The Stability Of The Filter With Jumps

In this paper, we consider the stability of the filter with jumps. Currently, the filter without jumps has been widely studied, especially when the signal is an ergodic Markov process. An important approach in dealing with such kind of problems is to obtain the representation of the conditional expectation of the filter through classical probabilistic techniques, such as the change of measure, martingale convergence, coupling, etc. Van Handel R. [23,24,25] didn’t rely on the ergodicity of the signal or the representation of the conditional expectation in his articles. Instead, he put forward the concepts of "Observability" and "Uniform observability" to express that the filter is stable when the observations are sufficiently informative.First, we introduce the results and methods of proving stability with "Observability" and "Uniform observability" by Van Handel R.; Then, considering Stephan B.’s filter with jumps [27] (driven by Brownian motion and Poisson process together), we reduce the requirements which ensure Ωr, the Radon-Nikodym density of Girsanov transformation, is a martingale to simplier forms. Furthermore, the filter is shown to be a Levy process by the Criterion Theorem of Levy process. At last, sufficient conditions for the stability of the filter with jumps are given, as well as concrete examples.