# 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 Josuè Nussbaumer (University Duisburg-Essen) Oct 29 Emiel Lorist (TU Delft) Nov 12 Sascha Nolte (University Duisburg-Essen) Nov 19 Nina Dörnemann (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 Duisburg-Essen) Dec 10 Tobias Hübner (University Duisburg-Essen) Dec 17 Jan 7 Tuan Anh Nguyen (University Duisburg-Essen) Jan 14 Benjamin Gess (MPI MiS Leipzig and University Bielefeld) Jan 21 Felix Lindner (University of Kassel) Jan 28

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 Josuè Nussbaumer (University 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 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 (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, Sørensen (1995) and Kessler, Sørensen (1999) in the case of discretely observed ergodic diffusion processes. In this talk we adapt the methodology of Kessler and Sørensen 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 Sørensen, 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.

When
Tuesdays, 16:15–17:15

Where
Room: WSC-S-U-3.02