Electronic Communications in Probability

Almost Sure Stability of Linear Itô-Volterra Equations with Damped Stochastic Perturbations

John Appleby

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Abstract

In this paper we study the a.s. convergence of all solutions of the Itô-Volterra equation \[ dX(t) = (AX(t) + \int_{0}^{t} K(t-s)X(s),ds)\,dt + \Sigma(t)\,dW(t) \] to zero. $A$ is a constant $d\times d$ matrix, $K$ is a $d\times d$ continuous and integrable matrix function, $\Sigma$ is a continuous $d\times r$ matrix function, and $W$ is an $r$-dimensional Brownian motion. We show that when \[ x'(t) = Ax(t) + \int_{0}^{t} K(t-s)x(s)\,ds \] has a uniformly asymptotically stable zero solution, and the resolvent has a polynomial upper bound, then $X$ converges to 0 with probability 1, provided \[ \lim_{t \rightarrow \infty} |\Sigma(t)|^{2}\log t= 0. \] A converse result under a monotonicity restriction on $|\Sigma|$ establishes that the rate of decay for $|\Sigma|$ above is necessary. Equations with bounded delay and neutral equations are also considered.

Article information

Source
Electron. Commun. Probab., Volume 7 (2002), paper no. 22, 223-234.

Dates
Accepted: 21 August 2002
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1463434789

Digital Object Identifier
doi:10.1214/ECP.v7-1063

Mathematical Reviews number (MathSciNet)
MR1952184

Zentralblatt MATH identifier
1027.60067

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations 34K20: Stability theory
Secondary: 45D05: Volterra integral equations [See also 34A12]

Keywords
Stochastic functional-differential equations Ito-Volterra equations uniform asymptotic stability almost sure stability pathwise stability simulated annealing

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Appleby, John. Almost Sure Stability of Linear Itô-Volterra Equations with Damped Stochastic Perturbations. Electron. Commun. Probab. 7 (2002), paper no. 22, 223--234. doi:10.1214/ECP.v7-1063. https://projecteuclid.org/euclid.ecp/1463434789


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