Stability of Nonlinear Filters and Branching Particle Approximations to The Filtering Problems

Various particle filters have been proposed and their convergence to the optimal filter are obtained for finite time intervals. However, uniform convergence results have been established only for discrete time filters. We prove the uniform convergence of a branching particle filter for continuous time setup when the optimal filter itself is exponentially stable.

The short interest rate process is modeled by an asymptotically stationary diffusion process. With the counting process observations, a filtering problem is formulated and its exponential stability is derived. Base on the stability result, the uniform convergence of a branching particle filter is proved.