möglichkeit zur Konstruktion stochastischer Integrale über elementare Prozesse vorgestellt werden. Der Schlüssel zum Beweis ist Levys Charakterisierung der Brownschen Bewegung.

August 04

Henry Wegener (Leibniz Universität Hannover)

Resultate über fast überall Konvergenz (u.a. Ergodentheorie)

July 16

Dr. Johannes Rauh (Max-Planck-Institut Leipzig)

Polytopes from Subgraph Statistics

For two graphs \(H\) and \(G\) let \(s(H,G)\) be the subgraph density of \(H\) in \(G\). For a family of graphs \(H_1,\dots,H_d\) we obtain a vector of subgraph densities for each graph \(G\). For fixed \(N\) I want to study the convex hull of all these vectors for all graphs on \(N\) nodes. This polytope plays an important role in the study of random graphs; for example, it equals the convex support of the corresponding exponential random graph model. In my talk I will present relations to the theory of graph limits (graphons), and I will discuss the example of the k-star polytope.

July 8

Ph.D. Stephen Tate (Ruhr-Universität Bochum)

The Combinatorics of Mayer's Theory of Cluster and Virial Expansions

In this talk, I introduce the cluster and virial expansions of Mayer in Statistical Mechanics. These are interpreted as weighted generating functions of connected and two-connected graphs, respectively, and arise as the functions describing the relationship between pressure and activity, respectively, density. In two particular models of Statistical Mechanics, one may obtain these two expansions independently of the generating function interpretation. When one compares the two methods of derivation, one finds some interesting combinatorial identities. This talk indicates an explanation of these identities, giving how the cancellations occur in the weighted generating functions.

June 17

Prof. Vladimir Panov (Higher School of Economics Moscow)

Maximal deviation distribution for projection estimates of Levy densities

This talk is devoted to projection estimates for Levy densities in both high and low frequency setup. After a short introduction to the theory of statistical inference for Levy processes, I will present new results concerning the convergence rates of the projection estimates. These results are based on some fresh ideas, which allow to reformulate the problem in terms of Gaussian processes.

June 3

Prof. Dr. Christoph Thäle (Ruhr-Universität Bochum)

Process-level Poisson approximation

I present a quantitative version of a process-level Poisson limit theorem, where distance is measured by an optimal transportation metric. To illustrate the flexibility of our result, we show applications to a number of limit theorems for classical U-statistics and also apply it to functionals arising in the context of stochastic geometry.

May 20

Prof. Dr. Christof Külske (Ruhr-Universität Bochum)

On nonergodic stochastic lattice systems with unique invariant measure

Is there an interacting particle system (IPS) on the 3-dimensional lattice which has a unique time-invariant measure but which does not attract all starting measures in the long time limit?

We provide a construction of such a system showing that the answer to this classical question is yes. The construction draws on properties of an underlying equilibrium measure of continuous spins showing phase transitions and is based on a heuristics of Maes and Shlosman. We also present results on similarities and differences between the non-ergodic continuous-time IPS and related versions of weak probabilistic cellular automata (PCA) which have simultaneous updates in discrete time with quasilocal rules.

May 13

M. Sc. Stephan Gufler (Goethe Universität Frankfurt am Main)

Lookdown representation for tree-valued Fleming-Viot processes

We construct the tree-valued Fleming-Viot process from the Lookdown model. The tree-valued Fleming-Viot process was introduced by Greven, Pfaffelhuber, and Winter [Probab. Theory Related Fields 2013]. We generalize this process to include the cases with multiple and simultaneous multiple reproduction events. In case that the associated coalescent comes down from infinity, the construction from the Lookdown model allows to read off a process with values in the space of measure-preserving isometry classes of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prohorov metric. This process has a.s. càdlàg paths with additional jumps at the times when old families become extinct.

May 6

Prof. Dr. Alexander Gushchin (Steklov Mathematical Institute Moskow)

Trajectorial counterparts of Doob's maximal inequalities and their applications

We present trajectorial counterparts of Doob's maximal inequalities for supermartingales and submartingales. From these inequalties, we obtain trajectorial counterparts of Doob's \(L^p\) and \(L \log L\) inequalities obtained by Acciaio, Beiglböck, Penkner, Schachermayer and Temme (2013). Another application of our inequalities is connected with the Azéma--Yor solution of the Skorokhod embedding problem. We show that this solution could be guessed from our inequalities. Moreover, we study the problem considered by Hobson (1998): to find an upper bound with respect to stochastic ordering for the maximum of a martingale with given initial and terminal distributions.

April 29

Dr. Paolo Di Tella (Humboldt-Universität zu Berlin)

The Chaotic Representation Property of Certain Families of Martingales

We investigate the chaotic representation property of certain families of square integrable martingales, which we call compensated-covariation stable families. First, we introduce the multiple integrals with respect to elements of a compensated-covariation stable family of martingales. The main result is that any compensated-covariation stable family of martingales which satisfies some further conditions possesses the chaotic representation property. As first examples, we consider continuous Gaussian families of martingales and independent families of compensated Poisson processes. Then we apply the result to the case of Lévy processes. We shall construct families of martingales relative to a Lévy filtration which possess the chaotic representation property. We give several examples including Teugels martingales.

April 22

Ph.D. Sandra Kliem (Universität Duisburg-Essen)

Modeling evolving phylogenies by means of marked metric measure spaces

In this talk, a model for evolving phylogenies, incorporating branching, mutation and competition is introduced. The state-space consists of marked tree-like metric measure (mmm)-spaces. The model arises as the limit of approximating finite population models with rates dependent on the individuals' traits and their genealogical distances.

The main focus of the talk will be on presenting the notion of mmm-spaces and to highlight their advantages in the given context. In particular, necessary and sufficient conditions for relative compactness of sets in mmm-spaces are explained. The route to verify these conditions to conclude the tightness of the approximating models from above is given.

A similar approximating model and its limit is treated in [Méléard and Tran, 2012] in the framework of nonlinear historical superprocess approximations. In the framework of mmm-spaces, work of [Depperschmidt, Greven, Pfaffelhuber and Winter, 2012--2013] introduces and studies tree-valued Fleming-Viot dynamics. During this talk, new ideas and challenges that arise from working with mmm-spaces in the context of evolving phylogenies are put into context of the above.

(This is joint work with Anita Winter.)


April 15

Dr. Wolfgang Löhr (Universität Duisburg-Essen)

Invariance principle for variable speed random walks on trees

Everyone knows that simple random walks on \(\mathbb{Z}\) converge, suitably rescaled, in path-space to Brownian motion on \(\mathbb{R}\). This has been generalised by Stone in 1963 to processes in "natural scale" on \(\mathbb{R}\), which are characterised by a speed measure \(\nu\). We call these processes speed-\(\nu\) motions and generalise Stone's observation that they depend continuously on \(\nu\) to the case of trees. More precisely, we show that whenever a locally compact \(\mathbb{R}\)-tree \(T_n\), together with a locally finite speed-measure \(\nu_n\) on \(T_n\), converges in the Gromov-vague topology to a limiting \(\mathbb{R}\)-tree \(T\) with measure \(\nu\), the speed-\(\nu_n\) motions converge to the speed-\(\nu\) motion under a mild lower mass-bound assumption. (joint work with Siva Athreya and Anita Winter)

Winter Semester 2013/14 (back to Contents)

Feb 19

Lisa Beck (Augsburg University)

Regularization by noise for the stochastic transport equation

We discuss several aspects of weak (\(L^\infty\)-) solutions to the stochastic transport equation \[ du = b \cdot Du dt + \sigma Du \circ dW_t \] with Stratonovich multiplicative noise. Here, \(b\) is a time dependent vector field (the drift), \(u\) is the unknown, \(\sigma\) a real number, and \(W_t\) a Brownian motion.

