Kolmogorov equations on the space of probability measures associated to the nonlinear filtering equation: the viscosity approach

TL;DR

This paper develops a viscosity solution framework for the backward Kolmogorov equation on the space of probability measures related to nonlinear filtering, proving existence, uniqueness, and a comparison theorem without relying on density assumptions.

Contribution

It introduces a novel viscosity approach to the Kolmogorov equation on measure spaces, establishing existence and uniqueness results directly on measures.

Findings

Proved existence of viscosity solutions for the equation.

Established a comparison theorem ensuring uniqueness.

Provided a new method avoiding density assumptions in measure-valued processes.

Abstract

We study the backward Kolmogorov equation on the space of probability measures associated to the Kushner-Stratonovich equation of nonlinear filtering. We prove existence and uniqueness in the viscosity sense and, in particular, we provide a comparison theorem. In the context of stochastic filtering it is natural to consider measure-valued processes that satisfy stochastic differential equations. In the literature, a classical way to address this problem is by assuming that these measure-valued processes admit a density. Our approach is different and we work directly with measures. Thus, the backward Kolmogorov equation we study is a second-order partial differential equation of parabolic type on the space of probability measures with compact support. In the literature only few results are available on viscosity solutions for this kind of problems and in particular the uniqueness is a very challenging issue. Here we find a viscosity solution and then we prove that it is unique via comparison theorem.