Maximum principle for forward-backward stochastic control system with random jumps and applications to finance

Both necessary and sufficient maximum principles for optimal control of stochastic system with random jumps consisting of forward and backward state variables are proved. The control variable is allowed to enter both diffusion and jump coefficients. The result is applied to a mean-variance portfolio selection mixed with a recursive utility functional optimization problem. Explicit expression of the optimal portfolio selection strategy is obtained in the state feedback form.