For the deterministic equation (\(\sigma = 0\)) it is well-known that multiple solutions may exist and that solutions may blow up from smooth initial data in finite time provided that the drift is not regular (basically less than Lipschitz in space). For the stochastic equation (\(\sigma \neq 0\)) instead, a suitable integrability condition on the drift is sufficient to prevent the formation of non-uniqueness and of singularities.

After a short review of some techniques for the deterministic equation, I will present in my talk two recent results for the stochastic equation, which concern the phenomenon of regularization by noise and which are part of a joint project with F. Flandoli, M. Gubinelli and M. Maurelli. More precisely, assuming merely the aforementioned integrability condition, we obtain in a first step conservation of Sobolev regularity of initial data by means of PDE techniques (and not via stochastic characteristics). In a second step this regularity is used to prove path-by-path uniqueness of solutions via a duality argument.

Warning: Non-standard time (Wednesday, 14:15) but usual venue (WSC-S-U-3.03).

Feb 11

Fabian Gerle (Ruhr-Universität Bochum)

Applications of Lindeberg's method in the theory of random matrices

Lindeberg's method for the proof of the central limit theorem is easily generalized and can be applied to elegantly proof limiting theorems for the empirical spectral distribution of certain types of random matrices.

In my talk about the subjects of my master thesis I will first give a sketch of Lindeberg's original proof of the central limit theorem from 1912. Following a work by Sourav Chatterjee we will see, how this argument can be extended to more general functions and other sequences of random variables and how this generalized Lindeberg principle can be applied in the theory of random matrices.

Using this generalized Lindeberg principle we will derive Lindebergesque conditions for the convergence of the empirical spectral distribution of Wigner and Wishart matrices in different setups.

Feb 4

Albert N. Shiryaev (Steklov Institute of Mathematics)

The concept of randomness: from von Mises' collective to Kolmogorov's complexity

Jan 28

Mareike Esser (Universität Bielefeld)

Single-crossover recombination and ancestral recombination trees

We consider the dynamics of the genetic composition of a population evolving under the evolutionary force of recombination. Recombination, which can be briefly described via an exchange of genetic material from maternal and paternal gene-sequences during sexual reproduction, gives rise to dependence problems and creates large state spaces. In continuous time the resulting system of non-linear differential equations could be solved in closed form. Inspired by this, we now investigate the stochastic counterpart. To this end, we trace back the ancestry of a single individual that has evolved in accordance to the Moran model with recombination. For \(N \to \infty\) the ancestry of a single individual is represented by a random binary tree which we call ancestral recombination tree. The probability of such trees can be derived via an inclusion-exclusion principle that leads to a decomposition of the trees into subtrees.

Jan 21

Alexander Kulikov (Moscow Institute of Physics and Technology)

Hedging volumetric risks using put options in commodity markets

Abstract: [pdf]

Jan 14

Loren Coquille (Universität Bonn)

Gaussian free field with disordered pinning on \(\mathbb{Z}^d\), \(d \geq 2\)

Dec 17

Angelika Rohde (Ruhr-Universität Bochum)

Accuracy of empirical projections of high-dimensional Gaussian matrices

Let \(X=C+\mathrm{E}\) with a deterministic matrix \(C\in\mathbb{R}^{M\times M}\) and \(\mathrm{E}\) some centered Gaussian \(M\times M\)-matrix whose entries are independent with variance \(\sigma^2\). In the present work, the accuracy of reduced-rank projections of \(X\) is studied. Non-asymptotic upper and lower bounds are derived, and favorable and unfavorable prototypes of matrices \(C\) in terms of the accuracy of approximation are characterized. The approach does not involve analytic perturbation theory of linear operators and allows for multiplicities in the singular value spectrum. Our main result is some general non-asymptotic upper bound on the accuracy of approximation which involves explicitly the singular values of \(C\), and which is shown to be sharp in various regimes of \(C\). The results are accompanied by lower bounds under diverse assumptions. Consequences on statistical estimation problems, in particular in the recent area of low-rank matrix recovery, are discussed.

Dec 10

Mikhail Urusov (Universität Duisburg-Essen)

On the processes that can be embedded in a geometric Brownian motion

A process is equivalent to a time-change of a geometric Brownian motion if and only if it is a nonnegative supermartingale.

Dec 3

Volker Krätschmer (Universität Duisburg-Essen)

Quasi-Hadamard differentiability of general risk functionals and its application to statistical inference

Nov 26

No seminar.

Nov 19

Patricia Alonso-Ruiz (Universität Ulm)

Dirichlet form and Laplacian on fractal quantum graphs via resistance forms

Resistance forms have turn out to be very useful in the study of analysis on fractals from an intrinsic point of view. Under a suitable choice of a measure, they provide a Dirichlet form on the space and thus a Laplacian on it.

In this talk we introduce fractal quantum graphs, which generalize classical quantum graphs and also include fractal sets like Hanoi attractors. We present in detail the construction of a Dirichlet form on any Hanoi attractor by means of resistance forms. We will also discuss the spectral asymptotics of the Laplacian associated to this Dirichlet form and if time allows it, we will point out some questions concerning the Einstein relation.

This is joint work with D.Kelleher and A.Teplyaev from the University of Connecticut.

Nov 8

Jordan Stoyanov (Newcastle University)

Moment (in)determinacy of probability distributions: recent progress

The discussion will be on heavy tailed probability distributions, one- or multi-dimensional, with finite all moments. One of the questions in the classical moment problem is about the uniqueness. Either such a distribution is uniquely determined by its moments (M-determinate), or it is nonunique (M-indeterminate). Our goal is to describe the current state of art in this area. A brief summary of known and widely used classical criteria (Cramer, Hausdorff, Carleman, Krein, ...) will go in parallel with the following very recent developments:

  1. Stieltjes classes for M-indeterminate distributions. Index of dissimilarity.
  2. New Hardy’s criterion for uniqueness. Multidimensional moment problem.
  3. Nonlinear transformations of random data and their moment (in)determinacy.

There will be several new results, hints for their proof, examples and counterexamples, and also open questions and conjectures.

Warning: Non-standard time (Friday, 14:15) but usual venue (WSC-S-U-3.03).

Nov 5

Johannes Ruf (University of Oxford)

Supermartingales as Radon-Nikodym densities, Novikov's and Kazamaki's criteria, and the distribution of explosion times

I will show how certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. I will review existence and (non-)uniqueness results for such measures. In the second part of my talk, I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. I will focus on the situation when jumps are present. If time permits, I will then illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation.

This presentation is based on joint papers with Nicolas Perkowski, Martin Larsson, and Ioannis Karatzas.

Oct 29

Anton Klimovsky (Universität Duisburg-Essen)

From Exchangeability to Ultrametricity (II)

I will review some representation results for exchangeable random arrays. As an application, I will sketch how these can be used to derive ultrametricity.

