# Mean-field optimal control problem of SDDEs driven by fractional Brownian motion

## TL;DR

This paper develops a framework for solving mean-field optimal control problems involving SDDEs driven by fractional Brownian motion with Hurst parameter greater than 0.5, using fractional White noise calculus and measure differentiation.

## Contribution

It introduces necessary and sufficient stochastic maximum principles for such control problems, overcoming the limitations of classical methods.

## Key findings

- Derived stochastic maximum principles for fractional Brownian motion-driven SDDEs.
- Solved an optimal consumption problem with delay.
- Addressed a linear-quadratic control problem with delay.

## Abstract

We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than one half. Stochastic optimal control problems driven by fractional Brownian motion can not be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional White noise calculus combined with some special tools related to the differentiation for functions of measures, we establish and prove necessary and sufficient stochastic maximum principles. To illustrate our study, we consider two applications: we solve a problem of optimal consumption from a cash flow with delay and a linear-quadratique (LQ) problem with delay.