Necessary and sufficient conditions of risk-sensitive optimal control and differential games for stochastic differential delayed equations

In this paper, we consider risk-sensitive optimal control and differential games for stochastic differential delayed equations driven by Brownian motion. The problems are related to robust stochastic optimization with delay due to the inherent feature of the risk-sensitive objective functional. For both problems, by using the logarithmic transformation of the associated risk-neutral problem, the necessary and sufficient conditions for the risk-sensitive maximum principle are obtained. We show that these conditions are 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 nonsmooth nonlinear transformation of the adjoint process of the associated risk-neutral problem. For applications, we consider the risk-sensitive linear-quadratic control and game problems with delay, and the optimal consumption and production game, for which we obtain explicit optimal solutions.