L\'evy processes linked to the lower-incomplete gamma function

TL;DR

This paper introduces new Lévy processes based on the lower-incomplete gamma function, providing finite-activity approximations to stable subordinators and exploring their applications in modeling anomalous diffusion.

Contribution

It defines a novel subordinator linked to the lower-incomplete gamma function and studies Lévy processes time-changed by these subordinators, including fractional Brownian motion.

Findings

The subordinator approximates stable processes with finite activity.

The tempered version overcomes infinite moments.

Time-changed fractional Brownian motion models sub-diffusive behavior.

Abstract

We start by defining a subordinator by means of the lower-incomplete gamma function. It can be considered as an approximation of the stable subordinator, easier to be handled thank to its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study L\'{e}vy processes time-changed by these subordinators, with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion gives a model of anomalous diffusion, which exhibits a sub-diffusive behavior.