The Maximum Principle for Partially Observed Optimal Control of Stochastic Differential Equations

Abstract

This paper concerns partially observed optimal control of possibly degenerate stochastic differential equations, with correlated noises between the system and the observation. The control is allowed to enter into all the coefficients. A general maximum principle is proved for the partially observed optimal control, and the relations among the adjoint processes are established. Adjoint vector fields, which are adapted to the past and present observations, are introduced as the solutions to some backward stochastic partial differential equations (BSPDEs), and their relations are established. Under suitable conditions, the adjoint processes are characterized in terms of the adjoint vector fields, their differentials and Hessians, along the optimal state process. Some other formulations of the partially observed stochastic maximum principle are then derived.

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