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

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} 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.