Xx. Liao et X. Mao, ALMOST SURE EXPONENTIAL STABILITY OF NEUTRAL STOCHASTIC DIFFERENTIAL-DIFFERENCE EQUATIONS, Journal of mathematical analysis and applications, 212(2), 1997, pp. 554-570
A neutral stochastic differential difference equation d[x(t) - G(x(t -
tau))] = f(t, x(t), x(t - tau)) dt + sigma(t, x(t), x(t - tau)) dw(t)
was introduced by V. B. Kolmanovskii and Y. R. Nosov (''Stability and
Periodic Modes of Control Systems with Aftereffect,'' Nauka, Moscow,
1981) several years ago. However, so far little is known about the alm
ost sure exponential stability for such equations and the aim of this
paper is to close this gap. The convergence of nonnegative special sem
imartingales established by R. Sh. Lipster and A. N. Shiryayev (''Theo
ry of Martingales,'' Kluwer Academic, Dordrecht, 1989) and the Ito for
mula will play a key role in this paper. (C) 1997 Academic Press.