Risk-sensitive Necessary and Sufficient Optimality Conditions and Financial Applications: Fully Coupled Forward-Backward Stochastic Differential Equations with Jump diffusion

Throughout this paper, we focused our aim on the problem of optimal control under a risk-sensitive performance functional, where the system is given by a fully coupled forward-backward stochastic differential equation with jump. The risk neutral control system has been used as preliminary step, where the admissible controls are convex, and the optimal solution exists. The necessary as well as sufficient optimality conditions for risk-sensitive performance are proved. At the end of this work, we illustrate our main result by giving an example of mean-variance for risk sensitive control problem applied in cash flow market.