Backward stochastic differential equations and optimal control of marked point processes

TL;DR

This paper investigates backward stochastic differential equations driven by marked point processes, establishing well-posedness, and applies them to solve and characterize optimal control problems and associated Hamilton-Jacobi-Bellman equations.

Contribution

It introduces a novel framework linking BSDEs with optimal control of non-Markovian point processes, including existence results and solution representations.

Findings

Proved well-posedness and continuous dependence of BSDE solutions.

Established existence of optimal controls using BSDEs.

Derived a stochastic Hamilton-Jacobi-Bellman equation with unique solutions.

Abstract

We study a class of backward stochastic differential equations (BSDEs) driven by a random measure or, equivalently, by a marked point process. Under appropriate assumptions we prove well-posedness and continuous dependence of the solution on the data. We next address optimal control problems for point processes of general non-markovian type and show that BSDEs can be used to prove existence of an optimal control and to represent the value function. Finally we introduce a Hamilton-Jacobi-Bellman equation, also stochastic and of backward type, for this class of control problems: when the state space is finite or countable we show that it admits a unique solution which identifies the (random) value function and can be represented by means of the BSDEs introduced above.