Stochastic averaging principle and stability for multi-valued McKean-Vlasov stochastic differential equations with jumps

TL;DR

This paper establishes a stochastic averaging principle and stability results for multi-valued McKean-Vlasov stochastic differential equations with jumps, extending classical Itô's formula and analyzing solution stability.

Contribution

It introduces an averaging principle for complex McKean-Vlasov equations with jumps and extends Itô's formula to this setting, providing stability analysis tools.

Findings

Solutions can be approximated by averaged equations in mean square sense.

Extended Itô's formula applicable to multi-valued McKean-Vlasov equations with jumps.

Proved exponential stability and boundedness of solutions using Lyapunov functions.

Abstract

In this paper, we consider the stochastic averaging principle and stability for multi-valued McKean-Vlasov stochastic differential equations with jumps. First, under certain averaging conditions, we are able to show that the solutions of the equations concerned can be approximated by solutions of the associated averaged multi-valued McKean-Vlasov stochastic differential equations with jumps in the sense of the mean square convergence. Second, we extend the classical It\^{o}'s formula from stochastic differential equations to multi-valued McKean-Vlasov stochastic differential equations with jumps. Last, as application of It\^{o}'s formula, we present the exponential stability of second moments, the exponentially 2-ultimate boundedness and the almost surely asymptotic stability for their solutions in terms of a Lyapunov function.