Zero-sum linear quadratic stochastic integral games and BSVIEs

TL;DR

This paper develops a comprehensive framework for zero-sum linear quadratic stochastic integral games governed by Volterra equations, providing conditions for saddle points, solving open problems, and exploring connections with backward stochastic Volterra equations.

Contribution

It introduces new conditions for saddle points in LQ stochastic Volterra games, allowing random coefficients and terminal terms, and studies the solvability of associated FBSVIEs.

Findings

Derived necessary and sufficient conditions for saddle points.

Solved open problems from Chen and Yong.

Established relations between FBSVIEs, BSFVIEs, and FSVIEs.

Abstract

This paper formulates and studies a linear quadratic (LQ for short) game problem governed by linear stochastic Volterra integral equation. Sufficient and necessary condition of the existence of saddle points for this problem are derived. As a consequence we solve the problems left by Chen and Yong in [3]. Firstly, in our framework, the term GX^2(T) is allowed to be appear in the cost functional and the coefficients are allowed to be random. Secondly we study the unique solvability for certain coupled forward-backward stochastic Volterra integral equations (FBSVIEs for short) involved in this game problem. To characterize the condition aforementioned explicitly, some other useful tools, such as backward stochastic Fredholm-Volterra integral equations (BSFVIEs for short) and stochastic Fredholm integral equations (FSVIEs for short) are introduced. Some relations between them are investigated. As a application, a linear quadratic stochastic differential game with finite delay in the state variable and control variables is studied.