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Summary: ALMOST SURE STABILITY OF LINEAR IT^O-VOLTERRA
EQUATIONS WITH DAMPED STOCHASTIC PERTURBATIONS
JOHN A. D. APPLEBY
Abstract. In this paper we study the a.s. convergence of all solutions of the
It^o-Volterra equation
dX(t) = (AX(t) +
t
0
K(t - s)X(s) ds) dt + (t) dW(t)
to zero. A is a constant d × d matrix, K is a d × d continuous and integrable
matrix function, is a continuous d × r matrix function, and W is an r-
dimensional Brownian motion. We show that when
x (t) = Ax(t) +
t
0
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
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