Generalized Girsanov Transform of Processes and Zakai Equation with Jumps

It is well known that the Girsanov transform (or, Girsanov's theorem) plays an important role in the stochastic analysis and this transform is closely related with the uniform integrability of local martingales. The ̄rst aim of this article is to give concrete, necessary and su±cient conditions of uniform integrability of positive local martingales with jumps. Then we shall apply Girsanov transform to Zakai equation (Zakai SDE) arisen from the ̄ltering problem of stochastic processes with jumps. Using Girsanov transform for L¶evy processes, Malliavin calculus could be applied to show the existence of smooth density of the ̄ltering measure. The second aim of this article is to show the uniqueness of solutions of Zakai equation. This is worthwhile from the fact that the solution of Zakai equation can be obtained from the ̄ltering measure by using Girsanov transform.