Exponential stability for nonlinear filtering of diffusion processes in a noncompact domain

The optimal nonlinear filtering problem for a diffusion process in a noncompact domain, observed in white noise, is considered. It is assumed that the process is ergodic, the diffusion coefficient is constant and the observation is linear. Using known bounds on the conditional density, it is shown that when the observation noise is sufficiently small, the filter is exponentially stable, and that the decay rate of the total variation distance between differently initialized filtering processes tends to infinity as the noise intensity approaches zero.