Optimal stopping of marked point processes and reflected backward stochastic differential equations

TL;DR

This paper introduces a class of reflected backward stochastic differential equations driven by marked point processes and Brownian motion, establishing existence, uniqueness, and their application to optimal stopping problems.

Contribution

It develops a new framework for RBSDEs driven by MPPs with non-explosive jumps, linking them to optimal stopping strategies.

Findings

Existence and uniqueness of solutions under certain conditions

Representation of the value function for optimal stopping

Characterization of optimal strategies

Abstract

We define a class of reflected backward stochastic differential equation (RBSDE) driven by a marked point process (MPP) and a Brownian motion, where the solution is constrained to stay above a given c\`adl\`ag process. The MPP is only required to be non-explosive and to have totally inaccessible jumps. Under suitable assumptions on the coefficients we obtain existence and uniqueness of the solution, using the Snell envelope theory. We use the equation to represent the value function of an optimal stopping problem, and we characterize the optimal strategy. Keywords: reflected backward stochastic differential equations, optimal stopping, marked point processes.