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

Wei Liu, Xuerong Mao

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

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.
Language English 1-20 20 Numerical Algorithms 9 Jun 2016 10.1007/s11075-016-0162-3 E-pub ahead of print - 9 Jun 2016

Fingerprint

Euler-Maruyama Method
Almost Sure Stability
Variable Step Size
Random variables
Stochastic Equations
Differential equations
Stopping Time
Random variable
Differential equation
Convergence Theory
Semimartingale
Euler

Keywords

• stopping time
• almost sure stability
• Euler-Maruyama
• variable stepsize
• semi-martingale convergence theory

Cite this

title = "Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations",
abstract = "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.",
keywords = "stopping time, almost sure stability, Euler-Maruyama, variable stepsize, semi-martingale convergence theory",
author = "Wei Liu and Xuerong Mao",
note = "The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-016-0162-3",
year = "2016",
month = "6",
day = "9",
doi = "10.1007/s11075-016-0162-3",
language = "English",
pages = "1--20",
journal = "Numerical Algorithms",
issn = "1017-1398",

}

In: Numerical Algorithms, 09.06.2016, p. 1-20.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations

AU - Liu, Wei

AU - Mao, Xuerong

N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-016-0162-3

PY - 2016/6/9

Y1 - 2016/6/9

N2 - 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.

AB - 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.

KW - stopping time

KW - almost sure stability

KW - Euler-Maruyama

KW - variable stepsize

KW - semi-martingale convergence theory

U2 - 10.1007/s11075-016-0162-3

DO - 10.1007/s11075-016-0162-3

M3 - Article

SP - 1

EP - 20

JO - Numerical Algorithms

T2 - Numerical Algorithms

JF - Numerical Algorithms

SN - 1017-1398

ER -