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

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.