Oct 22

Vladimir Panov (Universität Duisburg-Essen)

Exponential functionals of Lévy processes

The talk is devoted to the exponential functional \(\int_0^t e^{-\xi_s}ds\) of a Lévy process \(\xi_s\). This object is related to the generalized Ornstein-Uhlenbeck process and naturally arises in a broad class of applications. After the introduction to this research area, we will focus on the statistical aspects of the topic. In particular, we will present a new approach, which allows to infer on the characteristics of the Lévy process from the distribution of its exponential functional.

Sep 18

Eyal Neuman (Technion)

Pathwise Uniqueness of the Stochastic Heat Equations with Spatially Inhomogeneous White Noise

We study the solutions of the stochastic heat equation with spatially inhomogeneous white noise. This equation has the form \[ (*) \quad \frac{\partial}{\partial t} u(t,x) = \frac{1}{2}\Delta u(t,x) + \sigma(t,x,u(t,x))\dot{W} , \ \ t \geq 0, \ \ x\in \mathbb{R}. \] Here \( \Delta \) denotes the Laplacian and \( \sigma(t,x,u): \mathbb{R}_{+}\times\mathbb{R}^2\to\mathbb{R} \) is a continuous function with at most a linear growth in the \( u \) variable. We assume that the noise \( \dot{W} \) is a spatially inhomogeneous white noise on \( \mathbb{R}_{+}\times\mathbb{R} \). When \( \sigma(t,x,u)=\sqrt{u} \) such equations arise as scaling limits of critical branching particle systems which are known as catalytic super Brownian motion. In particular we prove pathwise uniqueness for solutions of \( (*) \) if \( \sigma \) is Hölder continuous of index \( \gamma>1-\frac{\eta}{2(\eta+1)} \) in \( u \). Here \( \eta\in(0,1) \) is a constant that defines the spatial regularity of the noise.

An extensive introduction to stochastic partial differential equations will be given in this talk.

Sep 17

Tal Orenshtein (Technische Universität München)

0-1 law for directional transience of one-dimensional excited random walks

We will discuss the following theorem which solves a problem posed by Kosygina and Zerner. For a one dimensional excited random walk in a stationary ergodic and elliptic cookie environment, the probability of being transient to the right (left) is either zero or one. (Joint work with Gideon Amir and Noam Berger.)

Summer Semester 2013 (back to Contents)

  • 16.07.2013, Yuri Kabanov (Université de Franche-Comté):
    On Essential Supremum and Essential Maximum with Respect to Random Partial Orders with Applications to Hedging of Contingent Claims under Transaction Costs
    Abstract: pdf
  • 09.07.2013, Siva Athreya (Indian Statistical Institute, Bangalore):
    One-dimensional Voter Model Interface Revisited
    Abstract: We consider the voter model on \(\mathbb{Z}\), starting with all 1's to the left of the origin and all 0's to the right of the origin. It is known that if the associated random walk kernel p has zero mean and a finite r-th moment for any r>3, then the evolution of the boundaries of the interface region between 1's and 0's converge in distribution to a standard Brownian motion \((B_t)_{t>0}\) under diffusive scaling of space and time. This convergence fails when p has an infinite r-th moment for any r0.
  • 02.07.2013, Anton Klimovsky (Universität Duisburg-Essen):
    From exchangeability to ultrametricity (I)
    Abstract: I will review representation results on exchangeable random arrays. As an application, I will try to sketch how these can be used to derive ultrametricity.
  • 25.06.2013, Wolfgang Löhr (Universität Duisburg-Essen):
    Tightness of Reversible Random Walks on a converging sequence of Discrete Measure Trees
    Abstract: For a (discrete) tree with locally finite measure \(\nu\), I define the \(\nu\)-reversible standard random walk on the tree and show that, whenever the underlying measure trees converge in an appropriate sense, the corresponding random walks are tight. This is the first step in proving that they converge to the Brownian motion on the (continuous) limiting \(\mathbb{R}\)-tree.
  • 18.06.2013, no talk because of Symposium Stochastic Analysis.
  • 11.06.2013, Sabine Jansen (Ruhr-Universität Bochum):
    Duality of Markov processes with respect to a function
    Abstract: There are several notions of duality for Markov processes. Duality with respect to a measure has been introduced in the context of potential theory, and there is by now a rich theory for this notion. Duality with respect to a function is a powerful tool in interacting particle systems and population genetics, but a general theory for this notion is still missing. This talk will present some first steps towards a systematic investigation of duality with respect to a function, with a focus on the functional analytic structures that are involved. The talk is based on joint work with Noemi Kurt (TU Berlin).
  • 07.06.2013, Fri, 14.15h: Zakhar Kabluchko (Universität Ulm):
    Nullstellenverteilung der Zufallspolynome
    Abstract: In diesem Vortrag werden wir die Nullstellen der Polynome mit zufälligen Koeffizienten betrachten. Unser Interesse gilt der Verteilung der Nullstellen dieser Polynome, wenn der Grad des Polynoms gegen \(\infty\) geht. Wir werden eine sehr allgemeine Methode zur Bestimmung der Nullstellenverteilung vorstellen und diese auf einige Beispiele von Zufallspolynomen anwenden. So werden wir etwa zeigen, dass die Nullstellen der Weyl-Polynome \[ P_n (z) = \sum_{k=0}^n \xi_k \frac{z^k}{\sqrt{k!}} \] für \(n\to\infty\) gleichverteilt auf dem Einheitskreis sind.
  • 04.06.2013, no talk; instead there is a talk on Friday, 14.15h.
  • 28.05.2013, Yannick Hoga (Universität Duisburg-Essen):
    Change point test for tail index for dependent data
    Abstract: The tail index of a distribution plays a central role in extreme value theory, because it determines the asymptotic distribution of e.g. the sample maximum. It is often used as a measure of tail thickness. However, often in time series from finance the assumption of tail index constancy over time seems unwarranted. This motivated Kim and Lee (Metrika 74:297-311, 2011) to propose a change point test for the tail index of stationary \(\beta\)-mixing random variables. In my master's thesis I was able to show that their test, which is based on comparing Hill's estimator from different time periods, can be successfully extended to \(\alpha\)-mixing random variables and even \(\mathcal{L}_{2}\)-E-NED sequences of random variables. The limiting distibution of the test statistic remains the same as in Kim an Lee. The \(\mathcal{L}_{2}\)-E-NED concept is due to Hill (Econometric Theory 26:1398-1436, 2010)and includes a wide range of processes, including among others \(\mathcal{L}_{2}\)-NED, ARFIMA, FIGARCH, explosive GARCH, nonlinear ARMA-GARCH, bilinear processes, and nonlinear distributed lags.
    A simulation study is implemented to investigate the empirical size and power of the test in finite samples. As an application, the data set of daily DJIA returns is examined for a possible change point in the tail index.
  • 14.05.2013, Sandra Kliem (Universität Duisburg-Essen):
    Past dependence in a population model: an article by Sylvie Méléard and Viet Chi Tran
    Abstract: This talk presents an overview of the model and proofs used in the article "Nonlinear historical superprocess approximations for population models with past dependence" by Sylvie Méléard and Viet Chi Tran, published 2012 in Electron. J. Probab.
    To quote their abstract: "We are interested in the evolving genealogy of a birth and death process with trait structure and ecological interactions. Traits are hereditarily transmitted from a parent to its offspring unless a mutation occurs. The dynamics may depend on the trait of the ancestors and on its past and allows interactions between individuals through their lineages."
    I shall introduce the model and give the main ideas of the proofs. This involves techniques such as nonlinear historical superprocesses, martingale problems and a tightness proof for a measure-valued càdlàg process.
  • 07.05.2013, Anita Winter (Universität Duisburg-Essen):
    Aldous's move on cladograms in the diffusion limit
    Abstract: A \(n\)-phylogenetic tree is a semi-labeled, unrooted and binary tree with \(n\) leaves labeled \(\{1,2,...,n\}\) and with \(n-2\) unlabeled internal leaves and positive edge lengths representing the time spans between common ancestors. In biological systematics \(n\)-phylogenetic trees are used to represent the revolutionary relationship between \(n\) species. If one does focus only on the kingship (that is taking all edge length of unit length), a more precise term is cladogram.
    Aldous constructed a Markov chain on cladograms and gave bounds on their mixing time. On the other hand, Aldous also gave a notion of convergence of cladograms which shows that the uniform cladogram with \(N\) leaves and edge length re-scaled by a factor of \(\frac{1}{\sqrt{N}}\) converges to the so-called Brownian continuum random tree (CRT) which is the tree ``below'' a standard Brownian excursion and can be thought of as the ``uniform'' tree. These two results suggest that if we re-scale edge lengths by a factor of \(\frac{1}{\sqrt{N}}\) and speeding up time by a factor of \(N^{\frac{3}{2}}\) the Aldous move on cladograms converges in some sense to a continuous tree-valued diffusion.
    The main emphasis of the talk is to give precise statements towards that direction. (joint work with Leonid Mytnik)
  • 30.04.2013, no talk (because of the Einweihungsfeier).
  • 23.04.2013, Wolfgang Löhr (Universität Duisburg-Essen):
    Convergence of Subtree Pruning Processes
    Abstract: In 1998, Aldous and Pitman constructed a tree-valued Markov chain by pruning off more and more subtrees above randomly chosen edges of a Galton-Watson tree. Recently, Abraham, Delmas and He considered a similar process, where the cut-points are chosen in a degree-dependent way. In the same spirit, prunings of continuous trees, such as the Brownian CRT or Lévy trees, were studied by various authors. No precise link has been given between the prunings of discrete and continuous trees.
    We provide a unified framework by regarding them as instances of the same Feller-continuous Markov process with different initial conditions. The state space, which we introduce, consists of measure R-trees with an additional pruning measure, which is not required to be locally finite. We obtain Skorohod convergence of some of the previously studied pruning processes. (joint work with Guillaume Voisin and Anita Winter)
  • 16.04.2013, no talk.
  • 09.04.2013, no talk.

