Systems Driven By Alpha-Stable Noises

Introduction and abstract. It has almost become a standard in stochastic mechanics applications of stochastic differential equations that the driving forces are modeled as Gaussian white noises, that is, as scalar or vector Brownian motion increments. However, this modeling may not always lead to responses that comply well with observed data. In particular the tails of the observed response distributions may even for linear systems be more fat than the tails obtained for Gaussian white noise input. Also the excitation may show jumps that cannot be modeled by Gaussian white noise. The paper supports the possibility of using the larger class of so-called α-stable white noises (Lévy noises for 0 < α < 2, Gaussian white noise for α = 2) to provide a better fit [4]. Lévy noise driven linear systems have responses that possess α-stable distributions with the same value of α as defined by the Lévy noise input. For α < 2 the absolute moments exist only up to the order α−, that is, the second order moment is infinite for any α < 2 and the mean does not exist if α ≤ 1. Alpha-stable noise and discussion of relevance. It is an elementary fact that any linear combination of n independent copies X 1 ,. .. , X n of a Gaussian random variable X is a Gaussian random variable. One may ask whether also some non-Gaussian distribution types satisfy the condition ∀a 1 ,. .. , a n ∈ R ∃c ∈ R : a 1 X 1 +.. .+a n X n = d cX, where = d means equal in distribution. A distribution with this property is said to be strictly stable. The characteristic function ψ(u) = E[e iuX ] obviously must satisfy the condition n