Uniqueness for Measure-Valued Equations of Nonlinear Filtering for Stochastic Dynamical Systems with L\'evy Noises

TL;DR

This paper proves pathwise and joint law uniqueness for the Zakai and Kushner-Stratonovich equations in nonlinear filtering of systems driven by Lévy noises, extending the theoretical understanding of measure-valued equations.

Contribution

It establishes the uniqueness of solutions for measure-valued equations in nonlinear filtering with Lévy noise, under general assumptions, advancing the theoretical framework.

Findings

Proved pathwise uniqueness for the Zakai equation.

Established uniqueness in joint law for the Zakai and Kushner-Stratonovich equations.

Extended uniqueness results to non-Gaussian systems with Lévy noises.

Abstract

In the article, Zakai and Kushner-Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system are considered. Moreover, we prove that under some general assumption, the Zakai equation has pathwise uniqueness and uniqueness in joint law, and the Kushner-Stratonovich equation is unique in joint law.