In this talk, I will show how the theory of rough stochastic differential equations (rough SDEs) — introduced by Friz, Hocquet, and Lê in 2021 — helps to establish the existence, uniqueness, or smoothness of solutions to certain rough partial differential equations (rough PDEs).
A key motivation comes from stochastic filtering, where the Zakai equation, an SPDE describing the unnormalized conditional density, can be reformulated as a rough PDE using rough path theory.
I will present results from [1], where we develop a solution theory for linear rough PDEs and derive a Feynman–Kac-type representation via rough SDEs. If time permits, I will briefly discuss how we extend Hörmander’s theory to the rough setting in [2] using Malliavin calculus.
[1] F.B., Peter K. Friz, Wilhelm Stannat, Parameter dependent rough SDEs with applications to rough PDEs, 2024 (arXiv:2409.11330)
[2] F.B., Michele Coghi, Torstein K. Nilssen, Malliavin calculus for rough stochastic differential equations, 2024 (arXiv:2402.12056)
In recent years, many models in mathematical physics have been encoded into graphical models, which are more accessible through the lens of probability theory. These graphical models often exhibit a natural percolation structure. One such model is the Random Loop Model introduced by Daniel Ueltschi. Peter Mühlbacher showed that the loop threshold for the Random Loop Model with θ=1 is larger than the percolation threshold. This is due to so-called blocking events in graphs with uniformly bounded degree. The proof primarily relies on a coupling method.
In my talk, I will introduce the model and the basic proof techniques. Furthermore, I will discuss a recent result where we generalize the method to obtain new results for general trees.
I will explain why the tree case differs from the case of a general graph. If time permits, I will use the Galton-Watson case to illustrate how the coupling in the proof works.
This talk is based on joint work with V. Betz, M. Kraft, B. Lees and C. Mönch
We study various formulation of zero-sum games between a singular-controller and a stopper with a finite-time horizon, where the underlying process is a multi-dimensional controlled stochastic differential equation evolving in an unbounded domain. We prove that such games admit a value and present an optimal strategy for the stopper. In some cases, we show the game's value is the maximal solution, in a suitable Sobolev class, of a variational inequality of 'min-max' type with both obstacle and gradient constraint. Under stricter assumptions, we provide an optimal strategy for the controller and establish a connection between the space derivative of the value function and the solution of an optimal stopping problem with absorption.
