Optimal Control of Partially Observable Diffusions

The problems considered are stochastic analogues of the problem of Lagrange in calculus of variations. The response to the control is assumed to be a diffusion process, and the controls admitted are based on partial observations of the current states of the response. The problem can then be phrased as one of optimally controlling the coefficients of linear second order parabolic equations. An existence theorem in the class of bounded, measurable controls and necessary conditions in terms of conditional expectations are obtained.