Terminal Perturbation Methods For Backward Stochastic Volterra Integral Equations

As a generalization of backward stochastic differential equations(BSDEs), we consider the following backward stochastic Volterra integral equations (BSVIEs):Yong firstly introduced the adapted M-solution of the BSVIE(0-1) in [39]. He proved the well-posedness of this kind of equations. Moreover, using Malliavin calculus, some regularity properties for this M-solutions are given. As an appli-cation, he solved the optimal controls of stochastic Volterra integral equations by presenting a Pontryagin type maximum principle.In our paper, we introduce a kind of stochastic optimization problem of equation(0-1). Using the Terminal perturbation methods and Ekeland's varia-tional principle, we obtain the variational inequality. Moreover, we will prove a maximum principle for stochastic optimization problem of BSVIEs.