On Two Multistable Extensions of Stable L,vy Motion and Their Semi-martingale Representations

We study two versions of multistable L,vy motion. Such processes are extensions of classical L,vy motion where the stability index is allowed to vary in time, a useful property for modeling non-increment stationary phenomena. We show that the two multistable L,vy 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 semi-martingales and provide semi-martingale decompositions.