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.

The alpha-Ford algebraic measure trees

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.

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.

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.

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.

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.

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.

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.

Felix Lindner (University of Kassel)
Strong approximation of constrained stochastic dynamics

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.

Gerónimo Rojas Barragán (University of Duisburg-Essen)
TBA

TBA