Almost sure asymptotic stabilizability for deterministic systems with wiener processes

In this paper, we discuss almost sure asymptotic stability for deterministic systems, which are randomized by adding one-dimensional standard Wiener processes. First, we clarify the difference between the deterministic Lyapunov stability theory and the stochastic stability theory proposed by Hasminskii, and show that local asymptotic stability in probability causes trouble in randomization problems. Second, we describe the Lyapunov stability theory based on almost sure stability, as proposed by Bardi and Cesaroni, and show that the stability is “almost the same” as the deterministic Lyapunov stability. Third, we survey randomization problems briefly, and show that Stratonovich integrals are valid for such problems based on one-dimensional Wiener processes. Finally, we derive the necessary conditions for stochastic Lyapunov functions to ensure almost sure stabilities and discuss the difference between Hasminskii´s stabilities and those of Bardi and Cesaroni via linear stochastic systems with numerical simulations.