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.
Original language | English |
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Pages (from-to) | 1390-1408 |
Number of pages | 19 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 458 |
Issue number | 2 |
Early online date | 18 Oct 2017 |
DOIs | |
Publication status | Published - 15 Feb 2018 |
Keywords
- stability
- hybrid stochastic differential functional equations
- Itô formula
- Brownian motion
- Markov chain