Two Barriers Reflected Backward Doubly Stochastic Differential Equation with generalized coefficients

Abstract

We study double reflected backward doubly stochastic differential equations (DRBDSDEs) with a generator fff of generalized growth and a square-integrable terminal condition. We introduce new local conditions on the generator f, assuming that it is left-Lipschitz continuous in the variable y and Lipschitz continuous in z. Under these assumptions, we establish the existence of solutions. Our results extend previous work on reflected backward doubly stochastic differential equations with two barriers. Although our main focus is on the double-reflected case, we also derive a new comparison theorem for DRBDSDEs. In particular, when the generator f is Lipschitz continuous in both y and z, we show that one can compare not only the Y-components of the solutions, but also the associated non-decreasing processes K+

As an application, we show that DRBDSDEs with two reflecting barriers are closely related to obstacle problems for semilinear stochastic partial differential equations (SPDEs) whose generators exhibit generalized growth.