Well-posedness and averaging principle for L\'evy-type McKean-Vlasov stochastic differential equations under local Lipschitz conditions

TL;DR

This paper proves the well-posedness and averaging principle for Le9vy-type McKean-Vlasov stochastic differential equations under local Lipschitz conditions, broadening applicability to equations with super-linear drifts.

Contribution

It establishes existence, uniqueness, and averaging results for McKean-Vlasov SDEs with weaker local Lipschitz conditions, allowing super-linear growth in drift.

Findings

Solutions exist and are unique under local Lipschitz conditions.

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

Results apply to a wider class of McKean-Vlasov SDEs with super-linear drifts.

Abstract

In this paper, we investigate a class of McKean-Vlasov stochastic differential equations under L\'evy-type perturbations. We first establish the existence and uniqueness theorem for solutions of the McKean-Vlasov stochastic differential equations by utilizing the Euler-like approximation. Then under some suitable conditions, we show that the solutions of McKean-Vlasov stochastic differential equations can be approximated by the solutions of the associated averaged McKean-Vlasov stochastic differential equations in the sense of mean square convergence. In contrast to the existing work, a novel feature is the use of a much weaker condition -- local Lipschitzian in the state variables, allowing for possibly super-linearly growing drift, but linearly growing diffusion and jump coefficients. Therefore, our results are suitable for a wider class of McKean-Vlasov stochastic differential equations.