We consider an infinite horizon optimal control problem for a continuoustime Markov chain X in a finite set I with noise-free partial observation. The observation process is defined as Yt = h(Xt), t ≥ 0, where h is a given map defined on I. The observation is noise-free in the sense that the only source of randomness is the process X itself. The aim is to minimize a discounted cost functional and study the associated value function V . After transforming the control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we provide a link between the value function v associated to the latter control problem and the original value function V . Then, we present two different characterizations of v (and indirectly of V ): on one hand as the unique fixed point of a suitably defined contraction mapping and on the other hand as the unique constrained viscosity solution (in the sense of Soner) of a HJB integro-differential equation. Under suitable assumptions, we finally prove the existence of an optimal control. projects Applicazioni innovative dei processi di punto marcato and Problemi di controllo ottimo con osservazione parziale: applicazioni dei processi di punto marcato and by MIUR-PRIN 2015 project Deterministic and stochastic evolution equations.

We consider an infinite horizon optimal control problem for a pure jump Markov process X, taking values in a complete and separable metric space I, with noisefree partial observation. The observation process is defined as Yt = h(Xt), t ≥ 0, where h is a given map defined on I. The observation is noise-free in the sense that the only source of randomness is the process X itself. The aim is to minimize a discounted cost functional. In the first part of the paper we write down an explicit filtering equation and characterize the filtering process as a Piecewise Deterministic Process. In the second part, after transforming the original control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we prove the equivalence of the original and separated problems through an explicit formula linking their respective value functions. The value function of the separated problem is also characterized as the unique fixed point of a suitably defined contraction mapping.

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