Winter Semester 2012/13 (back to Contents)

  • 06.02.2013, Wed, 16.15h in S-U-4.02: Moritz Kaßmann (Universität Bielefeld):
    Coercive nonlocal operators
    Abstract: We introduce a class of integro-differential operators which share several properties with the Laplace operator. We shortly discuss the corresponding jump processes. The emphasis of the talk is on regularity results and functional inequalities for linear and nonlinear equations driven by these nonlocal operators. The talk is based on joint works with Dyda, Felsinger and Rang.
  • 05.02.2013, no talk; instead there is a talk on Wednesday (together with the Analysis).
  • 29.01.2013, Patric Glöde (Universität Erlangen):
    Dynamics of Genealogical Trees for Autocatalytic Branching Processes
    Abstract: My talk will feature the dynamics of genealogical trees for autocatalytic branching processes. In such populations each individual dies at a rate depending on the total population size and, upon its death, produces a random number of offspring. I will consider finite as well as infinite populations. Formally, processes take values in the space of ultrametric measure spaces. The dynamics are characterised by means of martingale problems. Key issues are to prove well-posedness for the martingale problems and to find invariance principles linking finite and infinite populations. In fact, infinite populations arise as scaling limits of finite populations in the sense of weak convergence on path space with respect to the (polar) Gromov-weak topology. I will show that there is a close relationship between the genealogies of infinite autocatalytic branching processes and the Fleming-Viot process. I will also mention an abstract uniqueness result for martingale problems of skew product form which is of importance for my own processes but also applies to more general situations.
  • 22.01.2013, Martina Baar (Universität Bonn):
    Stochastic individual-based models of adaptive dynamics
    Abstract: In this talk, we study the limit behavior of a model for the Darwinian evolution in an asexual population characterized by a natural birth rate, a death rate due to age or competition and a probability of mutation at each birth event. The model is a stochastic, generic, individual-based model and belongs to the models of adaptive dynamics. We focus on the combination of the three main limits of the theory, large population size, rare mutations and small mutation effects, on the long-term evolution of the population. More precisely, we consider the following tree limit behaviors: first, only the limit of large population, second, the limit of large population and rare mutations, third the limit of large population and rare mutation and afterwards the one of small mutation. Finally we investigate the limit of large population, rare and small mutation simultaneously in one single step. We obtain for this limit that on a specific time scale coexistence of two traits cannot occur in the population process with monomorphic initial condition. In other words, the population stays essentially single modal centered around a trait, that evolves continuously.
  • 15.01.2013, Alexander Schnurr (Universität Dortmund):
    Quo vadis symbol (et ubi venis)?
    Abstract: Nach einer kurzen historischen Einführung führen wir das stochastische Symbol für die Klasse der homogenen Diffusionen mit Sprüngen (im Sinn von Jacod/Shiryaev) ein. Dabei handelt es sich um eine Verallgemeinerung des charakteristischen Exponenten eines Lévy-Prozesses. Durch das Symbol lassen sich 8 Indizes definieren, die den Blumenthal-Getoor Index verallgemeinern. Diese Indizes verwenden wir um Wachstums- und Hölderbedingungen der Pfade herzuleiten. Als Beispiele betrachten wir Lévy-getriebene stochastische Differentialgleichungen und den COGARCH Prozess, der dazu dient Prozesses in der Finanzmathematik zu modellieren. In der nahen Zukunft werden wir die technischen Hauptresultate dazu verwenden, weitere Feineigenschaften zu untersuchen.
  • 08.01.2013, no talk.
  • 18.12.2012, Olivier Hénard (Goethe Universität Frankfurt):
    Examples of intertwining relationships
    Abstract: When is a function of a Markov process Markov again? We will discuss the answer given by Rogers and Pitman (1981, AoP) and exemplify it by revisiting classical path decompositions as well as giving new ones within the framework of stochastic population models.
  • 11.12.2012, Anton Klimovsky (Universiteit Leiden):
    Complex Random Energy Model: Zeros and Fluctuations.
  • 04.12.2012, Alexa Manger (Universität Duisburg-Essen):
    Modeling evolving phylogenies in the context of phylodynamic patterns
    Abstract: We construct the evolution of phylogenies with mutation and selection.
  • 27.11.2012, Martin Hutzenthaler (Goethe Universität Frankfurt):
    Branching diffusions and genealogies in random environment
    Abstract: Individuals in a branching process in random environment (BPRE) branch independently of each other and the offspring distribution changes randomly over time. The variation in the offspring distribution models fluctuations in environmental conditions. The diffusion approximation of BPREs, which we denote as branching diffusion in random environment (BDRE), turns out to have strong analytical properties. We will explain important analogies with and differences to Feller's branching diffusion. In addition we focus on genealogies in random environment. The relative frequency of one BDRE within the sum of two BDREs in the same random environment turns out to be a Wright-Fisher diffusion with random selection. The genealogy of this well-known diffusion is unknown so far and we will present first ideas.
  • 20.11.2012, Jan H. Wirfs (Universität Ulm):
    Estimating the Ornstein-Uhlenbeck Stochastic Volatility Model using Characteristic Functions
    Abstract: Continuous-time stochastic volatility models are becoming more and more important in finance because of their adaptability to the most common effects in financial time series. In the talk, we discuss a stochastic volatility model where the volatility process is given by an Ornstein-Uhlenbeck process. Although the model seems to be feasible, the estimation is difficult, because of the inability to compute closed-form likelihood functions. Therefore, we discuss an estimation approach based on characteristic functions. Finally, we give some results of application.
  • 13.11.2012, Mikhail Urusov (Universität Duisburg-Essen):
    On the martingale property of exponential local martingales
    Abstract: The stochastic exponential \(Z\) of a continuous local martingale \(M\) is itself a continuous local martingale. We give a necessary and sufficient condition for the process \(Z\) to be a true martingale and for the process \(Z\) to be a uniformly integrable martingale in the case where \(M_t=\int_0^t b(Y_u)\,dW_u\), the process \(Y\) is a one-dimensional diffusion, and the process \(W\) is a Brownian motion. These conditions are deterministic and expressed only in terms of the function \(b\) and the drift and diffusion coefficients of \(Y\). This is a joint work with Aleksandar Mijatovi\'c.
  • 06.11.2012, Wolfgang Löhr (Universität Duisburg-Essen):
    Convergence of Random Walks on Discrete Trees to Brownian Motion on \(\mathbb{R}\)-Trees
    Abstract: Brownian motion can be defined on locally compact \(\mathbb{R}\)-trees via Dirichlet forms. We consider simple random walks on approximating discrete trees and their Dirichlet forms. Under some conditions, a generalised form of Mosco convergence of the forms can be used to show f.d.d. convergence of the associated processes, i.e. random walks on discrete trees to Brownian motion on \(\mathbb{R}\)-trees. Here, I give a short overview over relevant aspects of \(\mathbb{R}\)-trees, Dirichlet forms and Mosco convergence and show in a simplified setting how Mosco convergence can be proven. (Joint work with Siva Athreya and Anita Winter)
  • 30.10.2012, Anita Winter (Universität Duisburg-Essen):
    Tree valued spatial \(\Lambda\)-Cannings and \(\Lambda\)-Fleming-Viot dynamics
    Abstract: We study the evolution of genealogies for interacting spatially structured \(\Lambda\)-Cannings models which are also known as generalized Fleming-Viot processes. These are the limit processes of individual-based population models where individuals carry a type, and are replaced by descendants of possibly very sizable offspring. The spatial interaction is due to migration through geographic space. We show that the dual to these tree-valued spatial \(\Lambda\)-Cannings dynamics are tree-valued spatial \(\Lambda\)-coalescents, and conclude from here the convergence of the fixed time genealogies to the genealogy of an infinitely old population as time tends to infinity. Depending on the strength of migration the latter consists either of a single or of multiple families. We then study the populations on large tori in \(\mathbb{Z}^d\) with d greater than 2. Depending on the rescaling we find global features which are universal for all \(\Lambda\)-Cannings dynamics and local features which heavily depend on the measure \(\Lambda\).
  • 23.10.2012, Johannes Fiedler (Universität Duisburg-Essen):
    Occupation time large deviations for branching Brownian motion and super-Brownian motion in \(\mathbb{R}^3\)
    Abstract: We derive a large deviation principle (LDP) for the occupation time functional. In 1993, Ian Iscoe and Tzong-Yow Lee already proved such an LDP for test functions whose Lebesgue integral over the space \(\mathbb{R}^3\) is different from zero. However, if this integral is equal to zero, the LDP does not yield much information. Therefore, Jean-Dominique Deuschel and Jay Rosen dealt with this problem in 1997. They found out that the occupation time functional has to be multiplied by the fourth root of the time parameter in order to get a reasonable statement about the long-time behaviour. This quantity fulfills an LDP which depends on the structure of the test function.
  • 16.10.2012, no talk.

