带跳的完全耦合正倒向随机系统的非零和随机微分对策的变分公式及其应用

A variational formula for non-zero sum stochastic differential game of fully coupled forward-backward stochastic systems with jumps and some applications

本文主要研究由Brown运动和Poisson随机鞅测度共同驱动的完全耦合的正倒向随机系统的开环双人非零和随机微分对策问题.利用Hamilton函数和相应的对偶方程直接获得了性能指标的一个变分公式,其中对偶方程是一个线性正倒向随机微分方程,并且对经典的状态过程和性能指标的变分计算及其相应的Taylor展开均不需要考虑.作为应用,利用获得的变分公式在一个统一的框架下证明了开环Nash均衡点存在的一个必要条件(随机最大值原理)和一个充分条件(验证定理).本文中系统的控制区域要求是非空凸集,而且所有对手的可允许控制允许同时出现在状态方程的漂移项、扩散项和跳跃项.

In this paper, an open-loop two-person non-zero sum stochastic differential game is investigated for fully coupled forward-backward stochastic system driven by a Brownian motion and a Poisson random measure.A variational formula for the cost functionals is obtained directly in terms of the Hamiltonian and the associated adjoint system which is a linear FBSDEs and neither the variational systems nor the corresponding Taylor type expansions of the state process and the cost functional will be considered. As an application, one necessary condition （a stochastic maximum principle） and one sufficient condition （a verification theorem） for the existence of open-loop Nash equilibrium points are proved by the variation formula obtained in a unified way. The control domain need to be convex and the admissible controls for both players are allowed to appear in both the drift and diffusion of the state equations.