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 $V(\mathbf{x}) = ||\mathbf{x}||^p_Q = (\mathbf{x}^\top Q \mathbf{x})^{\frac{p}{2}}$, where the parameters are the positive definite matrix $Q$ and the number $p > 0$. We give several examples of our proposed method and show how it improves previous results.