Barbashin-Krasovskii theorem for stochastic differential equations

A system of stochastic differential equations $dX(t)=f(X)dt+ \sum_{i=1}^{k}g_i(X)dW_i(t)$ which has a zero solution $X=0$ is considered. It is assumed that there exists a positive definite function $V(x)$ such that the corresponding operator $LV$ is nonpositive. It is proved that if the set $\{M:~ LV=0\}$ does not include entire semitrajectories of the system almost surely, then the zero solution is asymptotically stable in probability.