Infinite horizon optimal control of mean-field delay system with semi-Markov modulated jump-diffusion processes

This paper describes the study of infinite horizon optimal control of stochastic delay differential equation with semi-Markov modulated jump-diffusion processes in which the control domain is not convex. In addition, the drift, diffusion, jump kernel term and cost functional are modulated by semi-Markov processes and expectation values of the state processes. Since the control domain is non-convex, the system exhibits non-guarantee to exist optimal control. Therefore, the concerned system is transformed into relaxed control model where the set of all relaxed controls forms a convex set, which gives the existence of optimal control. Further, stochastic maximum principle and necessary condition for optimality are established under convex perturbation technique for the relaxed model. Finally, an application of the theoretical study is shown by an example of portfolio optimization problem in financial market.