Governing stochastic equation for a self-similar random process

A stochastic differential equation is proposed, the solution of which is a characteristic random function that describes a stochastic process with Gaussian tails of distribution functions. The reciprocal to the characteristic random function describes a self-similar random process with a power spectrum inversely proportional to the frequency and with power-law a behavior of tails in the amplitude distribution functions. Gaussian tails for the characteristic distribution make it possible to evaluate its stability according to the formulas of classical statistics using the maximum of the Gibbs-Shannon entropy and, therefore, the stability of a random process given by an inverse function. (c) 2023 Elsevier B.V. All rights reserved.