The Zakai equation of nonlinear filtering for jump-diffusion observation: existence and uniqueness

TL;DR

This paper investigates the Zakai equation for nonlinear filtering in jump-diffusion systems, establishing existence, uniqueness, and equivalence with the Kushner-Stratonovich equation, thereby advancing the theoretical understanding of stochastic filtering.

Contribution

It proves the equivalence of strong uniqueness between the Zakai and Kushner-Stratonovich equations for jump-diffusion models, and establishes pathwise uniqueness using the Filtered Martingale Problem approach.

Findings

Proves equivalence of strong uniqueness between Zakai and Kushner equations.

Establishes pathwise uniqueness for solutions to the Zakai equation.

Discusses particular cases of the general filtering framework.

Abstract

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.