Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE

TL;DR

This paper develops a probabilistic Feynman-Kac representation for Hamilton-Jacobi-Bellman equations using a novel class of constrained BSDEs, enabling analysis of stochastic control problems without ellipticity assumptions.

Contribution

It introduces a new class of constrained BSDEs with jump components, providing a novel probabilistic representation for HJB IPDEs without requiring ellipticity.

Findings

Proved existence and uniqueness of minimal solutions for the constrained BSDEs.

Established a new probabilistic representation for nonlinear IPDEs of HJB type.

Derived a dual formula involving change of probability measures for the BSDE solutions.

Abstract

We aim to provide a Feynman-Kac type representation for Hamilton-Jacobi-Bellman equation, in terms of forward backward stochastic differential equation (FBSDE) with a simulatable forward process. For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for nonlinear integro-partial differential equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework, and importantly we do not make any ellipticity condition. Moreover, we state a dual formula of this BSDE minimal solution involving equivalent change of probability measures. This gives in particular an original representation for value functions of stochastic control problems including controlled diffusion coefficient.