Risk-sensitive stochastic maximum principle for forward-backward systems involving impulse controls

In this article, we study a risk-sensitive stochastic optimal control problem driven by forward-backward systems in which the control variable consists of two components: the continuous control and the impulse control. The control domain is assumed to be convex. We establish the maximum principle (i.e., necessary condition) for this kind of control problem. Under some additional assumptions, the necessary optimality conditions turn out to be sufficient. To explain the theoretical results, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and the optimal control is obtained.