  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.
