Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

Xuerong Mao, Lukasz Szpruch

Research output: Contribution to journalArticle

66 Citations (Scopus)

Abstract

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.
LanguageEnglish
Pages14-28
Number of pages15
JournalJournal of Computational and Applied Mathematics
Volume238
DOIs
Publication statusPublished - Jan 2013

Fingerprint

Strong Stability
Convergence of numerical methods
Implicit Method
Finance
Asymptotic stability
Strong Convergence
Stochastic Equations
Lipschitz
Euler
Numerical methods
Differential equations
Numerical Methods
Almost Sure Stability
Differential equation
Coefficient
Asymptotic Stability
Biology
Deduce
Approximation
Model

Keywords

  • super-linear growth
  • stochastic differential equation
  • strong convergence
  • backward Euler–Maruyama scheme
  • LaSalle principle
  • almost sure stability

Cite this

@article{e8de4b0863724ccd9264c07852bbd6da,
title = "Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients",
abstract = "We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.",
keywords = "super-linear growth, stochastic differential equation, strong convergence, backward Euler–Maruyama scheme, LaSalle principle, almost sure stability",
author = "Xuerong Mao and Lukasz Szpruch",
note = "I have added on the document. l I hope this copy of the document is ok",
year = "2013",
month = "1",
doi = "10.1016/j.cam.2012.08.015",
language = "English",
volume = "238",
pages = "14--28",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",

}

TY - JOUR

T1 - Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

AU - Mao, Xuerong

AU - Szpruch, Lukasz

N1 - I have added on the document. l I hope this copy of the document is ok

PY - 2013/1

Y1 - 2013/1

N2 - We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.

AB - We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.

KW - super-linear growth

KW - stochastic differential equation

KW - strong convergence

KW - backward Euler–Maruyama scheme

KW - LaSalle principle

KW - almost sure stability

UR - http://www.journals.elsevier.com/journal-of-computational-and-applied-mathematics

U2 - 10.1016/j.cam.2012.08.015

DO - 10.1016/j.cam.2012.08.015

M3 - Article

VL - 238

SP - 14

EP - 28

JO - Journal of Computational and Applied Mathematics

T2 - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -