Exact Moment Scaling from Multiplicative Noise

TL;DR

This paper derives exact analytical solutions for the moments of a broad class of diffusion processes with multiplicative noise, revealing how drift and diffusion coefficients influence moment scaling over time.

Contribution

It provides a general analytical framework for understanding moment dynamics in multiplicative noise processes with time-dependent parameters.

Findings

Analytical solutions for moments at all times.

Characterization of process evolution towards stationarity.

Relationship between coefficients and moment scaling.

Abstract

For a general class of diffusion processes with multiplicative noise, describing a variety of physical as well as financial phenomena, mostly typical of complex systems, we obtain the analytical solution for the moments at all times. We allow for a non trivial time dependence of the microscopic dynamics and we analytically characterize the process evolution, possibly towards a stationary state, and the direct relationship existing between the drift and diffusion coefficients and the time scaling of the moments.