# Existence and optimality conditions for relaxed mean-field stochastic control problems

## TL;DR

This paper investigates optimal control problems for mean-field stochastic differential equations with controls in both drift and diffusion, establishing existence and optimality conditions for relaxed controls involving measure-valued processes and martingale measures.

## Contribution

It extends the strict control framework to relaxed controls, proving existence of optimal controls and deriving necessary optimality conditions in this more general setting.

## Key findings

- Existence of optimal relaxed controls for mean-field stochastic systems.
- Derivation of necessary optimality conditions involving adjoint processes.
- Extension of classical results to measure-valued control processes.

## Abstract

We consider optimal control problems for systems governed by mean-field stochastic differential equations, where the control enters both the drift and the diffusion coefficient. We study the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is driven by an orthogonal martingale measure, whose covariance measure is the relaxed control. This is a natural extension of the original strict control problem, for which we prove the existence of an optimal control. Then, we derive optimality necessary conditions for this problem, in terms of two adjoint processes extending the known results to the case of relaxed controls.