Abstract
We compare two definitions of multistable Lévy motions. Such processes are extensions of classical Lévy motion where the stability index is allowed to vary in time. We show that the two multistable Lévy 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.
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