Summer Semester 2012 (back to Contents)

  • 17.07.2012, no talk.
  • 10.07.2012, Denis Belomestny (Universität Duisburg-Essen): Solving optimal stopping problems by empirical optimization and penalization.
  • 03.07.2012, Thomas Rippl (Universität Göttingen): Compact Support Property for Stochastic Heat Equation with Colored Noise
    Abstract: The stochastic heat equation is a stochastic partial differential equation arising in physics and biology, where the deterministic heat equation is influenced by some random effects. We will especially focus on the density-dependent branching case (biologically speaking)
    \( \partial_t u(t,x) = \frac{\Delta}{2} u(t,x) + u^\gamma (t,x) \dot{W}(t,x), \quad \hbox{for }t \geq 0, x \in R. \) Here \(W\) is a Gaussian noise, white in time and colored in space and \(\gamma \in (1/2,1)\) and \(u\) is the non-negative solution of this equation given initial data \(u(0,\cdot) = u_0(\cdot).\) Then we can show that given compact \(u_0\), the solution stays within a compact set almost surely and we can also describe the growth rate of the support. The proof technique was developed by Krylov in 1997 and omits the construction of the historical process.
  • 26.06.2012, no talk
  • 19.06.2012, Helmut Pitters (Universität Tübingen): A Correspondence between the Study of the Gene Tree and the Rook Theory
    Abstract: We consider a coalescent process with (values the equivalence relations of the natural numbers and) multiple collisions, restricted to the set \(\{1, \dots, n\} \times \{1, \dots, n\}\), where \(n>2\) is a fixed integer.
    In population genetics this process is used to model the genealogy of a sample of size n, drawn from a large population of haploid individuals. On the coalescent process we superimpose mutations. In the infinitely-many-sites model the chronologically ordered mutations, that each individual in the sample encounters, are often encoded in a rooted tree with n leaves, the so-called gene tree. We are interested in the distribution of the gene tree.
    Birkner and Blath gave a recursive expression for the gene tree distribution. We present a non-recursive expression for the gene tree distribution in terms of so-called boards, which are combinatorial objects that are studied in Rook Theory. Thus the question of the gene tree distribution is reduced mainly to the underlying combinatorial problem.
    As an application we obtain a combinatorial problem, that can be solved easily in the special case of the star-shaped coalescent, and yields an explicit formula for the gene tree distribution in this case.
  • 12.06.2012, no talk.
  • 05.06.2012, Wolfgang Löhr (Universität Duisburg-Essen): Convergence of metric measure trees with locally infinite measures.
    Abstract: Gromov-weak convergence induces a very useful topology on the space of metric measure spaces with finite measures. However, some natural measures on continuous trees, like the length measure, are typically locally infinite. We introduce and analyse a topology on the space of R-trees with two measures, one finite and one possibly locally infinite. This topology is useful to obtain convergence statements about tree-valued pruning processes. (Joint work with Guillaume Voisin and Anita Winter)
  • 29.05.2012, no talk.
  • 22.05.2012, Chiranjib Mukherjee (Universität Erlangen): Large deviations for Brownian intersection measure
    Abstract: We consider a number of independent Brownian motions in a bounded domain running until time t and look at their intersection set, i.e., points which are hit by all the motions before time t. This set supports a natural measure which counts the intensity of the intersections and is called the Brownian intersection measure. We derive the long time asymptotics of this measure in terms of a large deviation principle. The rate function is explicit and gives some direct meaning of the intersection measure as point-wise product of the occupation measures of individual Brownian motions, whose behavior is known by the celebrated Donsker-Varadhan theory (joint work with Wolfgang Koenig).
  • 15.05.2012, Uta Freiberg (Universität Siegen): Spectral dimension of V-variable fractal gaskets
    Abstract: The concept of V-variable fractals (developed by Barnsley, Hutchinson & Stenflo) allows describing new families of random fractals, which are intermediate between the notions of deterministic and of random fractals including random recursive as well as homogeneous random fractals. The parameter V describes the degree of "variability" of the realizations. Brownian motion and Laplacian can be constructed from the associated Dirichlet forms. The properties of these objects are modified by the degree of variability. We obtain the spectral dimension (i.e. the exponent of the leading term of the eigenvalue counting function of the Laplacian) by applying Kesten-Furstenberg techniques. If time allows, we sketch some results on Hausdorff and walk dimension.
  • 08.05.2012, Vladimir Osipov (Universität zu Köln): Clustering of p-closed sequences
    Abstract: The idea of p-closed sequences originated from the concept of Sieber-Richter pairs of periodic orbits appeared for the first time in quantum chaos theory. In the framework of the semiclassical approach the universal spectral correlations in the Hamiltonian systems with classical chaotic dynamics can be attributed to the systematic correlations between actions of periodic orbits which pass through approximately the same points of the phase space. We consider the problem in the simplest realization of chaotic dynamics on the Baker's map, and thus translate the idea of Sieber-Richter pairs onto symbolic sequences.
    Using the analogy with Sieber-Richter pairs we introduce a notion of p-close sequences. Roughly, two sequences are p-close if their local content (each subsequence of p close letters) is the same, while globally they are different, for instance two sequences with "glued" ends: [0010111] and [0011101] are 3-close to each other.
    The notion of p-closeness naturally leads to the notion of distance between sequences. As we demonstrate, this distance has ultrametric properties and thus all sequences of a given length can be distributed over clusters. In practical part of our work we study the distribution of cluster sizes in the limit of long sequences. The latter problem is equivalent to the combinatoric one of counting degeneracies in the length spectrum of the de Bruijn graphs, and has been solved by means of random matrix theory.
  • 01.05.2012, no talk.
  • 24.04.2012, Bence Mélykúti (Universität Freiburg): Searching for a diffusion process model for chemical reaction kinetics: the chemical Langevin equation and beyond
    Abstract: For introduction, I will present applications of stochastic processes in molecular biology. My objective is to examine the standard Ito stochastic differential equation (SDE) model for (bio)chemical reaction kinetics, the chemical Langevin equation (CLE). By the connection between the martingale problem and the existence of a weak solution for an SDE, one can formulate the CLE in alternative, weakly equivalent forms. I will explore what the minimum number of Brownian motions necessary for an equivalent formulation is and discuss the corresponding underlying geometrical structure. I will derive another formulation that speeds up numerical simulation. I will also show that in terms of first and second moments, the CLE appears to be the best Ito SDE model for reaction kinetics. I will highlight that in its original form, the variables of the CLE can become negative with positive probability. This leads to the open question whether there exists a simple and natural, nonnegativity-preserving, continuous-valued stochastic model for reaction kinetics.
  • 17.04.2012, Alexa Manger (Universität Duisburg-Essen): Introduction to Super-Brownian Motion
    Abstract: We will introduce the super-Brownian motion via its "discrete particle" counterpart, the branching Brownian motion. After rescaling we will get a martingale representation for the super-Brownian motion and deduce some properties.
  • 10.04.2012, no talk.

Winter Semester 2011/12 (back to Contents)

  • 01.02.2012, Wolfgang Löhr (Universität Duisburg-Essen): Aspects of Gromov-Weak Topology on the Space of Metric Measure Spaces (second part) Room V15 S04 D94 at 10:15
