Zero-sum linear quadratic stochastic integral games and BSVIEs

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.Comment: 27 page

Nonzero Sum Differential Game of Mean-Field BSDEs with Jumps under Partial Information

This paper is concerned with a kind of nonzero sum differential game of mean-field backward stochastic differential equations with jump (MF-BSDEJ), in which the coefficient contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. It is required that the control is adapted to a subfiltration of the filtration generated by the underlying Brownian motion and Poisson random measure. We establish a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a partial information linear-quadratic (LQ) game