Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form \begin{document} $V(\mathbf{x}) = \|\mathbf{x}\|_Q^p: = (\mathbf{x}^\top Q\mathbf{x})^{\frac{p}{2}}$ \end{document} , where the parameters are the positive definite matrix \begin{document} $Q$ \end{document} and the number \begin{document} $p>0$ \end{document} . We give several examples of our proposed method and show how it improves previous results.