IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v129y2019i8p2611-2653.html
   My bibliography  Save this article

Malliavin and Dirichlet structures for independent random variables

Author

Listed:
  • Decreusefond, Laurent
  • Halconruy, Hélène

Abstract

On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron–Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of U-statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions.

Suggested Citation

  • Decreusefond, Laurent & Halconruy, Hélène, 2019. "Malliavin and Dirichlet structures for independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2611-2653.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:8:p:2611-2653
    DOI: 10.1016/j.spa.2018.07.019
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918303892
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2018.07.019?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. Rhee, WanSoo T. & Talagrand, Michel, 1986. "Martingale inequalities and the Jackknife estimate of variance," Statistics & Probability Letters, Elsevier, vol. 4(1), pages 5-6, January.
    2. Nualart, David & Schoutens, Wim, 2000. "Chaotic and predictable representations for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 109-122, November.
    3. Arras, Benjamin & Swan, Yvik, 2017. "A stroll along the gamma," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3661-3688.
    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. H'el`ene Halconruy, 2021. "The insider problem in the trinomial model: a discrete-time jump process approach," Papers 2106.15208, arXiv.org, revised Sep 2023.

    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. Auguste Aman, 2012. "Reflected Generalized Backward Doubly SDEs Driven by Lévy Processes and Applications," Journal of Theoretical Probability, Springer, vol. 25(4), pages 1153-1172, December.
    2. Fan, Xiliang & Ren, Yong & Zhu, Dongjin, 2010. "A note on the doubly reflected backward stochastic differential equations driven by a Lévy process," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 690-696, April.
    3. Houdré, Christian, 1997. "The iterated jackknife estimate of variance," Statistics & Probability Letters, Elsevier, vol. 35(2), pages 197-201, September.
    4. Masafumi Hayashi, 2010. "Coefficients of Asymptotic Expansions of SDE with Jumps," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 17(4), pages 373-389, December.
    5. Kim, Mun-Chol & O, Hun, 2021. "A general comparison theorem for reflected BSDEs," Statistics & Probability Letters, Elsevier, vol. 173(C).
    6. Klimsiak, Tomasz, 2015. "Reflected BSDEs on filtered probability spaces," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4204-4241.
    7. Mohamed Otmani, 2009. "Reflected BSDE Driven by a Lévy Process," Journal of Theoretical Probability, Springer, vol. 22(3), pages 601-619, September.
    8. Choe, Hi Jun & Lee, Ji Min & Lee, Jung-Kyung, 2018. "Malliavin calculus for subordinated Lévy process," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 392-401.
    9. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    10. Mitsui, Ken-ichi & Tabata, Yoshio, 2008. "A stochastic linear-quadratic problem with Lévy processes and its application to finance," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 120-152, January.
    11. Evelina Shamarova & Rui S'a Pereira, 2013. "Hedging in a market with jumps - an FBSDE approach," Papers 1309.2211, arXiv.org, revised Aug 2017.
    12. Horst Osswald, 2009. "A Smooth Approach to Malliavin Calculus for Lévy Processes," Journal of Theoretical Probability, Springer, vol. 22(2), pages 441-473, June.
    13. Solé, Josep Lluís & Utzet, Frederic & Vives, Josep, 2007. "Canonical Lévy process and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 165-187, February.
    14. Jamshidian, Farshid, 2008. "On the combinatorics of iterated stochastic integrals," MPRA Paper 7165, University Library of Munich, Germany.
    15. Ankirchner, Stefan, 2008. "On filtration enlargements and purely discontinuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1662-1678, September.
    16. Lorenzo Mercuri & Andrea Perchiazzo & Edit Rroji, 2020. "Finite Mixture Approximation of CARMA(p,q) Models," Papers 2005.10130, arXiv.org, revised May 2020.
    17. Langovoy, Mikhail, 2011. "Algebraic polynomials and moments of stochastic integrals," Statistics & Probability Letters, Elsevier, vol. 81(6), pages 627-631.
    18. Niu, Liqun, 2008. "Some stability results of optimal investment in a simple Lévy market," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 445-452, February.
    19. El Otmani, Mohamed, 2008. "BSDE driven by a simple Lévy process with continuous coefficient," Statistics & Probability Letters, Elsevier, vol. 78(11), pages 1259-1265, August.
    20. Schoutens, Wim & Studer, Michael, 2003. "Short-term risk management using stochastic Taylor expansions under Lévy models," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 173-188, August.

    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:spapps:v:129:y:2019:i:8:p:2611-2653. 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/505572/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.