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Ornstein - Uhlenbeck Process Driven By $$\alpha$$ α -stable Process and Its Gamma Subordination

Author

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  • Janusz Gajda

    (University of Warsaw)

  • Aleksandra Grzesiek

    (Hugo Steinhaus Center Wroclaw University of Science and Technology)

  • Agnieszka Wyłomańska

    (Hugo Steinhaus Center Wroclaw University of Science and Technology)

Abstract

The variety and diversity of phenomena surrounding us and easy access to empirical data require either new and more complicated models that allow to capture features to resemble the data. In this paper, we study the Ornstein-Uhlenbeck (OU) process driven by $$\alpha -$$ α - stable Lévy process and delayed by the Gamma subordinator. The considered model captures the important features of the parent process, i.e, the OU process with heavy-tailed-based distribution, however, it also possesses some characteristics that are not adequate to the model without the subordination scenario. Thus, it can be very useful for real data with very specific behavior. The considered model can be considered as the natural extension of the variance Gamma process that arises as the ordinary Brownian motion time changed by the Gamma process. We demonstrate the probabilistic properties of the proposed model and indicate how the theoretical results could be applied for the estimation of the model’s parameters.

Suggested Citation

  • Janusz Gajda & Aleksandra Grzesiek & Agnieszka Wyłomańska, 2023. "Ornstein - Uhlenbeck Process Driven By $$\alpha$$ α -stable Process and Its Gamma Subordination," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-17, March.
    • Handle: RePEc:spr:metcap:v:25:y:2023:i:1:d:10.1007_s11009-023-09999-w
    DOI: 10.1007/s11009-023-09999-w
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