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 Electronic Communications in Probability > Vol. 7 (2002) > Paper 22 open journal systems 


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

John A. D. Appleby, Dublin City University


Abstract
In this paper we study the a.s. convergence of all solutions of the It^{o}-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 $dtimes d$ matrix, $K$ is a $dtimes d$ continuous and integrable matrix function, $Sigma$ is a continuous $dtimes r$ 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.


Full text: PDF

Pages: 223-234

Published on: August 21, 2002
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Electronic Communications in Probability. ISSN: 1083-589X