Itô calculus extended to systems driven by αα-stable Lévy white noises (a novel clip on the tails of Lévy motion)

The paper deals with probabilistic characterization of the response of non-linear systems under αα-stable Lévy white noise input. It is shown that, by properly selecting a clip in the probability density function of the input, the moments of the increments of Lévy motion process remain all of the same order (dt)(dt), like the increments of the Compound Poisson process. It follows that the Itô calculus extended to Poissonian input, may also be used for αα-stable Lévy white noise input processes. It is also shown that, when the clip on the tails of the probability of the increments of the Lévy motion approaches to infinity, the Einstein–Smoluchowsky equation is restored. Once these concepts are outlined extension to single oscillator is readily obtained. A discussion on the proper way to perform Monte Carlo simulation is also exploited.