Kolmogorov equations on spaces of measures associated to nonlinear filtering processes

TL;DR

This paper develops backward Kolmogorov equations on measure spaces related to nonlinear filtering, proving existence, uniqueness, and regularity of solutions without assuming densities, thus advancing the mathematical understanding of measure-valued filtering processes.

Contribution

It introduces a novel approach to Kolmogorov equations on measure spaces for filtering, avoiding density assumptions and establishing foundational results on solutions and derivatives.

Findings

Proved existence and uniqueness of classical solutions.

Developed Itô formulas for measure-valued processes.

Analyzed regularity of solutions with respect to initial data.

Abstract

We introduce and study some backward Kolmogorov equations associated to stochastic filtering problems. Measure-valued processed arise naturally in the context of stochastic filtering and one can formulate two stochastic differential equations, called Zakai and Kushner-Stratonovitch equation, that are satisfied by a positive measure and a probability measure-valued process respectively. The associated Kolmogorov equations have been intensively studied, mainly assuming that the measure-valued processes admit a density and then by exploiting stochastic calculus techniques in Hilbert spaces. Our approach differs from this since we do not assume the existence of a density and we work directly in the context of measures. We first formulate two Kolmogorov equations of parabolic type, one on a space of positive measures and one on a space of probability measures, and then we prove existence and uniqueness of classical solutions. In order to do that, we prove some intermediate results of independent interest. In particular, we prove It\^o formulas for the composition of measure-valued filtering processes and real-valued functions. Moreover we study the regularity of the solution to the filtering equations with respect to the initial datum. In order to achieve these results, proper notions of derivatives on space of positive measures have been introduced and discussed.