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In this paper, we study extended backward stochastic Volterra integralequations (EBSVIEs, for short). We establish the well-posedness under weakerassumptions than the literature, and prove a new kind of regularity propertyfor the solutions. As an application, we investigate, in the open-loopframework, a time-inconsistent stochastic recursive control problem where thecost functional is defined by the solution to a backward stochastic Volterraintegral equation (BSVIE, for short). We show that the corresponding adjointequations become EBSVIEs, and provide a necessary and sufficient condition foran open-loop equilibrium control via variational methods.

Extended backward stochastic Volterra integral equations and their applications to time-inconsistent stochastic recursive control problems