    Abstract: I introduce the algebra of polynomials and show that it is not dense but convergence determining. The second property is a direct consequence of a result due to Le Cam. This means that the embedding introduced in the first talk can be extended to a convex-linear, homeomorphic embedding of the space of probability measures on the space of mm-spaces into the space of distance matrix distributions.
    If time permits, I will also show the equivalence of Gromov-Prohorov and Gromov's Box-metric.
  • 31.01.2012, Volker Krätschmer (Universität Duisburg-Essen): Qualitative robustness of tail dependent statistical functionals
    Abstract: The main goal of the talk is to introduce a new notion of qualitative robustness that applies also to tail-dependent statistical functionals like L- and V-functionals, and that allows us to compare statistical functionals in regards to their degree of robustness. By means of new versions of the celebrated Hampel theorem, it will be shown that this degree of robustness can be characterized in terms of certain continuity properties of the statistical functionals. In applying the theoretical results, special attention will be payed to the class of distribution-invariant convex risk measures which has become a central concept as a building block in quantitative risk management. The talk is based on a joint work with Alexander Schied (University of Mannheim) and Henryk Zähle (University of Saarland).
  • 24.01.2012, Wolfgang Löhr (Universität Duisburg-Essen): Aspects of Gromov-Weak Topology on the Space of Metric Measure Spaces (first part)
    Abstract: I introduce the Gromov-Prohorov metric, Gromov weak topology and the homeomorphic (but not uniformly continuous!) embedding of the space of mm-spaces into the space of distance matrix distributions. I also discuss basic properties of the space, such as non-local-compactness, and simple examples.
  • 17.01.2012, Peter Seidel (Universität Erlangen): The spatial Moran model with mutation carrying the family structure.
    Abstract: In this talk we consider a finite, spatial and multi-type population model, the Moran model, in which individuals are located on different sites in a geographic space and having a genetic type. The evolution of these individuals is given by migration, mutation and resampling.
    Our aim is to incorporate the genealogy of the finite population by assigning to each individual beside its type and location its founding father. On the one hand we get a decomposition of the population into family clusters and on the other hand we prepare the construction of an historical model with ancestral lines and its dual.
  • 10.01.2012, Guillaume Voisin (Universität Duisburg-Essen): Non-selfsimilar fragmentation process by pruning a Lévy tree.
    Abstract: I consider a general Lévy tree. I construct a pruning procedure on the infinity nodes and on the skeleton of the Lévy tree such that the remaining subtrees are still Lévy trees. Then by cutting more and more the tree by the same procedure, we obtain a family of Lévy trees which are smaller and smaller, it is a fragmentation process. We can have some properties of this process but a complete characterization is still open.
  • 20.12.2011, Lisa Beck (Universität Bonn): Random perturbations of parabolic systems. Room V15S - V15 S03 C02 at 2pm
    Abstract: In the deterministic regularity theory for parabolic systems, there are both classes of systems, where all weak solutions are actually more regular, and examples of systems which admit solutions that develop singularities in finite time. In this talk we investigate the regularity of solutions under the effect of random perturbations. On the one hand, we present an extension of a regularity result (due to Kalita) to the stochastic setting. On the other hand we discuss a partial result that might allow to attack the (open) question whether or not stochastic noise might even prevent the emergence of singularities for some particular systems.
    (Joint work with F. Flandoli, Pisa)
  • 13.12.2011, Guillaume Voisin (Universität Duisburg-Essen): Pruning of a Lévy tree, on nodes and skeleton.
    Abstract: Lévy trees are random continuous trees defined by a functional of a Lévy process : the so-called height process. They also can be viewed as limited objects of Galton-Watson trees. I consider a general Lévy tree and I construct a pruning procedure on the infinite nodes and on the skeleton of the tree using some snakes. Then I prove that the pruned tree is again a Lévy tree defined by an explicit Lévy process.
    This procedure has already been studied for other cases : Brownian tree (Abraham & Serlet, 2004), Lévy without brownian part (Abraham & Delmas, 2008). There is also some related results on Galton-Watson trees : Aldous & Pitman (1998) or Abraham & Delmas & Hui (2011).
    (This is a joint work with Abraham & Delmas, 2011)
  • 06.12.2011, Roman Berezin (Technion - Israel): Survival behavior of the contact process with rapid stirring.
    Abstract: We study the limiting behaviour of an interacting particle system, under the rules of an exclusion process in a d-dimensional Euclidian space, and a contact process, when the branching speed is on a smaller scale than the exclusion process. It is known that the critical value for survival for such processes, starting from a single individual at the origin, is 1. However, how close the critical value is to 1, for large branching speed, remained an open problem posed by Konno. We show that this critical value is in fact 1+ θ/N, where N is the branching rate, and we find the particular constant θ.
  • 30.11.2011, no talk.
  • 22.11.2011, no talk.
  • 15.11.2011, Olivier Hénard (CERMICS - École des ponts): Generalized Fleming Viot process conditioned on non extinction of some types.
    Abstract: Using the lookdown construction of Donnelly and Kurtz, we condition a Generalized Fleming Viot process without mutation on non extinction of some types.
  • 08.11.2011, no talk.
  • 01.11.2011, no talk.
  • 25.10.2011, Sandra Kliem (Universität Duisburg-Essen) : Convergence of Rescaled Competing Species Processes to a Class of SPDEs
    Abstract: In this talk we construct a sequence of rescaled perturbations of voter processes in dimension d=1. We consider long-range interactions and show that the approximate densities converge to continuous space time densities which solve a class of SPDEs (stochastic partial differential equations), namely the heat equation with a class of drifts, driven by Fisher-Wright noise. If the initial condition of the limiting SPDE is integrable, weak uniqueness of the limits follows.
    The results obtained extend the results of Mueller and Tribe (1995) for the voter model by including perturbations. As an example we show that the new results cover spatial versions of the Lotka-Volterra model as introduced in Neuhauser and Pacala (1999) for parameters approaching one.
  • 18.10.2011, Anton Klimovsky (Universität Eindhoven) : Rather conventional Gaussian random fields in the light of spin glasses.
    Abstract: Spin glasses are paradigmatic models of complex stochastic systems with inhomogeneous interactions. While these models and related heuristics originate in theoretical physics, in the recent years spin glasses keep influencing combinatorial optimization, machine learning, information science (e.g., survey propagation, turbo codes), theoretical biology (e.g., Kauffmann's rugged fitness landscapes) and other sciences in a substantial way. This is not surprising, since spin glasses turn out to be a class of fundamental mathematical objects. In this talk, I report on our recent results concerning the analysis of exponential functionals of Gaussian random fields with stationary isotropic increments. The main tool is the rigorous spin glass techniques, which will be introduced along the way.
  • 11.10.2011, Alexa Manger (Universität Duisburg-Essen): Introduction to Super-Brownian Motion
    Abstract: We will introduce the super-Brownian motion via its "discrete particle" counterpart, the branching Brownian motion. After rescaling we will get a martingale representation for the super-Brownian motion and deduce some properties.


