Fabio Bugini (TU Berlin) – Rough stochastic differential equations and their applications to rough PDEs

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)