Almost sure and moment exponential stability of Euler–Maruyama discretizations for hybrid stochastic differential equations

Pang, Shaoning; Deng, Feiqi; Mao, Xuerong

doi:10.1016/j.cam.2007.01.003

Cited by 53 publications

(29 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is a continuation of the first author's earlier paper [17] jointly with Pang and Deng. It is concerned with the long time dynamics of numerical simulations of hybrid stochastic differential equations (SDEs).…”

Section: Introductionmentioning

confidence: 75%

“…[5,7,8,10,18,19]). More recently, the authors in [17] established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [17] is the global Lipschitz condition.…”

Section: Introductionmentioning

confidence: 99%

“…The key condition imposed in [17] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations

Mao

Shen

Gray

2011

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

This version is available at https://strathprints.strath.ac.uk/29078/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Accepted Manuscript A C C E P T E D M A N U S C R I P T AbstractThis is a continuation of the first author's earlier paper [17] jointly with Pang and Deng, in which the authors established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [17] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs.

show abstract

Section: Introductionmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations

Mao

Shen

Gray

2011

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, for nonlinear SDEs under the global Lipschitz condition, Higham, Mao, and Stuart [9] show that the exponential stability in mean square for the SDE is equivalent to the exponential stability in mean square of the numerical method (e.g., the EulerMaruyama and the stochastic theta method) for sufficiently small step sizes. For further developments in this area, we refer the reader to [5,8,19,22,24,27], for example, and the references therein.…”

mentioning

confidence: 99%

Almost Sure Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

Mao¹

2015

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

Abstract. This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschitz condition, we first show that the SDE is pth moment exponentially stable (for p ∈ (0, 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size Δt. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ∈ (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems, (P1) and (P2) listed in section 1.

show abstract

“…We refer to some of the works [8,10,13,17,21] and references therein. This paper is constructed as follows.…”

Section: Introductionmentioning

confidence: 99%

Almost sure stability of the Euler–Maruyama method with random variable stepsize for stochastic differential equations

Liu

Mao

2016

Numer Algor

Self Cite

View full text Add to dashboard Cite

In this paper, the Euler-Maruyama (EM) method with random variable stepsize is studied to reproduce the almost sure stability of the true solutions of stochastic differential equations.Since the choice of the time step is based on the current state of the solution, the time variable is proved to be a stopping time. Then the semimartingale convergence theory is employed to obtain the almost sure stability of the random variable stepsize EM solution. To our best knowledge, this is the first paper to apply the random variable stepsize (with clear proof of the stopping time) to the analysis of the almost sure stability of the EM method.

show abstract

Almost sure and moment exponential stability of Euler–Maruyama discretizations for hybrid stochastic differential equations

Cited by 53 publications

References 22 publications

Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations

Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations

Almost Sure Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

Almost sure stability of the Euler–Maruyama method with random variable stepsize for stochastic differential equations

Contact Info

Product

Resources

About