Almost sure exponential stability of multi-dimensional stochastic systems driven by a symmetric $α$-stable process

Abstract

This article investigates the stability of a class of multidimensional stochastic systems driven by a symmetric $α$-stable process with $1<α<2$. These driving processes have infinite second moments, which prevents the direct application of standard second-moment methods and may lead to divergent jump integrals in the generators of quadratic Lyapunov functions. We establish criteria for $p$th-moment exponential stability and almost sure exponential stability. A numerical example illustrates the theoretical results.