Stochastic Lyapunov method
Let \((\Omega, {\mathcal F}, P)\) be a complete probability space, and let \(\{{\mathcal F}_t \subset {\mathcal F}\}\) be an increasing family of \(\sigma\)-sub-algebras adopted to a standard \(m\)-dimensional Wiener process \(W\). The authors consider solutions to the stochastic differential equation (*) \(dx = f(x(t)) dt + \sigma (x(t)) dW(t)\), \(x \in \mathbb{R}^n\), starting from \(x(0)\). They study conditions under which the inequality \(\forall t \geq 0\) \(V(x(t)) \leq \omega (t)\) holds almost everywhere. Here \(\omega (\cdot)\) is a solution to the differential equation \(\omega' = - \varphi (\omega)\) and \(V : \mathbb{R}^n \to \mathbb{R} \cup \{+ \infty\}\). On the basis of such inequalities a lot of asymptotic behaviors of \(V\) along the solutions to (*) is obtained.
scientific article; zbMATH DE number 822781
scientific article