Abstract
In this article, we are concerned with the non-linear filtering problem for the signal process of Markov type with the observation of discontinuous semimartingales. Through the extended Girsanov transform of observation processes, we consider processes of unnormalized filtering measures so that the filtering problem reduces to analyze Markov processes with generators affected by observations. This way makes it possible to use the Malliavin calculus to show the existence of smooth densities of filtering measures. Processes of unnormalized filtering measures are characterized by linear functional SDE (Zakai equations) whose coefficients are integrodifferential operators. In the case where noises of the signal and the observation are independent, by a simple replacement of solutions, the associated Zakai equation can be transformed to a parabolic integro-differential equation with coefficients affected by the observation. This transform would be useful to prove the uniqueness of solutions to Zakai equations. It is possible to generalize these results to some cases where noises of the signal and the observation are correlated. Under regularity conditions for the existence of stochastic flows, by certain replacement of solutions using stochastic flows, those Zakai equations also can be transformed to parabolic integro-differential equations.
