The Zakai Equation of Nonlinear Filtering for Jump-Diffusion Observations: Existence and Uniqueness

Abstract

In this paper we study a 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∈[0,T], the σ-algebra \(\mathcal{F}^{Y}_{t}:= \sigma\{ Y_{s}: s\leq t\}\) provides all the available information about the signal Xt. The central goal of stochastic filtering is to characterize the filter, πt, which is the conditional distribution of Xt, given the observed data \(\mathcal{F}^{Y}_{t}\). In Ceci and Colaneri (Adv. Appl. Probab. 44(3):678–701, 2012) it is proved that π is the unique probability measure-valued process satisfying a nonlinear stochastic equation, the so-called Kushner-Stratonovich equation (in short KS equation). In this paper the aim is to improve the hypotheses to obtain the KS equation and describe the filter π in terms of the unnormalized filter ϱ, which is solution of a linear stochastic differential equation, the so-called Zakai equation. We prove the equivalence between strong uniqueness of the solution of the KS equation and strong uniqueness of the solution of the Zakai one and, as a consequence, we deduce pathwise uniqueness of the solution of the Zakai equation by applying the Filtered Martingale Problem approach (Kurtz and Ocone in Ann. Probab. 16:80–107, 1988; Ceci and Colaneri in Adv. Appl. Probab. 44(3):678–701, 2012). To conclude, we discuss some particular models.