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( […]

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## Stochastic Invariant Set

- Post author By Math Dev
- Post date March 26, 2020
- No Comments on Stochastic Invariant Set

- Tags $ 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 Lyapuno, i.e. (i) $\Phi$ is positive definite. (ii) $L\Phi\left( x\right) $ is not necessary to be nonpositive in $C$ but $L\Phi\left( x\right) <0$, Let us consider a stochastic differential equation (SDE), where $L$ is the infinitesimal generator of the SDE. How can I prove that $C$ is an invariant set with respect to the solutions of the SDE?