Volume 26, 2020
|Number of page(s)||47|
|Published online||03 March 2020|
Stochastic filtering and optimal control of pure jump Markov processes with noise-free partial observation*
Department of Statistics and Quantitative Methods, University of Milano-Bicocca,
via Bicocca degli Arcimboldi 8,
** Corresponding author: firstname.lastname@example.org
Accepted: 2 April 2019
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 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. 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.
Mathematics Subject Classification: 93E11 / 93E20 / 60J25 / 60J75
Key words: Stochastic filtering / partial observation control problem / pure jump processes / piecewise-deterministic Markov processes / Markov decision processes
© EDP Sciences, SMAI 2020
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