A stochastic maximum principle for risk-sensitive mean-field-type control

In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle for optimal control of stochastic differential equations of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. For a general action space a Peng's type stochastic maximum principle is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type under linear stochastic dynamics with exponential quadratic cost function. Explicit characterizations are given for both mean-field free and mean-field risk-sensitive models.