Well-posedness of reflected BSDEs with default time and irregular barrier: An application to optimal control

TL;DR

This paper establishes existence and uniqueness for reflected BSDEs with default time and irregular barriers, and links solutions to optimal stopping problems under certain conditions.

Contribution

It introduces a modified penalization method for proving well-posedness and characterizes solutions as value functions of optimal stopping problems.

Findings

Existence and uniqueness of solutions under Lipschitz conditions.

Characterization of solutions as optimal stopping value functions.

Extension to irregular barriers with semi-continuity assumptions.

Abstract

We consider a reflected backward stochastic differential equations with default time and an optional barrier in a filtration generated by a one-dimensional Brownian motion and a defaultable process. We suppose that the barrier have trajectories with left and right finite limits. We provide the existence and uniqueness result when the coefficient is scholastic Lipschitz by using a modified penalization method. Under an additional assumption of right-upper semi-continuity along stopping times on the trajectories of the barrier, we characterize the state process for such RBSDEs as the value function of an optimal stopping problem associated with a non-linear $f$-expectation.