Doubly reflected BSDEs driven by a Lévy process

In this paper, we show the existence and uniqueness of the solution for a class of doubly reflected backward stochastic differential equations driven by a Lévy process (DRBSDELs in short) by means of the penalization method as well as the fixed point theorem. In addition, we obtain the comparison theorem for the solutions of DRBSDELs. As an application, we give a probabilistic formula for the viscosity solution of a class of partial differential–integral equations (PDIEs in short) with two obstacles.