On the stochastic maximum principle for manifold-valued processes

The aim of this paper is to obtain a stochastic maximum principle (MP) for controlled processes taking values in a compact manifold. The process is specified by its generator, which is that of a nondegenerate diffusion with control appearing in the drift term. Feedback controls based on complete observations are used. The process is constructed using the "horizontal lifting" technique of Eells and Elworthy and the MP is derived from the MP for controlled stochastic differential equations which is formulated in the first part of the paper. The evolution of the "adjoint variable", which here is a 1-form valued process, is shown to be related to an intrinsic operator on the manifold, namely the Laplacian of de Rham-Kodaira.