Backward stochastic Volterra integral equations and some related problems

Backward stochastic Volterra integral equations (BSVIEs, for short) are introduced. The existence and uniqueness of adapted solutions are established. A duality principle between linear BSVIEs and (forward) stochastic Volterra integral equations is obtained. As applications of the duality principle, a comparison theorem is proved for the adapted solutions of BSVIEs, and a Pontryagin type maximum principle is established for an optimal control of stochastic integral equations.