A Dynamic Programming Formulation for the Nonlinear Filter

TL;DR

This paper introduces a dynamic programming approach to solve a class of nonlinear filtering problems formulated as BSDE-constrained stochastic control problems, building on previous dual control methods.

Contribution

It develops a dynamic programming principle for BSDE-constrained stochastic control, enabling a new solution method for nonlinear filtering problems.

Findings

Derived a DP-based solution for nonlinear filtering

Extended dual control formulation with DP approach

Provides a new framework for solving BSDE-constrained problems

Abstract

This paper build on our recent work where we presented a dual stochastic optimal control formulation of the nonlinear filtering problem [1]. The constraint for the dual problem is a backward stochastic differential equations (BSDE). The solution is obtained via an application of the maximum principle (MP). In the present paper, a dynamic programming (DP) principle is presented for a special class of BSDE-constrained stochastic optimal control problems. The principle is applied to derive the solution of the nonlinear filtering problem.