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 Itô´s processes. An illustrative example is demonstrated in terms of the maximum principle and the filtering.