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Spectral representations of quasi-infinitely divisible processes

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  • Passeggeri, Riccardo

Abstract

This work is divided in three parts. First, we introduce quasi-infinitely divisible (QID) random measures and formulate spectral representations. Second, we introduce QID stochastic integrals and present integrability conditions, continuity properties and spectral representations. Finally, we introduce QID processes, i.e. stochastic processes with QID finite dimensional distributions. For example, a process X is QID if there exist two ID processes Y and Z such that X+Y=dZ with Y independent of X. The class of QID processes is strictly larger than the class of ID processes. We provide spectral representations and Lévy–Khintchine formulations for potentially all QID processes. Many examples are presented.

Suggested Citation

  • Passeggeri, Riccardo, 2020. "Spectral representations of quasi-infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1735-1791.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:3:p:1735-1791
    DOI: 10.1016/j.spa.2019.05.014
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    References listed on IDEAS

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    1. Horn, Roger A. & Steutel, F. W., 1978. "On multivariate infinitely divisible distributions," Stochastic Processes and their Applications, Elsevier, vol. 6(2), pages 139-151, January.
    2. Horowitz, Joseph, 1986. "Gaussian random measures," Stochastic Processes and their Applications, Elsevier, vol. 22(1), pages 129-133, May.
    3. David Berger, 2019. "On quasi‐infinitely divisible distributions with a point mass," Mathematische Nachrichten, Wiley Blackwell, vol. 292(8), pages 1674-1684, August.
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