Probability Seminar Essen
Probability Seminar Essen
Covers a wide range of topics in Probability and its applications.
Oct 8 | Máté Gerencsér (IST Austria) |
Oct 22 | Josué Nussbaumer (University of Duisburg-Essen) |
Oct 29 | Emiel Lorist (TU Delft) |
Nov 12 | Sascha Nolte (University of Duisburg-Essen) |
Nov 19 | Nina Dörnemann (Ruhr University Bochum) |
Nov 21 | Nicole Hufnagel (TU Dortmund) Non-standard time: Thursday! in room N-U-2.04 |
Nov 26 | Kilian Hermann (TU Dortmund) |
Dec 3 | Fabian Gerle (University of Duisburg-Essen) |
Dec 10 | Tobias Hübner (University of Duisburg-Essen) |
Jan 14 | Benjamin Gess (MPI MiS Leipzig and Bielefeld University) |
Jan 21 | Felix Lindner (University of Kassel) |
Jan 28 | Gerónimo Rojas Barragán (University of Duisburg-Essen) |
Abstracts:
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) Joint work with Mark Veraar |
Nov 12 |
Sascha Nolte (University of Duisburg-Essen) 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. |
Nov 19 |
Nina Dörnemann (Ruhr University Bochum) 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 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. |
Nov 26 |
Kilian Hermann (TU Dortmund) 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) 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 This is work in progress. |
Dec 10 |
Tobias Hübner (University of Duisburg-Essen) 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. |
Jan 14 |
Benjamin Gess (MPI MiS Leipzig and Bielefeld University) 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 21 |
Felix Lindner (University of Kassel) In this talk I present a strong approximation result for a class of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by higher-index stochastic differential-algebraic equations, involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, whose coefficients are in general not globally Lipschitz continuous and of super-linear growth. Strong convergence is established for a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Concrete examples for the considered systems are bead-rod chain models used in molecular dynamics as well as spatially discretized models for the dynamics of inextensible fibers in turbulent flows as occurring, e.g, in the spunbond production process of non-woven textiles. The talk is based on joint work with Holger Stroot, ITWM Kaiserslautern. |
Jan 28 |
Gerónimo Rojas Barragán (University of Duisburg-Essen) TBA |