Convergence of martingale solutions to the hybrid slow-fast system

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.

Figures (3)

• Figure 1: Expectation of solutions $x^\varepsilon(t)$, $\bar{x}(t)$ to the original slow system (\ref{['sec4-1']}) and the averaged system (\ref{['sec4-11']}), and error curve. $(a)$$D=0.1$, $x_0=1,y_0=0$, $\varepsilon=0.01$, $(b)$$D=0.1$, $x_0=1,y_0=0$, $\varepsilon=0.001$.
• Figure 2: Expectation of solutions $x^\varepsilon(t)$, $\bar{x}(t)$ to the original slow system (\ref{['sec4-2']}) and the averaged system (\ref{['sec4-12']}), and error curve. $(a)$$D=0.1$, $x_0=1,y_0=0$, $\varepsilon=0.2$, $(b)$$D=0.1$, $x_0=1,y_0=0$, $\varepsilon=0.001$.
• Figure 3: Expectation of solutions $x^\varepsilon(t)$ and $\bar{x}(t)$ to the original slow system (\ref{['sec4-3']}) and the averaged system (\ref{['sec4-31']}), and error curve. $(a)$$D=0.1$, $\lambda=0.5$, and $Z_j \sim N(0,1)$, $x_0=1,y_0=0$, $\varepsilon=0.2$, $(b)$$D=0.1$, $\lambda=0.5$, and $Z_j \sim N(0,1)$, $x_0=1,y_0=0$, $\varepsilon=0.001$.

Theorems & Definitions (12)

• Definition 2.1
• Lemma 2.2
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Remark 3.7
• Example 4.1