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

   
@article{ECP1063,
	author = {John Appleby},
	title = {Almost Sure Stability of Linear Ito-Volterra Equations with Damped Stochastic Perturbations},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {7},
	year = {2002},
	keywords = {Stochastic functional-differential equations, Ito-Volterra equations, uniform asymptotic stability, almost sure stability, pathwise stability, simulated annealing.},
	abstract = {In this paper we study the a.s. convergence of all solutions of the Ito-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.},
	pages = {no. 22, 223-234},
	issn = {1083-589X},
	doi = {10.1214/ECP.v7-1063},    
        url = {http://ecp.ejpecp.org/article/view/1063}}