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Summary: pth
MEAN INTEGRABILITY AND ALMOST SURE
ASYMPTOTIC STABILITY OF SOLUTIONS OF IT^O-VOLTERRA
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^o-Volterra equation. Under
Lipschitz conditions on the state-dependent functions, and with continuity and
integrability required of the kernels, it is shown that any solution which is p-th
mean integrable for some p 2 is p-th mean-asymptotically stable, and also
p-th mean integrable and asymptotically stable, almost surely. If there is no
delay-dependent 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^o-Volterra equations, where the delay is
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