A Mean-field Stochastic Control Problem with Partial Observations

TL;DR

This paper introduces a new class of mean-field stochastic control problems with partial observations, addressing path-dependence and nonlinearity in the conditional law, and establishes foundational theoretical results including well-posedness and optimality conditions.

Contribution

It develops a novel framework for mean-field control with partial observations, proving well-posedness and deriving a Pontryagin maximum principle with new mean-field BSDEs.

Findings

Established well-posedness of the state-observation dynamics.

Proved the Pontryagin stochastic maximum principle for the problem.

Characterized the adjoint equations as new mean-field BSDEs.

Abstract

In this paper we are interested in a new type of {\it 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.