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arXiv · 1910.00660

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

Benjamin Cooper Boniece·Gustavo Didier·Farzad Sabzikar
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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.

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Benjamin Cooper Boniece, Gustavo Didier, Farzad Sabzikar. 2019-10-01. On fractional L\'evy processes: tempering, sample path properties and stochastic integration. https://arxiv.org/abs/1910.00660

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