Polynomial asymptotic stability of damped stochastic differential equations
The paper studies the polynomial convergence of solutions of a scalar nonlinear It\^{o} stochastic differential equation\[dX(t) = -f(X(t))\,dt + \sigma(t)\,dB(t)\] where it is known, {\it a priori}, that $\lim_{t\rightarrow\infty} X(t)=0$, a.s. The intensity of the stochastic perturbation $\sigma$ i...
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University of Szeged
2004-08-01
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Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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doaj-0c787fdd8fb846e2a714ca1f59f65e452021-07-14T07:21:18ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38751417-38752004-08-012003213310.14232/ejqtde.2003.6.2189Polynomial asymptotic stability of damped stochastic differential equationsJohn Appleby0D. Mackey1School of Mathematical Sciences, Dublin City University, Dublin 9, IrelandCMDE, School of Mathematical Sciences, Dublin City University, Dublin 9, IrelandThe paper studies the polynomial convergence of solutions of a scalar nonlinear It\^{o} stochastic differential equation\[dX(t) = -f(X(t))\,dt + \sigma(t)\,dB(t)\] where it is known, {\it a priori}, that $\lim_{t\rightarrow\infty} X(t)=0$, a.s. The intensity of the stochastic perturbation $\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\lim_{x\rightarrow 0}\mbox{sgn}(x)f(x)/|x|^\beta = a$, for some $\beta>1$, and $a>0$.We study two asymptotic regimes: when $\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\sigma\equiv0$). When $\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=189 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
John Appleby D. Mackey |
spellingShingle |
John Appleby D. Mackey Polynomial asymptotic stability of damped stochastic differential equations Electronic Journal of Qualitative Theory of Differential Equations |
author_facet |
John Appleby D. Mackey |
author_sort |
John Appleby |
title |
Polynomial asymptotic stability of damped stochastic differential equations |
title_short |
Polynomial asymptotic stability of damped stochastic differential equations |
title_full |
Polynomial asymptotic stability of damped stochastic differential equations |
title_fullStr |
Polynomial asymptotic stability of damped stochastic differential equations |
title_full_unstemmed |
Polynomial asymptotic stability of damped stochastic differential equations |
title_sort |
polynomial asymptotic stability of damped stochastic differential equations |
publisher |
University of Szeged |
series |
Electronic Journal of Qualitative Theory of Differential Equations |
issn |
1417-3875 1417-3875 |
publishDate |
2004-08-01 |
description |
The paper studies the polynomial convergence of solutions of a scalar nonlinear It\^{o} stochastic differential equation\[dX(t) = -f(X(t))\,dt + \sigma(t)\,dB(t)\] where it is known, {\it a priori}, that $\lim_{t\rightarrow\infty} X(t)=0$, a.s. The intensity of the stochastic perturbation $\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\lim_{x\rightarrow 0}\mbox{sgn}(x)f(x)/|x|^\beta = a$, for some $\beta>1$, and $a>0$.We study two asymptotic regimes: when $\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\sigma\equiv0$). When $\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven. |
url |
http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=189 |
work_keys_str_mv |
AT johnappleby polynomialasymptoticstabilityofdampedstochasticdifferentialequations AT dmackey polynomialasymptoticstabilityofdampedstochasticdifferentialequations |
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1721303891434274816 |