Anticipated backward doubly stochastic differential equations

In this paper, we deal with a new type of differential equations called anticipated backward doubly stochastic differential equations (anticipated BDSDEs). The coefficients of these BDSDEs depend on the future value of the solution (Y,Z). We obtain the existence and uniqueness theorem and a comparison theorem for the solutions of these equations. Besides, as an application, we also establish a duality between the anticipated BDSDEs and the delayed doubly stochastic differential equations (delayed DSDEs).