Robustness of Zakai's equation via Feynman-Kac representations

: We propose to study the sensitivity of the optimal filter to its initialization, by looking at the distance between two differently initialized filtering processes in terms of the ratio between two simple Feynman-Kac integrals in the product space. We illustrate, by considering two simple examples, how this approach may be employed to study the asymptotic decay rate, as the difference between the growth rates of the two integrals. We apply asymptotic methods, such as large deviations, to estimate these growth rates. The examples we consider are the linear case, where we recover known results, and a case where the drift term in the state process is nonlinear. In both cases, only the small noise regime and only one-dimensional diffusions are studied. KEYWORDS: Non-linear filtering; Feynman-Kac; Large deviations. 1 Introduction Zakai's equation of nonlinear filtering theory represents the solution of the filtering problem consisting of evaluating the conditional law of a Markov process..