The obstacle problem of integro-partial differential equations with applications to stochastic optimal control/stopping problem

Abstract

This paper is devoted to existence and uniqueness of minimal mild super solutions to the obstacle problem governed by integro-partial differential equations. We first study the well-posedness and local Lipschitz regularity of Lp solutions (p ≥ 2) to reflected forward-backward stochastic differential equations (FBSDEs) with jump and lower barrier. Then we show that the solutions to reflected FBSDEs provide a probabilistic representation for the mild super solution via a nonlinear Feynman–Kac formula. Finally, we apply the results to study stochastic optimal control/stopping problems.