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Backward doubly stochastic Volterra integral equations and applications to optimal control problems
Yufeng Shi, Jiaqiang Wen, Jie Xiong · 2019-06-25 · via math.PR updates on arXiv.org

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

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