# On the relaxed mean-field stochastic control problem

## TL;DR

This paper investigates relaxed control problems for mean-field stochastic differential equations, demonstrating the existence of optimal relaxed controls and conditions under which they are strict, with key insights into their mathematical structure.

## Contribution

It introduces a novel framework for relaxed controls governed by orthogonal martingale measures and proves the existence and approximation of optimal controls.

## Key findings

- Relaxed controls are governed by orthogonal martingale measures instead of Brownian motion.
- Existence of optimal relaxed controls is established.
- Under convexity, optimal controls are strict.

## Abstract

This paper is concerned with optimal control problems for systems governed by mean-field stochastic differential equation, in which the control enters both the drift and the diffusion coefficient. We prove that the relaxed state process, associated with measure valued controls, is governed by an orthogonal martingale measure rather that a Brownian motion. In particular, we show by a counter example that replacing the drift and diffusion coefficient by their relaxed counterparts does not define a true relaxed control problem. We establish the existence of an optimal relaxed control, which can be approximated by a sequence of strict controls. Moreover under some convexity conditions, we show that the optimal control is realized by a strict control.