### Abstract

*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.

Language | English |
---|---|

Pages | 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 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Mathematical Analysis and Applications*,

*458*(2), 1390-1408. DOI: 10.1016/j.jmaa.2017.10.042

}

*Journal of Mathematical Analysis and Applications*, vol. 458, no. 2, pp. 1390-1408. DOI: 10.1016/j.jmaa.2017.10.042

**Almost sure exponential stability of hybrid stochastic functional differential equations.** / Song, Minghui; Mao, Xuerong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Almost sure exponential stability of hybrid stochastic functional differential equations

AU - Song,Minghui

AU - Mao,Xuerong

PY - 2018/2/15

Y1 - 2018/2/15

N2 - 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.

AB - 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.

KW - stability

KW - hybrid stochastic differential functional equations

KW - Itô formula

KW - Brownian motion

KW - Markov chain

UR - http://www.sciencedirect.com/science/journal/0022247X?sdc=1

U2 - 10.1016/j.jmaa.2017.10.042

DO - 10.1016/j.jmaa.2017.10.042

M3 - Article

VL - 458

SP - 1390

EP - 1408

JO - Journal of Mathematical Analysis and Applications

T2 - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -