Mean-field SDE driven by a fractional Brownian motion and related stochastic control problem

TL;DR

This paper investigates mean-field stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2, deriving a Pontryagin maximum principle and establishing conditions for optimal control.

Contribution

It introduces a Pontryagin maximum principle for mean-field SDEs driven by fractional Brownian motion and proves sufficiency of the necessary conditions under generalized assumptions.

Findings

Derived a Pontryagin maximum principle for the problem.

Established the associated backward stochastic differential equation.

Proved sufficiency of the optimality conditions under certain assumptions.

Abstract

We study a class of mean-field stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ and a related stochastic control problem. We derive a Pontryagin type maximum principle and the associated adjoint mean-field backward stochastic differential equation driven by a classical Brownian motion, and we prove that under certain assumptions, which generalise the classical ones, the necessary condition for the optimality of an admissible control is also sufficient.