Infinite horizon mean-field type relaxed optimal control with Lévy processes

Abstract

This paper investigates the problem of infinite horizon optimal control of stochastic differential equation with Teugels martingales associated with Lévy processes under non convex control domain. Proposed model of mean-field dynamical system is considered with the expectation values of state processes which are included explicitly in drift, diffusion, jump kernel and cost functional terms. In general, the assumption of non convex control domain does not guarantee the existence of optimal control. Therefore, the concerned system is transformed into relaxed control model, where set of all relaxed controls forms a convex set and exhibits the existence of optimal control. Moreover, stochastic maximum principle and necessary condition for optimality are established under convex perturbation technique for the proposed relaxed model. Finally, an application of the theoretical study is demonstrated by an example of portfolio optimization problem in financial market.