 
Summary: pth
MEAN INTEGRABILITY AND ALMOST SURE
ASYMPTOTIC STABILITY OF SOLUTIONS OF IT^OVOLTERRA
EQUATIONS
JOHN A. D. APPLEBY
Abstract. This paper studies the pathwise asymptotic stability and integra
bility of the zero solution of a finite dimensional It^oVolterra equation. Under
Lipschitz conditions on the statedependent functions, and with continuity and
integrability required of the kernels, it is shown that any solution which is pth
mean integrable for some p 2 is pth meanasymptotically stable, and also
pth mean integrable and asymptotically stable, almost surely. If there is no
delaydependent term in the volatility, the same result can be shown for p 1.
Examples which illustrate the usefulness of these results are presented, and
extensions to other classes of functional differential equations discussed.
1. Introduction
Much research in recent years has focussed on the almost sure exponential as
ymptotic stability of solutions of stochastic differential equations and stochastic
delay differential equations with bounded delay, with several recent monographs
appearing by Mao [11, 12]. However, less attention has been devoted to the almost
sure exponential asymptotic stability of It^oVolterra equations, where the delay is
