On fractional L\'evy processes: tempering, sample path properties and stochastic integration

TL;DR

This paper introduces two new classes of tempered fractional Lévy processes that model anomalous diffusion with semi-long range dependence, analyzing their sample path properties and developing a stochastic integration framework for them.

Contribution

The paper defines the tempered fractional Lévy processes of the first and second kinds, expanding the class of models for anomalous diffusion with non-Gaussian, finite-variance properties.

Findings

TFLP and TFLP II exhibit semi-long range dependence.

Sample path properties are rigorously established.

A new stochastic integration framework is developed for these processes.

Abstract

We define two new classes of stochastic processes, called tempered fractional L\'{e}vy process of the first and second kinds (TFLP and TFLP $I\!I$, respectively). TFLP and TFLP $I\!I$ make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional L\'{e}vy process. Accordingly, the increment processes of TFLP and TFLP $I\!I$ display semi-long range dependence. We establish the sample path properties of TFLP and TFLP $I\!I$. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP $I\!I$, which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.