Convergence of martingale solutions to the hybrid slow-fast system

TL;DR

This paper investigates the weak convergence of a complex slow-fast stochastic system with jumps and Markovian switching, using a novel combination of perturbed test functions and time discretization methods.

Contribution

It introduces a new approach combining perturbed test functions and time discretization to handle convergence analysis in systems with jumps and Markovian switching.

Findings

Established weak convergence results for the hybrid system.

Extended the analysis to systems without Markovian switching.

Numerical simulations confirm theoretical findings.

Abstract

This paper is devoted to studying the weak convergence for a slow-fast system with jumps modulated by Markovian switching regimes with the martingale method. However, due to the coexistence of fast component and Markovian switching regimes, the martingale method and perturbed test functions can not be applied directly. In this situation, a combination of perturbed test functions and the time discretization is applied efficiently. And the choice of appropriate perturbed test functions, which are related to the averaged coefficients, plays a decisive role. Our results also cover the case of slow-fast system without Markovian switching regimes. Finally, some examples are presented,and numerical simulations are carried out to observe a good agreement.