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

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.