# Stability of Mc Kean-Vlasov stochastic differential equations and applications

## TL;DR

This paper investigates the existence, uniqueness, and stability of solutions to McKean-Vlasov stochastic differential equations, and explores their applications in mean-field control and approximation of relaxed controls by strict controls.

## Contribution

It extends stability and existence results for MVSDEs under Osgood conditions and demonstrates the equivalence of relaxed and strict control problems in this context.

## Key findings

- Established stability properties with respect to initial data and coefficients.
- Proved existence and uniqueness of solutions under weaker conditions.
- Showed relaxed and strict control problems have the same value function.

## Abstract

We consider Mc Kean-Vlasov stochastic differential equations (MVSDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. This type of SDEs was studied in statistical physics and represents the natural setting for stochastic mean-field games. We will first discuss questions of existence and uniqueness of solutions under an Osgood type condition improving the well known Lipschitz case. Then we derive various stability properties with respect to initial data, coefficients and driving processes, generalizing known results for classical SDEs. Finally, we establish a result on the approximation of the solution of a MVSDE associated to a relaxed control by the solutions of the same equation associated to strict controls. As a consequence, we show that the relaxed and strict control problems have the same value function. This last property improves known results proved for a special class of MVSDEs, where the dependence on the distribution was made via a linear functional. Key words: Mc Kean-Vlasov stochastic differential equation -- Stability -- Martingale measure - Wasserstein metric -- Existence -- Mean-field control -- Relaxed control.