Stability for multivalued McKean-Vlasov stochastic differential equations

TL;DR

This paper establishes existence, uniqueness, and stability results for multivalued McKean-Vlasov stochastic differential equations, extending classical Itô's formula and analyzing solution stability using Lyapunov functions.

Contribution

It introduces new existence and uniqueness results for multivalued McKean-Vlasov SDEs with non-Lipschitz coefficients and extends Itô's formula to this setting.

Findings

Proved existence and uniqueness of strong solutions.

Extended Itô's formula to multivalued McKean-Vlasov SDEs.

Demonstrated asymptotic stability of solutions using Lyapunov functions.

Abstract

The work concerns multivalued McKean-Vlasov stochastic differential equations. First of all, we prove the existence and uniqueness of strong solutions for multivalued McKean-Vlasov stochastic differential equations with non-Lipschitz coefficients. Then, the classical It\^{o}'s formula is extended to that for multivalued McKean-Vlasov stochastic differential equations. Finally, the asymptotic stability of second moments and the almost surely asymptotic stability for their solutions in terms of a Lyapunov function are shown.