Optimal switching problem for marked point process and systems of reflected BSDE

TL;DR

This paper addresses an optimal switching problem driven by a marked point process and Brownian motion, introducing a system of reflected BSDEs to characterize the value function under a family of probability measures.

Contribution

It formulates a novel optimal switching framework with marked point processes, establishing well-posedness of a coupled reflected BSDE system and linking it to the value function.

Findings

Proved well-posedness of the reflected BSDE system using Picard iteration.

Established a comparison theorem for BSDEs driven by marked point processes and Brownian motion.

Represented the optimal value function via the solution to the reflected BSDE system.

Abstract

We formulate an optimal switching problem when the underlying filtration is generated by a marked point process and a Brownian motion. Each mode is characterized by a different compensator for the point process, and thus by a different probability $\mathbb{P}^i$, which form a dominated family. To each strategy $\mathbf{a}$ of switching times and actions then corresponds a compensator and a probability $\mathbb{P}^\mathbf{a}$, and the reward is calculated under this probability. To solve this problem, we define and study a system of reflected BSDE where the obstacle for each equation depends on the solution to the others. The main assumption is that the point process is non explosive and quasi-left continuous. We prove wellposedness of this system through a Picard iteration method, and then use it to represent the optimal value function of the switching problem. We also obtain a comparison theorem for BSDE driven by marked point process and Brownian motion. Keywords: reflected backward stochastic differential equations, optimal stopping, optimal switching, marked point processes.