The averaging method for multivalued SDEs with jumps and non-Lipschitz coefficients

Wei Mao,Xuerong Mao,Surong You,Liangjian Hu

doi:10.3934/dcdsb.2019039

Abstract

In this paper, we study the averaging principle for multivalued SDEs with jumps and non-Lipschitz coefficients. By the Bihari's inequality and the properties of the concave function, we prove that the solution of averaged multivalued SDE with jumps converges to that of the standard one in the sense of mean square and also in probability. Finally, two examples are presented to illustrate our theory.

Highlights

In this paper, we are concerned with the averaging principle of the following equation: ∫dx(t) + A(x(t))dt ∋ f (t, x(t))dt + g(t, x(t))dw(t) + h(t, x(t−), v)N, (1)Z where A is a multi-valued maximal monotone operator defined on Rn and w(t) is an m dimensional Brownian motion, N is the counting measure of a stationary Poisson point process with characteristic measure π on some measurable space (Z, B(Z)).2010 Mathematics Subject Classification
Where A is a multi-valued maximal monotone operator defined on Rn and w(t) is an m dimensional Brownian motion, N is the counting measure of a stationary Poisson point process with characteristic measure π on some measurable space (Z, B(Z))
By the Bihari’s inequality and our proposed conditions, we prove that the solution of the averaged equation converges to that of the standard equation in the mean square sense

Summary

Introduction

We are concerned with the averaging principle of the following equation: ∫dx(t) + A(x(t))dt ∋ f (t, x(t))dt + g(t, x(t))dw(t) + h(t, x(t−), v)N (dt, dv), (1)Z where A is a multi-valued maximal monotone operator defined on Rn and w(t) is an m dimensional Brownian motion, N is the counting measure of a stationary Poisson point process with characteristic measure π on some measurable space (Z, B(Z)).2010 Mathematics Subject Classification. It is very important for us to establish the averaging principle of the multivalued SDEs with jumps (1) under some weaker conditions. We shall study the averaging principle for multi-valued SDEs with jumps.

Results

Conclusion