Summer Semester 2011 (back to Contents)

  • 04.08.2011, Lior Bary-Soroker (Universität Duisburg-Essen): Galois and Probability
    Abstract: Évariste Galois died in a dual at the age of 20 on May 31, 1832. The mathematics he managed to do until that led, a decade after his death, to Modern Algebra. In this talk I will try to explain what is Galois theory. In particular I hope to give some insights to the famous theorem of Galois saying that the general equation of degree 5 has no root formula. If time permits, I'll discuss two connections with probability (or one, or zero but then we can discuss on coffee). As tempting as it is, I'm not going to discuss history, if one is interested in the story of his life, there are many books on this subjects, or wikipedia.....
  • 15.06.2011, Anja Sturm (Universität Göttingen): Long-term behavior of subcritical contact processes
    Abstract: We consider the long-time behavior of the law of a contact process started with a single infected site, distributed according to counting measure on the lattice. This distribution is related to the configuration as seen from a typical infected site and gives rise to the definition of so-called eigenmeasures, which are possibly infinite measures on the set of non empty configurations that are preserved under the dynamics up to a multiplicative constant. We show that contact processes on general countable groups have in the subcritical regime a unique spatially homogeneous eigenmeasure. We also discuss possible applications of this result, in particular regarding the behavior of the exponential growth rate of the process as a function of its death rate. This is joint work with Jan Swart (UTIA Prague)
  • 08.06.2011,
  • 01.06.2011, Siva Athreya (Indian Statistical Institute, Bangalore): Blowup and Conditionings of $\psi$-super Brownian Exit Measures
    Abstract: We extend earlier results on conditioning of super-Brownian motion to general branching rules. We obtain representations of the conditioned process, both as an $h$-transform, and as an unconditioned superprocess with immigration along a branching tree. Unlike the finite-variance branching setting, these trees are no longer binary, and strictly positive mass can be created at branch points. This construction is singular in the case of stable branching. We analyze this singularity first by approaching the stable branching function via analytic approximations. In this context the singularity of the stable case can be attributed to blowup of the mass created at the first branch of the tree. Other ways of approaching the stable case yield a branching tree that is different in law. To explain this anomaly we construct a family of martingales whose backbones have multiple limit laws.
  • 18.05.2011, Wolfgang Löhr (Universität Duisburg-Essen): Measures and Continuous Functions on the Space of Metric Measure Spaces
    Abstract: We introduce the space of metric measure spaces (mm-spaces) and its embedding into the space of distance matrix distributions. This classical embedding can be extended to the space of measures on the space of mm-spaces, which has been shown recently by Depperschmidt, Greven and Pfaffelhuber, and will be shown here in a different way. We also explain that the space of mm-spaces is not locally compact and the algebra of polynomials is not dense in the space of bounded continuous functions.
  • 04.05.2011, Andrej Fischer (Universität Köln): Stochastic tunneling in a two-locus system with recombination
  • 06.04.2011, Anita Winter (Universität Duisburg-Essen): Coalescent processes arising in the study of diffusive clustering
    Abstract: We study the spatial coalescent on $\Z^2$. In our setting, partition elements are located at the sites of $Z^2$ and undergo local delayed coalescence and migration. The system starts in either locally finite configurations or in configurations containing countably many partition elements per site. Our goal is to determine the longtime behavior with an initial population of countably many individuals per site restricted to a box $\Lambda^{\alpha,t}:=[-t^{\alpha/2}, t^{\alpha/2}]^2 \cap \Z^2$ and observed at time $t^\beta$ with $1 \geq \beta \geq \alpha\ge 0$. We study both asymptotics, as $t\to\infty$, for a fixed value of $\alpha$ as the parameter $\beta\in[\alpha,1]$ varies and for a fixed $\beta$, as the parameter $\alpha\in [0,\beta]$ varies. This exhibits the genealogical structure of the mono-type clusters arising in 2-dimensional Moran and Fisher-Wright systems. A new random object, the so-called {\em coalescent with rebirth}, is constructed via a look-down procedure and shown to arise in the space-time limit of the coalescent restricted to $\Lambda^{\alpha,t}$ with $\alpha\in[0,1]$ and observed at time time goes to infinity. (this is joint work with Andreas Greven and Vlada Limic)


