  This paper is concerned with the nonlinear filtering problem for a general
Markovian partially observed system (X,Y), whose dynamics is modeled by
correlated jump-diffusions having common jump times. At any time t, the
sigma-algebra generated by the observation process Y provides all the available
information about the signal X. The central goal of stochastic filtering is to
characterize the filter which is the conditional distribution of X, given the
observed data. It has been proved in Ceci-Colaneri (2012) that the filter is
the unique probability measure-valued process satisfying a nonlinear stochastic
equation, the so-called Kushner-Stratonovich equation (KS-equation). In this
paper the aim is to describe the filter in terms of the unnormalized filter,
which is solution to a linear stochastic differential equation, called the
Zakai equation. We prove equivalence between strong uniqueness for the solution
to the Kushner Stratonovich equation and strong uniqueness for the solution to
the Zakai one and, as a consequence, we deduce pathwise uniqueness for the
solutions to the Zakai equation by applying the Filtered Martingale Problem
approach (Kurtz-Ocone (1988), Kurtz-Nappo (2011), Ceci-Colaneri (2012)). To
conclude, we discuss some particular cases.
