Almost sure exponential stability of hybrid stochastic functional differential equations

Minghui Song, Xuerong Mao

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10 Citations (Scopus)
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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.
Original languageEnglish
Pages (from-to)1390-1408
Number of pages19
JournalJournal of Mathematical Analysis and Applications
Issue number2
Early online date18 Oct 2017
Publication statusPublished - 15 Feb 2018


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


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