Lyapunov exponents for finite state nonlinear filtering

Consider the Wonham optimal filtering problem for a finite state ergodic Markov process in both discrete and continuous time, and let a be the noise intensity for the observation. We examine the sensitivity of the solution with respect to the filter's initial conditions in terms of the gap between the first two Lyapunov exponents of the Zakai equation for the unnormalized conditional probability. This gap is studied in the limit as sigma --> 0 by techniques involving considerations of nonlinear filtering and the stochastic Feynman-Kac formula. Conditions are given for the limit to be either negative or -infinity. Asymptotic bounds are derived in the latter case.