Electronic Communications in Probability

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

John Appleby

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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