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This paper formulates and studies a linear quadratic (LQ for short) gameproblem governed by linear stochastic Volterra integral equation. Sufficientand necessary condition of the existence of saddle points for this problem arederived. 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 costfunctional and the coefficients are allowed to be random. Secondly we study theunique solvability for certain coupled forward-backward stochastic Volterraintegral equations (FBSVIEs for short) involved in this game problem. Tocharacterize the condition aforementioned explicitly, some other useful tools,such as backward stochastic Fredholm-Volterra integral equations (BSFVIEs forshort) and stochastic Fredholm integral equations (FSVIEs for short) areintroduced. Some relations between them are investigated. As a application, alinear quadratic stochastic differential game with finite delay in the statevariable and control variables is studied.

Zero-sum linear quadratic stochastic integral games and BSVIEs