Tempered Fractional Multistable Motion and Tempered Multifractional Stable Motion

TL;DR

This paper introduces two new classes of stochastic processes, tempered fractional multistable motion and tempered multifractional stable motion, which incorporate exponential tempering to model data with dependence and variable jump intensity.

Contribution

It extends existing fractional multistable and multifractional stable motions by adding exponential tempering, and analyzes their fundamental properties and potential applications.

Findings

Analyzed scaling properties and tail behaviors.

Established sample path regularity and local Hölder exponents.

Identified semi-long-range dependence structures.

Abstract

This work defines two classes of processes, that we term {\it tempered fractional multistable motion} and {\it tempered multifractional stable motion}. They are extensions of fractional multistable motion and multifractional stable motion, respectively, obtained by adding an exponential tempering to the integrands. We investigate certain basic features of these processes, including scaling property, tail probabilities, absolute moment, sample path properties, pointwise H\"{o}lder exponent, H\"{o}lder continuity of quasi norm, (strong) localisability and semi-long-range dependence structure. These processes may provide useful models for data that exhibit both dependence and varying local regularity/intensity of jumps.