Necessary condition for optimal control of fully coupled forward-backward stochastic system with random jumps

One kind of fully coupled forward-backward stochastic control system with random jumps is considered here. Necessary condition of Pontraygin´s type maximum principle for the optimal control is derived. The control domain is not assumed to be convex and the control variable appears neither in the diffusion nor the jump coefficient of the forward equation. A linear quadratic stochastic optimal control problem is discussed as an illustrate example.