Nonlinear Filtering for Jump Diffusion Observations

Abstract

We deal with the filtering problem of a general jump diffusion process,X, when the observation process,Y, is a correlated jump diffusion process having common jump times withX. In this setting, at any timetthe σ-algebraprovides all the available information aboutXt, and the central goal is to characterize the filter, πt, which is the conditional distribution ofXtgiven observations. To this end, we prove that πtsolves the Kushner-Stratonovich equation and, by applying the filtered martingale problem approach (see Kurtz and Ocone (1988)), that it is the unique weak solution to this equation. Under an additional hypothesis, we also provide a pathwise uniqueness result.