Fractional Diffusion Equation for a Power-Law-Truncated Levy Process

TL;DR

This paper introduces a fractional diffusion equation modeling truncated Levy flights with a power-law cutoff, providing a mathematical framework for processes that transition from Levy to Gaussian behavior with heavy tails.

Contribution

It presents a novel fractional diffusion equation for truncated Levy processes, including a closed-form characteristic function and analysis of the probability distribution's behavior.

Findings

The process exhibits a crossover from Levy to Gaussian distribution.

The probability density function has a slowly converging Gaussian core with heavy tails.

A closed-form characteristic function for the process is derived.

Abstract

Truncated Levy flights are stochastic processes which display a crossover from a heavy-tailed Levy behavior to a faster decaying probability distribution function (pdf). Putting less weight on long flights overcomes the divergence of the Levy distribution second moment. We introduce a fractional generalization of the diffusion equation, whose solution defines a process in which a Levy flight of exponent alpha is truncated by a power-law of exponent 5 - alpha. A closed form for the characteristic function of the process is derived. The pdf of the displacement slowly converges to a Gaussian in its central part showing however a power law far tail. Possible applications are discussed.