A PDE approach to quantitative asymptotic stability of the optimal filter in the linear detectable setting
The filtering problem asks to estimate a signal process, the initial state of which is unknown, given its partial and noisy observations. From a probabilistic perspective, the best estimate at each time is the optimal filter, which is the conditional law of the signal given all observations up to the time. The problem of asymptotic stability of the optimal filter, which asks whether the filter initialized with the incorrect initial law converges to the correctly initialized filter, is of fundamental practical and theoretical importance. This paper studies the quantitative asymptotic stability of the optimal filter for signal-observation models with linear drift and additive Brownian noise. When the initial law is additionally assumed to be Gaussian, the optimal filter reduces to the celebrated Kalman-Bucy filter, for which the detectability condition on the signal and observation matrices is the sharp condition ensuring asymptotic stability. In the linear setting where the detectability condition holds but the initial law need not be Gaussian, we make the following contributions by studying the Zakai equation satisfied by the unnormalized filter. First, we study the asymptotic behavior of the deterministic part of the Zakai equation in the general case with nonlinear drift, and place the detectability condition in this context. Second, in the case of linear drift, we characterize the solution of the Zakai equation (hence the optimal filter) by a gauge transformation that separates the finite-dimensional stochastic dynamics from the infinite-dimensional deterministic dynamics. Third, we use this representation to establish asymptotic exponential filter stability in the $χ^p$-divergence (with respect to a reference optimal filter), the total variation norm, and the $2$-Wasserstein distance.