We employ an ergodic theory argument to demonstrate the foundations of ubiquity of Lévy stable self-similar processes in physics and present a class of models for anomalous and nonextensive diffusion. A relationship between stationary and self-similar models is clarified. The presented stochastic integral description of all Lévy stable processes could provide new insights into the mechanism underlying a range of self-similar natural phenomena. Finally, this effect is illustrated by self-similar approach to financial modelling.

PACS numbers: 05.40.-a, 02.50.Ey, 05.20.-y, 05.45.-a