Asymptotic Stability in Nonlinear Filtering via Stochastic Control and Reflection Couplings

The aim of this thesis is the study of filter stability in a nonlinear setting. As is well-known from the literature, the characterization of the conditional law of the signal can be stated as a stochastic optimal control problem. Starting from the results in Ramon van Handel's PhD thesis, in which the author proves an exponential stability bound under strong convexity assumptions, we consider a weakened notion of convexity and exploit recent results on the propagation of weak convexity along HJB equations to derive an exponential stability bound in the 1-Wasserstein distance in a relaxed framework. The proof uses well-established results on the contraction rates of diffusions.