Adaptive stochastic filtering

We reconsider the problem of controlling the solution of a stochastic DE whose drift depends on an unknown r.v. of given a priori distribution, with a view to minimizing a cost functional of standard type (final value plus running cost), and with control adapted to the solution itself or to noisy observations. This kind of problem goes back to A.A. Feldbaum, and has been considered by J.-M. Bismut, A. Bensoussan, and R. Rishel. This previous work was based on variational methods, and led to a formulation of the minimization problem in terms of an adjoint equation. Here, for various examples, we try directly to arrive at "backward" functional equations necessarily linking optimal controls with observations.