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On the joint distribution of the supremum functional and its last occurrence for subordinated linear Brownian motion

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  • Fotopoulos, Stergios
  • Jandhyala, Venkata
  • Wang, Jun

Abstract

In this communication, a convenient Laplace transform of the bivariate supremum and the last time the supremum is attained, is established when the underlying Lévy process is subordinate Brownian motion with drift. Explicit integral representations of the Laplace transform of the joint supremum and the last time it occurred are derived in terms of the Lévy–Khintchine exponent of the subordinator Laplace exponent. As an example, a subordinator with exponential Lévy measure is exploited.

Suggested Citation

  • Fotopoulos, Stergios & Jandhyala, Venkata & Wang, Jun, 2015. "On the joint distribution of the supremum functional and its last occurrence for subordinated linear Brownian motion," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 149-156.
    • Handle: RePEc:eee:stapro:v:106:y:2015:i:c:p:149-156
    DOI: 10.1016/j.spl.2015.07.018
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    References listed on IDEAS

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    1. Kim, Panki & Song, Renming & Vondraček, Zoran, 2013. "Potential theory of subordinate Brownian motions with Gaussian components," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 764-795.
    2. Kim, Panki & Song, Renming & Vondracek, Zoran, 2009. "Boundary Harnack principle for subordinate Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1601-1631, May.
    3. Kwaśnicki, Mateusz & Małecki, Jacek & Ryznar, Michał, 2013. "First passage times for subordinate Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1820-1850.
    4. Kuznetsov, A., 2013. "On the density of the supremum of a stable process," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 986-1003.
    5. Kim, Panki & Lee, Yunju, 2013. "Oscillation of harmonic functions for subordinate Brownian motion and its applications," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 422-445.
    6. Bertoin, J. & Doney, R. A., 1994. "Cramer's estimate for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 363-365, December.
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    Cited by:

    1. Stergios B. Fotopoulos & Venkata K. Jandhyala & Alex Paparas, 2021. "Some Properties of the Multivariate Generalized Hyperbolic Laws," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 187-205, February.
    2. Stergios B. Fotopoulos & Alex Paparas & Venkata K. Jandhyala, 2020. "Multivariate generalized hyperbolic laws for modeling financial log‐returns: Empirical and theoretical considerations," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 36(5), pages 757-775, September.

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