Risk-sensitive maximum principle for stochastic optimal control of mean-field type Markov regime-switching jump-diffusion systems

We consider the risk-sensitive optimal control problem for mean-field type Markov regime-switching jump-diffusion systems driven by Brownian motions and Poisson jumps with (Markovian) switching coefficients. The system is coupled with its mean-filed term, that is, the expected value of the state process, and the objective functional is of the risk-sensitive type. Our problem is closely related to the mean-field type robust optimization problem for a general class of stochastic jump systems due to the inherent feature of the risk-sensitive objective functional. By establishing the logarithmic transformations of the associated equivalent singular risk-neutral control problem, we obtain the risk-sensitive maximum principle type necessary and sufficient conditions for optimality, where the sufficient condition requires an additional convexity assumption. The risk-sensitive maximum principle in this article is characterized as the variational inequality, together with the first- and second-order (mean-field type) adjoint processes as well as the auxiliary first-order adjoint process. Unlike the risk-neutral and mean-field free cases, the additional adjoint equation is induced due to the mean-field coupling term and the risk-sensitive logarithmic transformation. We apply the risk-sensitive maximum principle of this article to the risk-sensitive linear-quadratic problem, for which an explicit optimal solution is obtained.