Backward Stochastic Differential Equations and Backward Stochastic Volterra Integral Equations with Anticipating Generators

TL;DR

This paper investigates backward stochastic Volterra integral equations (BSVIEs) with anticipating generators, establishing well-posedness under certain conditions and extending results to path-dependent cases, highlighting key differences from backward stochastic differential equations (BSDEs).

Contribution

It demonstrates the well-posedness of BSVIEs with anticipating generators and extends the analysis to path-dependent BSVIEs, revealing that anticipation cannot always be avoided.

Findings

Well-posedness of BSVIEs with anticipating generators is established.

Extension of results to path-dependent BSVIEs.

Anticipating generators are generally unavoidable in path-dependent BSVIEs.

Abstract

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