Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes

TL;DR

This paper establishes a probabilistic representation of the value function in optimal control of pure jump Markov processes using constrained backward stochastic differential equations, linking control problems with nonlinear Feynman-Kac formulas.

Contribution

It introduces a novel constrained BSDE framework for representing the value function in jump process control problems, extending existing probabilistic methods.

Findings

Proves existence and uniqueness of solutions for constrained BSDEs.

Establishes a nonlinear Feynman-Kac formula for the value function.

Shows the equivalence of original and auxiliary control problems.

Abstract

We consider a classical finite horizon optimal control problem for continuous-time pure jump Markov processes described by means of a rate transition measure depending on a control parameter and controlled by a feedback law. For this class of problems the value function can often be described as the unique solution to the corresponding Hamilton-Jacobi-Bellman equation. We prove a probabilistic representation for the value function, known as nonlinear Feynman-Kac formula. It relates the value function with a backward stochastic differential equation (BSDE) driven by a random measure and with a sign constraint on its martingale part. We also prove existence and uniqueness results for this class of constrained BSDEs. The connection of the control problem with the constrained BSDE uses a control randomization method recently developed in the works of I. Kharroubi and H. Pham and their co-authors. This approach also allows to prove that the value function of the original non-dominated control problem coincides with the value function of an auxiliary dominated control problem, expressed in terms of equivalent changes of probability measures.