Journal
MATHEMATICS OF CONTROL SIGNALS AND SYSTEMS
Volume 29, Issue 1, Pages -Publisher
SPRINGER LONDON LTD
DOI: 10.1007/s00498-016-0181-6
Keywords
Backward stochastic differential equations; Optimal control problems; Semi-Markov processes; Marked point processes
Categories
Funding
- GNAMPA Project Controllo ottimo di processi di punto e controllo impulsivo: il metodo dell'equazioni retrograde con salti
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In the present work, we employ backward stochastic differential equations (BSDEs) to study the optimal control problem of semi-Markov processes on a finite horizon, with general state and action spaces. More precisely, we prove that the value function and the optimal control law can be represented by means of the solution of a class of BSDEs driven by a semi-Markov process or, equivalently, by the associated random measure. We also introduce a suitable Hamilton-Jacobi-Bellman (HJB) equation. With respect to the pure jump Markov framework, the HJB equation in the semi-Markov case is characterized by an additional differential term. a. Taking into account the particular structure of semi-Markov processes, we rewrite the HJB equation in a suitable integral form which involves a directional derivative operator D related to. a. Then, using a formula of Ito type tailor-made for semi-Markov processes and the operator D, we are able to prove that a BSDE of the above-mentioned type provides the unique classical solution to the HJB equation, which identifies the value function of our control problem.
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