# Almost sure exponential stability of hybrid stochastic functional differential equations

Research output: Research - peer-review › Article

This paper is concerned with the almost sure exponential stability of the

*n*-dimensional nonlinear hybrid stochastic functional differential equation (SFDE) dx(t)=f(ψ_{1}(x_{t},t),r(t),t)dt+g(ψ_{2}(x_{t},t),r(t),t)dB(t), where x_{t}={x(t+u):−τ≤u≤0} is a C([−τ,0];Rn)C([−τ,0];R^{n})-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 | |

State | Published - 15 Feb 2018 |

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