Rough stochastic filtering

Abstract

This article is concerned with the well-posedness of the "filtering equations", due to Zakai and Kushner-Stratonovich, arising in nonlinear stochastic filtering. In general situations, notably in correlated diffusion models and when signal coefficients depend on the observation process, the well-posedness is a difficult problem, mainly due to conflicting martingale structures of the involved forward and backward equations. Crisan-Pardoux (2024) address this classical problem with BSPDE techniques, Du et al. (2013), a Sobolev-based approach that however requires increasingly strong regularity assumptions in high dimensions. In this work, we take a new mixed rough stochastic perspective which allows us to derive well-posed rough counterparts of the filtering equations. Importantly, the rough filtering equations are seen, upon randomization, to coincide with the classical filtering equations. Our framework yields well-posedness (existence, uniqueness, stability) under dimension-independent regularity assumptions, providing a robust and conceptually unified solution to a longstanding problem in stochastic filtering theory. To illustrate the flexibility of the method, we also treat rough versions of the classical Kalman-Bucy filter, with characteristics described by a new class of RDEs of rough Riccati type.