Almost sure exponential stability of hybrid stochastic functional differential equations

Minghui Song, Xuerong Mao

Research output: Contribution to journalArticle

Abstract

This paper is concerned with the almost sure exponential stability of the n -dimensional nonlinear hybrid stochastic functional differential equation (SFDE) dx(t)=f(ψ1(xt,t),r(t),t)dt+g(ψ2(xt,t),r(t),t)dB(t), where xt={x(t+u):−τ≤u≤0} is a C([−τ,0];Rn)C([−τ,0];Rn)-valued process, B(t)B(t) is an m -dimensional Brownian motion while r(t) is a Markov chain. We show that if the corresponding hybrid stochastic differential equation (SDE) dy(t)=f(y(t),r(t),t)dt+g(y(t),r(t),t)dB(t) is almost surely exponentially stable, then there exists a positive number τ⁎ such that the SFDE is also almost surely exponentially stable as long as τ<τ⁎. We also describe a method to determine τ⁎ which can be computed numerically in practice.
LanguageEnglish
Pages1390-1408
Number of pages19
JournalJournal of Mathematical Analysis and Applications
Volume458
Issue number2
Early online date18 Oct 2017
DOIs
StatePublished - 15 Feb 2018

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Almost Sure Exponential Stability
Stochastic Functional Differential Equations
Asymptotic stability
Differential equations
Stochastic Equations
Brownian motion
Markov chain
Brownian movement
Differential equation
Markov processes

Keywords

  • stability
  • hybrid stochastic differential functional equations
  • Itô formula
  • Brownian motion
  • Markov chain

Cite this

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abstract = "This paper is concerned with the almost sure exponential stability of the n -dimensional nonlinear hybrid stochastic functional differential equation (SFDE) dx(t)=f(ψ1(xt,t),r(t),t)dt+g(ψ2(xt,t),r(t),t)dB(t), where xt={x(t+u):−τ≤u≤0} is a C([−τ,0];Rn)C([−τ,0];Rn)-valued process, B(t)B(t) is an m -dimensional Brownian motion while r(t) is a Markov chain. We show that if the corresponding hybrid stochastic differential equation (SDE) dy(t)=f(y(t),r(t),t)dt+g(y(t),r(t),t)dB(t) is almost surely exponentially stable, then there exists a positive number τ⁎ such that the SFDE is also almost surely exponentially stable as long as τ<τ⁎. We also describe a method to determine τ⁎ which can be computed numerically in practice.",
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Almost sure exponential stability of hybrid stochastic functional differential equations. / Song, Minghui; Mao, Xuerong.

In: Journal of Mathematical Analysis and Applications, Vol. 458, No. 2, 15.02.2018, p. 1390-1408.

Research output: Contribution to journalArticle

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