In this talk, we focus on the well-posedness of the Zakai and Kushner-Stratonovich equations in nonlinear stochastic filtering, specifically considering a correlated noise setting with observation-dependent coefficients. While existing BSPDE techniques, due to Crisan and Pardoux (2024), rely on dimension-dependent regularity demands, we introduce a novel rough stochastic filtering framework that bypasses this limitation.
We derive rough counterparts of the filtering equations and establish well-posedness under dimension-independent regularity assumptions, using duality techniques based on backward rough PDEs. Upon randomization, our rough equations recover the classical ones. This shows that, contrary to common belief, rough path techniques can actually require less regularity than classical stochastic approaches. If time allows, we also show a rough stochastic version of the Kalman-Bucy filter.

Joint work with Peter K. Friz, Khoa Lê, and Huilin Zhang (arXiv:2509.11825).