In   this article, a class of backward stochastic Volterra integro-differential   equations (BSVIDEs) is introduced and studied. It is worthy mentioning that   the proposed BSVIDEs can not be covered by the existing backward stochastic   Volterra integral equations (BSVIEs), and they also have the nice flow   property such that Ito's formula becomes quite applicable. It is found that   BSVIDEs can provide a neat sufficient condition for the solvability of BSVIEs   with generator depending on the diagonal value of the solutions. As   applications, the optimal control problems in terms of maximum principles and   linear quadratic control problems of optimal control for forward stochastic   Volterra integro-differential equations (FSVIDE) are investigated. In   contrast with the BSVIEs in current literature, some interesting phenomena   and advantages of BSVIDEs are revealed.

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