Intrinsic methods in filter stability

Abstract. The purpose of this article is to survey some intrinsic methods for studying the stability of the nonlinear filter. By ‘intrinsic ’ we mean methods which directly exploit the fundamental representation of the filter as a con-ditional expectation through classical probabilistic techniques such as change of measure, martingale convergence, coupling, etc. Beside their conceptual appeal and the additional insight gained into the filter stability problem, these methods allow one to establish stability of the filter under weaker conditions compared to other methods, e.g., to go beyond strong mixing signals, to reveal connections between filter stability and classical notions of observability, and to discover links to martingale convergence and information theory. 1. Inroduction Consider a pair of random sequences (X,Y) = (Xn, Yn)n∈Z+, where the signal component Xn takes values in a Polish space 1 S and the observation component Yn takes values in Rp for some p ≥ 1. The classical filtering problem is to compute the conditional distribution πn(·) = P(Xn ∈ · |F Y 0,n), (1.1) where FYk,n stands for the σ-algebra of events generated by Ym, k ≤ m ≤ n (sim-ilarly, we will use below the σ-algebra FXk,n generated by Xm, k ≤ m ≤ n). Once πn is found, the optimal mean square estimate of f(Xn) can be calculated a