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Anticipated backward doubly stochastic differential equations. (English) Zbl 1329.60203
Summary: 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).

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
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