Stochastic Maximum Principle for Mean-Field Type Optimal Control Under Partial Information

This technical note is concerned with a partially observed optimal control problem, whose novel feature is that the cost functional is of mean-field type. Hence determining the optimal control is time inconsistent in the sense that Bellman's dynamic programming principle does not hold. A maximum principle is established using Girsanov's theorem and convex variation. Some nonlinear filtering results for backward stochastic differential equations (BSDEs) are developed by expressing the solutions of the BSDEs as some Ito's processes. An illustrative example is demonstrated in terms of the maximum principle and the filtering.