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

Author

Listed:
  • Alessandro Gregorio

    (“Sapienza” University of Rome)

  • Francesco Iafrate

    (“Sapienza” University of Rome)

Abstract

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.

Suggested Citation

  • Alessandro Gregorio & Francesco Iafrate, 2024. "Path Dynamics of Time-Changed Lévy Processes: A Martingale Approach," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3246-3280, November.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01361-1
    DOI: 10.1007/s10959-024-01361-1
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    References listed on IDEAS

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    1. Meerschaert, Mark M. & Toaldo, Bruno, 2019. "Relaxation patterns and semi-Markov dynamics," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2850-2879.
    2. Kei Kobayashi, 2011. "Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 24(3), pages 789-820, September.
    3. Marjorie Hahn & Kei Kobayashi & Sabir Umarov, 2012. "SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations," Journal of Theoretical Probability, Springer, vol. 25(1), pages 262-279, March.
    4. Mauro Politi & Taisei Kaizoji & Enrico Scalas, 2011. "Full characterization of the fractional Poisson process," Papers 1104.4234, arXiv.org.
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