Path Dynamics of Time-Changed Lévy Processes: A Martingale Approach

Lévy processes time-changed by inverse subordinators have been intensively studied in the last years. Their importance in connection with non-local operators and semi-Markov dynamics is well understood, but, in our view, several questions remain open concerning the probabilistic structure of such processes. The time-changed Lévy processes are particularly useful to describe complex systems with fractional and/or anomalous dynamics. The purpose of our work is to analyze the features of the sample paths of such processes, focusing on a martingale-based approach. We introduce the fractional Poisson random measure as the main tool for dealing with the jump component of time-changed càdlàg processes. Further, the fractional random measure is an interesting and novel topic in itself, and thus, it is thoroughly analyzed in the paper. A central role in our analysis is then played by fractional Poisson integrals (involving the aforementioned fractional Poisson measure) which allow a useful description of the random jumps. We investigate these stochastic integrals and the martingale property of their compensated counterpart. Therefore, we are able to obtain a semimartingale representation of time-changed processes analogous to the celebrated Lévy–Itô decomposition. Finally, an approximation scheme of such random processes will be discussed.