 
Summary: EXPONENTIAL ASYMPTOTIC STABILITY OF LINEAR
IT^OVOLTERRA EQUATIONS WITH DAMPED STOCHASTIC
PERTURBATIONS
JOHN A. D. APPLEBY AND ALAN FREEMAN
Abstract. This paper studies the convergence rate of solutions of the linear
It^o  Volterra equation
(0.1) dX(t) = AX(t) +
t
0
K(t  s)X(s) ds dt + (t) dW(t)
where K and are continuous matrixvalued functions defined on R+, and
(W(t))t0 is a finitedimensional standard Brownian motion. It is shown that
when the entries of K are all of one sign on R+, that (i) the almost sure
exponential convergence of the solution to zero (ii) the pth mean exponential
convergence of the solution to zero (for all p > 0), and (iii) the exponential
integrability of the entries of the kernel K, the exponential square integrability
of the entries of noise term , and the uniform asymptotic stability of the
solutions of the deterministic version of (0.1) are equivalent. The paper extends
a result of Murakami which relates to the deterministic version of this problem.
1. Introduction
