The Stochastic Maximum Principle for Risk-Sensitive Optimal Control with Delay and Applications

In this paper, we consider risk-sensitive optimal control for stochastic differential delayed equations. We use the logarithmic transformation of the associated risk-neutral problem to obtain the risk-sensitive maximum principle with delay, which is a necessary condition for optimality. We show that the risk-sensitive maximum principle with delay is characterized in terms of the variational inequality and the coupled anticipated backward stochastic differential equations (ABSDEs). The coupled ABSDEs consist of the first-order adjoint equation and an additional scalar ABSDE, where the latter is induced due to the non-smooth nonlinear transformation of the adjoint process of the associated risk-neutral problem. We also show that under an additional condition, the corresponding risk-sensitive maximum principle becomes sufficient. For applications, we consider the risk-sensitive linear-quadratic control problem with delay and the risk-sensitive optimal consumption problem with delay, for which we obtain the explicit optimal solutions.