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.
