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In this paper existence of solutions to the Bellman equation corresponding to risk sensitive control of partially observed discrete time Markov processes is shown; this in turn leads to the existence of optimal strategies. The method used in the paper is based on discounted risk sensitive approximation

The paper deals with a possible approach to the problem of finite-dimensional filters in the nonlinear case, when the signal is a diffusion process and the observations are corrupted by additive white noise. The approach considers a sequence of finite-dimensional recursive filters that approximate the actual optimal one. The approximating filters are given in terms of functionals of continuous-time Markov chains that converge weakly to the original diffusion. These functionals can be recursively computed via a finite-dimensional Zakai equation, for which the solution is given in terms of a robust input-output relation.