Mean field backward doubly stochastic Volterra integral equations and their applications

Abstract

This paper deals with a class of mean field backward doubly stochastic Volterra integral equations (MFBDSVIEs, for short). Firstly, we prove the existence and uniqueness results in the sense of M-solutions for MFBDSVIEs as well as a comparison theorem is obtained. Based on comparison theorem, we will study the existence of minimal solution with linear growth condition. Secondly, two duality principles between linear mean field forward doubly stochastic Volterra integral equations (MFFDSVIEs, for short) and MFBDSVIEs are derived. As an application for the duality principles, we show a maximum principle of Pontryagin type for an optimal control problem of MFFDSVIEs.