A generalized existence theorem of two barriers reflected backward doubly stochastic differential equation

We deal with double reflected backward doubly stochastic differential equations (DRBDSDEs) with a generalized growth generator f and a square integrable terminal datum. We introduce a new local conditions on the generator f where is left-Lipischitz in y and Lipischitz in z then we show that they ensure the existence of solutions. Our work go beyond the previous results on reflected backward doubly stochastic differential equations with two barriers. Although we are focused on double reflected case, our comparaison theorem is new for double reflected backward doubly stochastic differential equations too in the case where the generator f is Lipischitiz in y and z, we show that we can compare not only the components Y’s but also the non-decreasing processes K+’s of the solutions. As application, the reflected backward doubly stochastic differential equations with two barriers provide connections with the following obstacle problem of semilinear stochastic partial differential equations (SPDEs) with generalized growth gernerator.