Integro-differential optimality equations for the risk-sensitive control of piecewise deterministic Markov processes

• Published in: Mathematical methods of operations research (Heidelberg, Germany) Germany), 2021-04, Vol.93 (2), p.327-357
• Main Authors: Costa, O. L. V., Dufour, F.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Business and Management; Calculus of Variations and Optimal Control; Optimization; Differential equations; Iterative algorithms; Markov analysis; Markov processes; Mathematics; Mathematics and Statistics; Operations research; Operations Research/Decision Theory; Optimization; Optimization and Control; Original Article; Risk management
• ISSN: 1432-2994; 1432-5217
• DOI: 10.1007/s00186-020-00732-8

In this paper we study the minimization problem of the infinite-horizon expected exponential utility total cost for continuous-time piecewise deterministic Markov processes with the control acting continuously on the jump intensity λ and on the transition measure Q of the process. The action space is supposed to depend on the state variable and the state space is considered to have a frontier such that the process jumps whenever it touches this boundary. We characterize the optimal value function as the minimal solution of an integro-differential optimality equation satisfying some boundary conditions, as well as the existence of a deterministic stationary optimal policy. These results are obtained by using the so-called policy iteration algorithm, under some continuity and compactness assumptions on the parameters of the problem, as well as some non-explosive conditions for the process.