Backward doubly SDEs and SPDEs with superlinear growth generators

Abstract

We deal with multidimensional backward doubly stochastic differential equations (BDSDEs) with a superlinear growth generator and a square integrable terminal datum. We introduce new local conditions on the generator and then show that they ensure the existence and uniqueness as well as the stability of solutions. Our work goes beyond the previous results on the subject. Although we are focused on multidimensional case, the uniqueness result we establish is new in one-dimensional too. As an application, we establish the existence and uniqueness of probabilistic solutions to some semilinear stochastic partial differential equations (SPDEs) with superlinear growth gernerator. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.