Large deviation principle for a two-time-scale McKean-Vlasov model with jumps

TL;DR

This paper establishes a large deviation principle for a complex two-time-scale McKean-Vlasov system with jumps, using variational methods and averaging principles to analyze the asymptotic behavior of the system.

Contribution

It introduces a novel approach combining variational framework and Khasminskii-type averaging to handle the challenges of law dependence in the controlled system.

Findings

Large deviation principle is proved for the system.

The limit relates to the Dirac measure of an ODE solution.

Method effectively handles jumps and law dependence.

Abstract

This work focus on the large deviation principle for a two-time scale McKean-Vlasov system with jumps. Based on the variational framework of the McKean-Vlasov system with jumps, it is turned into weak convergence for the controlled system. Unlike general two-time scale system, the controlled McKean-Vlasov system is related to the law of the original system, which causes difficulties in qualitative analysis. In solving this problem, employing asymptotics of the original system and a Khasminskii-type averaging principle together is efficient. Finally, it is shown that the limit is related to the Dirac measure of the solution to the ordinary differential equation.