Almost sure exponential stability of numerical solutions for stochastic delay differential equations

Fuke Wu, Xuerong Mao, Lukasz Szpruch

Research output: Contribution to journalArticle

87 Citations (Scopus)

Abstract

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma.
LanguageEnglish
Pages681-697
Number of pages17
JournalNumerische Mathematik
Volume115
Issue number4
DOIs
Publication statusPublished - Jun 2010

Fingerprint

Almost Sure Exponential Stability
Stochastic Delay Differential Equations
Asymptotic stability
Differential equations
Numerical Solution
Borel-Cantelli Lemma
Chebyshev's inequality
Semimartingale
Convergence Theorem
Numerical methods
Exact Solution
Numerical Methods
Moment

Keywords

  • numerical solutions
  • stochastic delay
  • differential equations

Cite this

@article{3aea55280313461189cbf1647802645f,
title = "Almost sure exponential stability of numerical solutions for stochastic delay differential equations",
abstract = "Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma.",
keywords = "numerical solutions , stochastic delay , differential equations",
author = "Fuke Wu and Xuerong Mao and Lukasz Szpruch",
year = "2010",
month = "6",
doi = "10.1007/s00211-010-0294-7",
language = "English",
volume = "115",
pages = "681--697",
journal = "Numerische Mathematik",
issn = "0029-599X",
number = "4",

}

Almost sure exponential stability of numerical solutions for stochastic delay differential equations. / Wu, Fuke; Mao, Xuerong; Szpruch, Lukasz.

In: Numerische Mathematik, Vol. 115, No. 4, 06.2010, p. 681-697.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Almost sure exponential stability of numerical solutions for stochastic delay differential equations

AU - Wu, Fuke

AU - Mao, Xuerong

AU - Szpruch, Lukasz

PY - 2010/6

Y1 - 2010/6

N2 - Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma.

AB - Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma.

KW - numerical solutions

KW - stochastic delay

KW - differential equations

U2 - 10.1007/s00211-010-0294-7

DO - 10.1007/s00211-010-0294-7

M3 - Article

VL - 115

SP - 681

EP - 697

JO - Numerische Mathematik

T2 - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 4

ER -