On two multistable extensions of stable L\'evy motion and their semimartingale representation

TL;DR

This paper compares two types of multistable Lévy motions with varying stability indices, revealing their distinct properties and establishing their semimartingale decompositions.

Contribution

It introduces and analyzes two multistable Lévy motions, demonstrating their differences and proving their semimartingale properties with explicit decompositions.

Findings

One multistable Lévy motion is a pure-jump Markov process.

The other multistable Lévy motion is neither Markov nor pure-jump.

Both processes are semimartingales with explicit decompositions.

Abstract

We compare two definitions of multistable L\'evy motions. Such processes are extensions of classical L\'evy motion where the stability index is allowed to vary in time. We show that the two multistable L\'evy motions have distinct properties: in particular, one is a pure-jump Markov process, while the other one satisfies neither of these properties. We prove that both are semimartingales and provide semimartingale decompositions.