Irregular barrier reflected BSDEs driven by a Lévy process

Abstract

We consider reflected backward stochastic differential equations driven by Teugels martingales associated with a Lévy process, in which the barrier process is optional with regulated trajectories (i.e., trajectories with left and right finite limits), which is assumed to be right upper semi-continuous. We prove the existence and uniqueness of such equations by using the predictable representations for Lévy processes due to Nualart and Schoutens, and some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales. We also discuss the case where the barrier is assumed to be completely irregular, and we establish an infinitesimal characterization of the solution in terms of a value process to an extension of the optimal stopping problem.