On fractional L\'evy processes: tempering, sample path properties and stochastic integration
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 , respectively). TFLP and TFLP 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 display semi-long range dependence. We establish the sample path properties of TFLP and TFLP . 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 , which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.
Cite
@article{arxiv.1910.00660,
title = {On fractional L\'evy processes: tempering, sample path properties and stochastic integration},
author = {Benjamin Cooper Boniece and Gustavo Didier and Farzad Sabzikar},
journal= {arXiv preprint arXiv:1910.00660},
year = {2019}
}