Anticipated backward stochastic Volterra integral equations and their applications to nonzero-sum stochastic differential games

TL;DR

This paper develops a comprehensive theory for anticipated backward stochastic Volterra integral equations (BSVIEs), including well-posedness and comparison theorems, and applies these to derive a maximum principle for nonzero-sum stochastic differential games involving delay Volterra integral equations.

Contribution

It introduces a more general class of anticipated BSVIEs with new theoretical results and applies them to establish a maximum principle for complex stochastic differential games.

Findings

Established well-posedness and comparison theorems for anticipated BSVIEs.

Proved regularity results of solutions using Malliavin calculus.

Derived a maximum principle for nonzero-sum stochastic differential games.

Abstract

In [J. Wen, Y. Shi, Stat. Probab. Lett. 156 (2020) 108599] the authors first introduced a kind of anticipated backward stochastic Volterra integral equations (anticipated BSVIEs, for short). By virtue of the duality principle, it is found in this paper that the anticipated BSVIEs can be applied to the study of stochastic differential games. Naturally, in order to develop the related theories and applications of BSVIEs, in this paper we deeply investigate a more general class of anticipated BSVIEs whose generator includes both pointwise and average time-advanced functions. In theory, the well-posedness and the comparison theorem of anticipated BSVIEs are established, and some regularity results of adapted M-solutions are proved by applying Malliavin calculus, which cover the previous results for BSVIEs. Further, using linear anticipated BSVIEs as the adjoint equation, we present the maximum principle for the nonzero-sum differential game system of stochastic delay Volterra integral equations (SDVIEs, for short) for the first time. As one of the applications of the principle, a Nash equilibrium point of the linear-quadratic differential game problem of SDVIEs is obtained.