# Global maximum principle for mean-field forward-backward stochastic systems with delay and application to finance

## TL;DR

This paper establishes a new maximum principle for optimal control of mean-field forward-backward stochastic systems with delay, providing tools for financial applications and extending classical results to delay systems.

## Contribution

It introduces a novel estimate and develops a maximum principle for mean-field systems with delay, including anticipated backward equations and matrix-valued adjoint systems.

## Key findings

- Derived a new estimate for delay control systems.
- Established a stochastic maximum principle for mean-field systems with delay.
- Applied the theory to a financial mean-field game example.

## Abstract

The purpose of this paper is to explore the necessary conditions for optimality of mean-field forward-backward delay control systems. A new estimate is proved, which is a powerfultool to deal with the optimal control problems of mean-field type with delay. Different from the classical situation, in our case the first-order adjoint system is an anticipated mean-field backward stochastic differential equation, and the second-order adjoint system is a system of matrix-valued process, not mean-field type.With the help of two adjoint systems, the second-order expansion of the variation of the state $Y$ is proved, and therewith the Peng's stochastic maximum principle. As an illustrative example, we apply our result to the mean-field game in Finance. Although we just investigate the case of one pointwise delay for convenience, but our method is adequate for analysing the case of pointwise delay.