Backward doubly stochastic Volterra integral equations and their applications

In this paper, we introduce a new class of equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). First, the well-posedness of BDSVIEs in the sense of introduced M-solutions is established. Second, a comparison theorem of BDSVIEs is proved. As an application of the comparison theorem, we derive the existence of solutions of BDSVIEs with continuous coefficients. Third, a duality principle between linear forward doubly stochastic Volterra integral equations (FDSVIEs, for short) and BDSVIEs is obtained. Moreover, by virtue of the duality principle, a maximum principle of Pontryagin type is established for an optimal control problem of FDSVIEs.;In this paper, we introduce a new class of equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). First, the well-posedness of BDSVIEs in the sense of introduced M-solutions is established. Second, a comparison theorem of BDSVIEs is proved. As an application of the comparison theorem, we derive the existence of solutions of BDSVIEs with continuous coefficients. Third, a duality principle between linear forward doubly stochastic Volterra integral equations (FDSVIEs, for short) and BDSVIEs is obtained. Moreover, by virtue of the duality principle, a maximum principle of Pontryagin type is established for an optimal control problem of FDSVIEs.