Uniqueness and superposition of the space-distribution dependent Zakai equations

TL;DR

This paper develops and analyzes space-distribution dependent Zakai equations related to nonlinear filtering of McKean-Vlasov SDEs, establishing uniqueness and a superposition principle with Fokker-Planck equations.

Contribution

It introduces the first formulation of space-distribution dependent Zakai and Kushner-Stratonovich equations, proving their uniqueness and linking them to Fokker-Planck equations.

Findings

Pathwise uniqueness of strong solutions established.

Superposition principle between Zakai and Fokker-Planck equations proved.

Conditions for existence of weak solutions to Fokker-Planck equations provided.

Abstract

The work concerns the space-distribution dependent Zakai equations from nonlinear filtering problems of McKean-Vlasov stochastic differential equations with correlated noises. First of all, we establish the space-distribution dependent Kushner-Stratonovich equations and the space-distribution dependent Zakai equations. Then, the pathwise uniqueness of their strong solutions is shown. Finally, we prove a superposition principle between the space-distribution dependent Zakai equations and space-distribution dependent Fokker-Planck equations. As a by-product, we give some conditions under which space-distribution dependent Fokker-Planck equations have weak solutions.