Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control

TL;DR

This paper develops an elementary method to solve backward stochastic differential equations driven by marked point processes, enabling numerical solutions and applying the results to an optimal control problem involving such processes.

Contribution

It introduces a new approach that reduces the problem to deterministic differential equations, bypassing martingale representation, and applies it to optimal control with marked point processes.

Findings

Proved existence and uniqueness under Lipschitz conditions.

Reduced BSDEs to deterministic differential systems.

Applied method to an optimal control problem.

Abstract

We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter-examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.