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A mean-field stochastic control problem with partial observations. (English) Zbl 1380.93282

Summary: In this paper, we are interested in a new type of mean-field, non-Markovian stochastic control problems with partial observations. More precisely, we assume that the coefficients of the controlled dynamics depend not only on the paths of the state, but also on the conditional law of the state, given the observation to date. Our problem is strongly motivated by the recent study of the mean field games and the related McKean-Vlasov stochastic control problem, but with added aspects of path-dependence and partial observation. We shall first investigate the well-posedness of the state-observation dynamics, with combined reference probability measure arguments in nonlinear filtering theory and the Schauder fixed-point theorem. We then study the stochastic control problem with a partially observable system in which the conditional law appears nonlinearly in both the coefficients of the system and cost function. As a consequence, the control problem is intrinsically “time-inconsistent”, and we prove that the Pontryagin stochastic maximum principle holds in this case and characterize the adjoint equations, which turn out to be a new form of mean-field type BSDEs.

MSC:

93E20 Optimal stochastic control
49K45 Optimality conditions for problems involving randomness
93E11 Filtering in stochastic control theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
49K35 Optimality conditions for minimax problems
47N70 Applications of operator theory in systems, signals, circuits, and control theory
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