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Stochastic Invariant Set

Let us consider a stochastic differential equation (SDE), $ dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}% $ and a compact set $C\subset\mathbb{R}^{n}$. Given a stochastic Lyapunov function $\Phi\left( x_{t}\right) $ for this SDE with respect to $C$, i.e. (i) $\Phi$ is positive definite. (ii) $L\Phi\left( x\right) $ is not necessary to be nonpositive in $C$ but $L\Phi\left( […]