A General Maximum Principle for Partially Observed Risk-Sensitive Optimal Control Problem of Mean-Field Type

We study a general maximum prinicple for a kind of partially observed risk-sensitive optimal control problem of mean-field type, where the system contains the mathematical expectation of state. With the use of Girsanov's theorem and the spike variational, we are able to derive a general maximal principle for the aforementioned problem. Both the first-and second-order adjoint equations are mean-field backward stochastic defferential equations (BSDEs). Moreover, a risk parameter $\kappa$ has a significant impact on adjoint equations and variational inequalities.