A Second-Order Necessary Condition for Risk-Sensitive Mean-Field Type Control*

In this paper, we investigate the second-order necessary condition for risk-sensitive mean-field type control. A class of new split first-order adjoint equations is brought in, which demonstrates more superiorities in analysing the transformation of the second-order adjoint variables. By means of constructing a new transformation of the second-order adjoint variables, the second-order stochastic maximum principle of risk-sensitive mean-field type singular control is proved. Finally, an illustrative example is provided.