Nonlinear Filtering of Partially Observed Systems arising in Singular Stochastic Optimal Control

TL;DR

This paper develops nonlinear filtering equations for systems influenced by processes with bounded variation, enabling analysis of singular stochastic control problems with partially observed signals.

Contribution

It derives Zakai and Kushner-Stratonovich equations for such systems using a reference probability approach, addressing challenges posed by the process  and establishing uniqueness results.

Findings

Derived Zakai equation for the unnormalized filter

Proved uniqueness of the filtering process under certain conditions

Facilitated analysis of stochastic control with singular controls

Abstract

This paper deals with a nonlinear filtering problem in which a multi-dimensional signal process is additively affected by a process $\nu$ whose components have paths of bounded variation. The presence of the process $\nu$ prevents from directly applying classical results and novel estimates need to be derived. By making use of the so-called reference probability measure approach, we derive the Zakai equation satisfied by the unnormalized filtering process, and then we deduce the corresponding Kushner-Stratonovich equation. Under the condition that the jump times of the process $\nu$ do not accumulate over the considered time horizon, we show that the unnormalized filtering process is the unique solution to the Zakai equation, in the class of measure-valued processes having a square-integrable density. Our analysis paves the way to the study of stochastic control problems where a decision maker can exert singular controls in order to adjust the dynamics of an unobservable It\^o-process.