A risk-sensitive maximum principle: the case of imperfect state observation

The risk-sensitive maximum principle for optimal stochastic control derived by the author in an earlier work (System Control Letters, vol.15, 1990) is restated. This is an immediate generalization of the classic Pontryagin principle, to which it reduces in the deterministic case, and is expressed immediately in terms of observables. It is derived on the assumption that the criterion function is the exponential of an additive cost function, and is exact under linear-quadratic Gaussian assumptions, but is otherwise valid as a large deviation approximation. The principle is extended to the case of imperfect state observation after preliminary establishment of a certainty-equivalence principle. The derivation yields as byproduct a large-deviation version of the updating equation for nonlinear filtering. The development is heuristic. It is believed that the mathematical arguments given are the essential ones, and provide a self-contained treatment at this level