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Polynomial asymptotic stability of damped stochastic differential equations.

Appleby, John A.D. and Mackey, Dana (2004) Polynomial asymptotic stability of damped stochastic differential equations. In: The 7th Colloquium on the Qualitative Theory of Differential Equations, 14 - 18 July, 2003, Szeged, Hungary.

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Abstract

The paper studies the polynomial convergence of solutions of a scalar nonlinear Itˆo stochastic differential equation dX(t) = −f(X(t)) dt + (t) dB(t) where it is known, a priori, that limt!1 X(t) = 0, a.s. The intensity of the stochastic perturbation is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function f obeys limx!0 sgn(x)f(x)/|x| = a, for some > 1, and a > 0. We study two asymptotic regimes: when tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when 0). When decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.

Item Type:Conference or Workshop Item (Paper)
Event Type:Conference
Refereed:Yes
Uncontrolled Keywords:polynomial asymptotic stability; almost sure asymptotic stability; simulated annealing; diffusion process;
Subjects:Mathematics > Differential equations
Mathematics > Stochastic analysis
DCU Faculties and Centres:DCU Faculties and Schools > Faculty of Science and Health > School of Mathematical Sciences
Official URL:http://www.math.u-szeged.hu/ejqtde/
Use License:This item is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 3.0 License. View License
ID Code:17
Deposited On:26 Oct 2006 by DORAS Administrator. Last Modified 29 Jan 2009 16:40

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