ALMOST SURE EXPONENTIAL STABILITY FOR A CLASS OF STOCHASTIC DIFFERENTIAL-EQUATIONS WITH APPLICATIONS TO STOCHASTIC FLOWS

The objective of this paper is to use the Lyapunov function to study the almost sure exponential stability of the stochastic differential equation phi(t) = x + integral-t/o F(phi(S), ds) where F(x, t) is a continuous C-semimartingale with spatial parameter x. This equation includes many important stochastic systems, for example, the classical Ito equation. More importantly, our result can be employed to study the bound of the Lyapunov exponent of stochastic flows