Mean-square stability properties of an adaptive time-stepping SDE solver

Lamba, Harbir; Seaman, T.

doi:10.1016/j.cam.2005.07.007

Cited by 8 publications

(9 citation statements)

References 27 publications

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“…In this work, we consider long time dynamics, and in particular focus on exponential mean-square stability. Our work therefore builds on the well known and highly informative analysis for deterministic problems and its more recent extension to SDEs [2,[4][5][6][7][8]10,11,[14][15][16].…”

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confidence: 99%

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

2007

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Positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with Markovian switching. Euler-Maruyama and implicit theta-method discretisations are shown to capture exponential mean-square stability for all sufficiently small time-steps under appropriate conditions. Moreover, the decay rate, as measured by the second moment Lyapunov exponent, can be reproduced arbitrarily accurately. New finite-time convergence results are derived as an intermediate step in this analysis. We also show, however, that the mean-square A-stability of the theta method does not carry through to this switching scenario. The proof techniques are quite general and hence have the potential to be applied to other numerical methods.

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confidence: 99%

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

2007

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“…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

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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.

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“…We look for conditions under which positive results of the backward Euler-Maruyama (BEM) method can be derived in the small step size setting. Our work therefore builds on the well known and highly informative analysis for deterministic problems and its more recent extension to SDEs [2,3,4,5,7,9,10,11,18,19].…”

Section: Introductionmentioning

confidence: 99%

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

Mao

Shen

Gray

2011

Journal of Computational and Applied Mathematics

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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.

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Mean-square stability properties of an adaptive time-stepping SDE solver

Cited by 8 publications

References 27 publications

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

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

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

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