Author： Liao X.X. Balas X.M.J.
Publisher： Academic Press
ISSN： 0022-247X
Source： Journal of Mathematical Analysis and Applications, Vol.212, Iss.2, 1997-08, pp. : 554-570
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Abstract
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 almost sure exponential stability for such equations and the aim of this paper is to close this gap. The convergence of nonnegative special semimartingales established by R. Sh. Lipster and A. N. Shiryayev ("Theory of Martingales," Kluwer Academic, Dordrecht, 1989) and the Ito formula will play a key role in this paper.
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