Non-Markovian optimal stopping problems and constrained BSDEs with jump

TL;DR

This paper introduces a novel representation of non-Markovian optimal stopping problems using constrained backward stochastic differential equations with jumps, linking it to an auxiliary control problem for the point process intensity.

Contribution

It provides an alternative BSDE formulation involving a jump process and a sign-constrained process, clarifying its relation to classical reflected BSDEs and connecting to a controlled intensity optimization.

Findings

Equivalent value representation via constrained BSDEs with jumps

Connection established between the new BSDE and classical reflected BSDEs

Optimal stopping value matches an auxiliary control problem with controlled jump intensity

Abstract

We consider a non-Markovian optimal stopping problem on finite horizon. We prove that the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a stochastic integral having a one-jump point process as integrator and an (unknown) process with a sign constraint as integrand. This provides an alternative representation with respect to the classical one given by a reflected BSDE. The connection between the two BSDEs is also clarified. Finally, we prove that the value of the optimal stopping problem is the same as the value of an auxiliary optimization problem where the intensity of the point process is controlled.