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Backward doubly stochastic Volterra integral equations (BDSVIEs, for short)are introduced and studied systematically. Well-posedness of BDSVIEs in thesense of introduced M-solutions is established. A comparison theorem forBDSVIEs is proved. By virtue of the comparison theorem, we derive the existenceof solutions for BDSVIEs with continuous coefficients. Furthermore, a dualityprinciple between linear (forward) doubly stochastic Volterra integral equation(FDSVIE, for short) and BDSVIE is obtained. A Pontryagin type maximum principleis also established for an optimal control problem of FDSVIEs.

Backward doubly stochastic Volterra integral equations and applications to optimal control problems