IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v106y2015icp149-156.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715215002515
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2015.07.018?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    3. 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.
    4. 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.
    5. Bertoin, J. & Doney, R. A., 1994. "Cramer's estimate for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 363-365, December.
    6. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Grzywny, Tomasz & Kwaśnicki, Mateusz, 2018. "Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 1-38.
    2. Chen, Zhen-Qing & Wang, Jie-Ming, 2022. "Boundary Harnack principle for diffusion with jumps," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 342-395.
    3. 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.
    4. Asghari, N.M. & Dȩbicki, K. & Mandjes, M., 2015. "Exact tail asymptotics of the supremum attained by a Lévy process," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 180-184.
    5. Mijatović, Aleksandar & Pistorius, Martijn, 2015. "Buffer-overflows: Joint limit laws of undershoots and overshoots of reflected processes," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 2937-2954.
    6. Griffin, Philip S., 2020. "General tax structures for a Lévy insurance risk process under the Cramér condition," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1368-1387.
    7. Jérôme Spielmann, 2018. "Classification of the Bounds on the Probability of Ruin for Lévy Processes with Light-tailed Jumps," Working Papers hal-01597828, HAL.
    8. Griffin, Philip S. & Maller, Ross A. & Roberts, Dale, 2013. "Finite time ruin probabilities for tempered stable insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 478-489.
    9. J'er^ome Spielmann, 2017. "Classification of the Bounds on the Probability of Ruin for L{\'e}vy Processes with Light-tailed Jumps," Papers 1709.10295, arXiv.org, revised Feb 2018.
    10. Liu, Rongli & Ren, Yan-Xia & Song, Renming, 2022. "Convergence rate for a class of supercritical superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 286-327.
    11. Chaumont, Loïc & Rivero, Víctor, 2007. "On some transformations between positive self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1889-1909, December.
    12. Arista, Jonas & Rivero, Víctor, 2023. "Implicit renewal theory for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 262-287.
    13. Palmowski, Zbigniew & Pistorius, Martijn, 2009. "Cramér asymptotics for finite time first passage probabilities of general Lévy processes," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1752-1758, August.
    14. Baurdoux, E.J. & Palmowski, Z. & Pistorius, M.R., 2017. "On future drawdowns of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2679-2698.
    15. Griffin, Philip S. & Maller, Ross A. & Schaik, Kees van, 2012. "Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 382-392.
    16. Boxma, Onno & Kella, Offer & Mandjes, Michel, 2023. "On fluctuation-theoretic decompositions via Lindley-type recursions," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 316-336.
    17. Griffin, Philip S., 2022. "Path decomposition of a reflected Lévy process on first passage over high levels," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 29-47.
    18. Ren, Yan-Xia & Song, Renming & Sun, Zhenyao, 2020. "Limit theorems for a class of critical superprocesses with stable branching," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4358-4391.
    19. 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.
    20. Kim, Kyung-Youn & Kim, Panki, 2014. "Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C1,η open sets," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3055-3083.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:106:y:2015:i:c:p:149-156. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.