Mean-Field Stochastic Control with Elephant Memory in Finite and Infinite Time Horizon

Nacira Agram, Bernt Øksendal

Published 2018-04-26Version 1

Our purpose of this paper is to study stochastic control problem for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history in both cases, finite and infinite time horizon. - In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem. - For infinite horizon, we derive sufficient and necessary maximum principles. As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.