Winter Semester 2010/11 (back to Contents)

  • 10.02.2011, Anita Winter (Universität Duisburg-Essen): A multitype branching model with local self-regulation
    Abstract: We consider a spatial multi-type branching model in which individuals migrate in $Z^d$ according to random walks and reproduce according to a branching mechanism which can be sub-, super- or critically depending on carrying capacities and the local intensity of individuals of the different types. In this talk we will focus on the diffusion limit of small mass, locally many individuals and rapid reproduction in the exchangeable set-up where non of the parameters involved in the model are type dependent. In Etheridge (2006) it has been shown that there are parameter regimes allowing for survival in all dimensions. In this talk we present duality relations which allow for monotonicity statements in the parameters with regards whether or not the different surviving types can coexist. (joint work with Andreas Greven, Peter Pfaffelhuber, Anja Sturm and Iljana Zähle)
  • 25.01.2011, Alexa Manger (Universität Duisburg-Essen): Association of Ito processes
    Abstract: Association is a special kind of positive dependence. In the special case of It\^o processes we find conditions for the association and we can conclude the association of their hitting times which gives applications in risk management
  • 18.01.2011, Anita Winter (Universität Duisburg-Essen): Brownian motion on real trees
    Abstract: The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on any given locally compact real tree equipped with a Radon measure. We specify a criterion under which the Brownian motion is recurrent or transient. (this is joint work with Siva Athreya and Michael Eckhoff)
  • 11.01.2011, Guillaume Voisin (Universität Duisburg-Essen): Local time of a diffusion in a Levy environment
    Abstract: Diffusions in random environment can be viewed as a limit in time and space of random walks in discrete random environment. In the recurrent case, discrete and continuous diffusions have localization properties. The local time process on some well chosen points of the medium gives a better idea of this localization. We get the asymptotic law of the local time process at the favorite point of the diffusion.
  • 14.12.2010, Vladimir Osipov (Universität Duisburg-Essen): Ultra-metric models of protein conformational dynamics
  • 07.12.2010, Andre Depperschmidt (Hausdorff Zentrum Bonn): Tree-valued Fleming-Viot process with mutation and selection
    Abstract: In population genetics Moran models are used to describe the evolution of types in a population of a fixed size N. The type of individuals may change due to mutation. Furthermore, due to selection the offspring distribution of an individual depends on its current type. As N tends to infinity the empirical distribution of types converges to the Fleming-Viot process. At fixed times the genealogy of such populations can be constructed using the ancestral selection graph (ASG) of Krone and Neuhauser, which generalizes the Kingman coalescent. As the population evolves its genealogy evolves as well. We construct a tree-valued version of the Fleming-Viot process with mutation and selection (TFVMS) using a well-posed martingale problem. This extends the construction of the neutral tree-valued process given in (Greven, Pfaffelhuber and Winter, 2010). For existence we use approximating tree-valued Moran models and for uniqueness a Girsanov-type theorem on marked measure spaces, the state spaces of TFVMS. Furthermore we study the long-time behavior of TFVMS using duality. Finally, in a concrete example, we compare the Laplace transforms of pairwise genealogical distances in equilibrium of TFVMS and the neutral tree-valued process. This is joint work with Andreas Greven and Peter Pfaffelhuber.
  • 30.11.2010, Lorenz Pfeiffroth (TU München): Frogs in a random environment on Z
    Abstract: The frog model in a fixed environment can be described as follows. Let G be a graph and take one vertex as origin. Initially there is a number of sleeping frogs at each vertex except the origin. At the origin there is one active frog which jumps according to a random walk on $G$. If an active frog jumps to a vertex where sleeping frogs are, they get awake and move according to the same random walk, independently from everything else. The idea of this model is that every active frog has some information and it shares it with the sleeping frogs for the first time when they meet. Alves, Machado and Popov proved a recurrence criterion if the graph is Z^d or T_d and the underlying random walk is a symmetric simple random walk. The first time other underlying random walks were investigated was by Gantert and Schmidt in 2008. The random walk was a simple random walk in Z with drift to the right. In the first part of this talk we consider a more general setting of underlying random walks. I.e. the only assumption for our random walk is that he is transient to the right. The question, we are interested in, is if the origin is visited infinitely often by active frogs with probability 1 or not. This is not a trivial question in this setting because all random walks in this model won't eventually visit the negative integers. But intuitively spoken if there are enough frogs on the positive integers, which will be activated surely, the change of visiting the negative integers is increasing and thus also the origin. So we expect if there are enough frogs on the right of the origin the model will be recurrent. We give a necessary and sufficient condition that this will happened. Also we show that our result is a generalization of the model, which Gantert and Schmidt investigate, and present a 0-1 law for this model. Now the question naturally arise is if we take the jumping probability random, can we derive analogue conditions for the recurrence of such a model. The second part of this talk deals with that kind of problem. We give recurrence criteria for such a model. If we take the starting configuration of sleeping frogs also as random, we derive a 0-1 law too and show that the recurrence of such a model only depends on the distribution of the starting configuration and it does not depend on the distribution of the jumping probability of the underlying random walk. In the last part I sketch the proof of the recurrence criteria for a frog model in a fixed and random environment, respectively.
  • 23.11.2010, no talk because of the mini-workshop on dualities at the Hausdorff Center Bonn
  • 16.11.2010, Monika Meise (Universität Duisburg-Essen): Shape restricted smoothing
  • 09.11.2010, no talk because of the SFB/TR 12 meeting
  • 02.11.2010, Wolfgang Löhr (Universität Duisburg-Essen): Complexity Measures of Discrete-Time Stochastic Processes, Continuity and Ergodic Decomposition II
    Abstract: Continuation of the talk from 19.10.2010
  • 26.10.2010, Anton Klimovsky (Hausdorff-Zentrum Bonn): Universal macroscopic behavior of evolving genealogies of spatial Lambda-Fleming-Viot processes
    Abstract: We consider a class of stochastic processes -- the so-called spatial Lambda-Fleming-Viot processes -- that describe the evolution of the genealogies in the spatially extended populations with migration and occasionally large (i.e., comparable to the population size) reproduction events. What reproduction mechanisms can be observed in these processes on the macroscopic level? We argue that, in the regime when the migration mechanism mixes the spatially extended population well, the macroscopic reproduction behavior is rather universal and is described by the Kingman coalescent. Joint work in progress with A. Greven and A. Winter.
  • 19.10.2010 , Wolfgang Löhr (Universität Duisburg-Essen): Complexity Measures of Discrete-Time Stochastic Processes, Continuity and Ergodic Decomposition
    Abstract: In complex system sciences, one tries to quantify different kinds of ``complexity'' of processes. The resulting complexity measures are then used for data analysis and modelling. In my work, I provide a rigorous mathematical framework for one of these complexity measures, namely statistical complexity, and some related quantities. As a main result, I obtain functional properties such as lower-semi continuity and behaviour under ergodic decomposition. An important tool is the prediction process introduced by Frank Knight in 1975.

Former Organizers:

Martin Hutzenthaler
Anton Klimovsky
Sandra Kliem
Wolfgang Löhr
Mikhail Urusov

Guillaume Voisin
Anita Winter