Lyapunov functions and almost sure exponential stability of stochastic differential equations based on semimartingales with spatial parameters

The objective of this paper is to use the Lyapunov function to study the almost sure exponential stability of the stochastic differential equation \[\varphi _t = x + \int_0^t {F(\varphi _x ,ds)} ,\] where $F(x,t)$ is a continuous C-semim irtingaie with spatial parameter x. This equation includes many important stochastic systems, for example, the classical Ito equation. More importantly, the result can be employed to study the bound of the Lyapunov exponent of stochastic flows.