Path dynamics of time-changed Lévy processes: a martingale approach

Iafrate Francesco, Sapienza University
De Gregorio Alessandro, Sapienza University

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 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 dealing with the jump component of time-changed càdlàg processes. A central role in our analysis is then played by fractional Poisson integrals which allow an useful description of the random jumps. We investigate these integrals and the martingale property of their compensated counterpart. Therefore, we are able to obtain a semi-martingale representation of time-changed processes analogous to the celebrated Lévy-Ito decomposition. Some future extensions are presented. Based on joint work with Alessandro De Gregorio.

Area: CS38 - Subordination and time-changed stochastic processes (Alessandro De Gregorio)

Keywords: fractional Poisson random measures, fractional compound Poisson processes, semimartingale decomposition, time-changed compensators, squared integrable martingales, stopped filtration

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