# Mean-field FBSDE and optimal control

## TL;DR

This paper develops a stochastic maximum principle for optimal control problems involving mean-field forward-backward stochastic differential equations, and demonstrates its application in a mean-field risk minimization portfolio problem.

## Contribution

It introduces a new stochastic maximum principle for mean-field FBSDEs and applies it to solve a portfolio optimization problem with mean-field risk.

## Key findings

- Derived necessary and sufficient optimality conditions.
- Solved a portfolio risk minimization problem using the maximum principle.
- Demonstrated the applicability of the theoretical results.

## Abstract

We study optimal control for mean-field forward backward stochastic differential equations with payoff functionals of mean-field type. Sufficient and necessary optimality conditions in terms of a stochastic maximum principle are derived. As an illustration, we solve an optimal portfolio with mean-field risk minimization problem.