Weak order in averaging principle for stochastic differential equations with jumps

Bengong Zhang,Li Wan,Hongbo Fu,Jicheng Liu

doi:10.1186/s13662-018-1638-3

Abstract

In this paper, we deal with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equations. Under suitable conditions, we expand the weak error in powers of timescale parameter. We prove that the rate of weak convergence to the averaged dynamics is of order 1. This reveals that the rate of weak convergence is essentially twice that of strong convergence.

Highlights

1 Introduction We consider a two-time-scale system of jump-diffusion stochastic differential equation of the form dXt = a Xt, Yt dt + b Xt dBt + c Xt– dPt, X0 = x, 1 dYt = f Xt, Yt dt + √1 g Xt, Yt dWt + h Xt, Yt– dNt, Y0 = y, (1.1) (1.2)
A simplified equation, which is independent of the fast variable and possesses the essential features of the system, is highly desirable
To derive the averaged dynamics of system (1.1)–(1.2), we introduce the fast motion equation with a frozen slow component x ∈ Rn of the form dYtx = f x, Ytx dt + g x, Ytx dWt + h x, Ytx– dNt, Y0x = y, (1.3)

Summary

Introduction

To derive the averaged dynamics of system (1.1)–(1.2), we introduce the fast motion equation with a frozen slow component x ∈ Rn of the form dYtx = f x, Ytx dt + g x, Ytx dWt + h x, Ytx– dNt, Y0x = y,. In [11], it was shown that, under the stated conditions, the slow motion Xt converges strongly to the solution Xt of the averaged equation with jumps. By averaging the coefficient a with respect to the invariant measure μx we can define the Rn-valued mapping a (x) := a(x, y)μx(dy), x ∈ Rn. Due to assumption (A1), it is to check that a (x) is twice differentiable with bounded derivatives, and it is Lipschitz-continuous:.

Now we can introduce the effective dynamical system

Rn ds

If we define ςtx