Stochastic Optimal Control Of Systems Driven By Stochastic Differential Equations Of Mean-Field Type With Irregular Drift Coefficients

Abstract

The main objective of this work is to maximize a performance functional subjected to a controlled stochastic di erential equation of mean- eld type using the stochastic maximum principle approach. The controlled mean- eld stochastic di erential equation has a non smooth drift and is driven by a one dimensional Brownian motion. We started by rst showing that, considering a corresponding sequence of mean- eld stochastic di erential equations with a smooth drift coe cient, the corresponding sequence of solutions will converge to the solution of the mean- eld stochastic di erential equation. We study the representation of the stochastic (Sobolev) di erential ow, via a time-space local time integration argument. Lastly, we look at a control problem where the state process follows the dynamics of a mean- eld stochastic di erential equation. Since the drift coe cient is non smooth, we characterize the optimal control through an approximate performance functional which is derived using the Ekeland's variational principle. Afterwards, we pass to the limit and prove convergence of the stochastic maximum principle.