Backward stochastic differential equations and backward stochastic Volterra integral equations with anticipating generators

For a backward stochastic differential equation (BSDE, for short), when the generator is not progressively measurable, it might admit adapted solutions, shown by an example. However, Volterra integral equations (BSVIEs, generators are allowed to be anticipating. This gives, among other things, essential difference between BSDEs and BSVIEs. Under some proper conditions, well-posedness of such BSVIEs established. Further, results extended path-dependent BSVIEs, in which can depend on future paths unknown processes. An additional finding that general, situation anticipating avoidable, adaptedness condition similar imposed anticipated Peng−Yang [22] necessary.