IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v28y2015i2d10.1007_s10959-013-0522-z.html
   My bibliography  Save this article

Stochastic Analysis for Obtuse Random Walks

Author

Listed:
  • Uwe Franz

    (UFC)

  • Tarek Hamdi

    (IPEST, Université de Carthage)

Abstract

We present a construction of the basic operators of stochastic analysis (gradient and divergence) for a class of discrete-time normal martingales called obtuse random walks. The approach is based on the chaos representation property and discrete multiple stochastic integrals. We show that these operators satisfy similar identities as in the case of the Bernoulli random walks. We prove a Clark–Ocone-type predictable representation formula, obtain two covariance identities and derive a deviation inequality. We close the exposition by an application to option hedging in discrete time.

Suggested Citation

  • Uwe Franz & Tarek Hamdi, 2015. "Stochastic Analysis for Obtuse Random Walks," Journal of Theoretical Probability, Springer, vol. 28(2), pages 619-649, June.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:2:d:10.1007_s10959-013-0522-z
    DOI: 10.1007/s10959-013-0522-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-013-0522-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-013-0522-z?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. J. Jacod & A.N. Shiryaev, 1998. "Local martingales and the fundamental asset pricing theorems in the discrete-time case," Finance and Stochastics, Springer, vol. 2(3), pages 259-273.
    2. Stéphane Attal & Ameur Dhahri, 2010. "Repeated Quantum Interactions and Unitary Random Walks," Journal of Theoretical Probability, Springer, vol. 23(2), pages 345-361, June.
    Full references (including those not matched with items on IDEAS)

    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. Romain Blanchard & Laurence Carassus & Miklós Rásonyi, 2018. "No-arbitrage and optimal investment with possibly non-concave utilities: a measure theoretical approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(2), pages 241-281, October.
    2. Irle, A., 2004. "A measure-theoretic approach to completeness of financial markets," Statistics & Probability Letters, Elsevier, vol. 68(1), pages 1-7, June.
    3. Michał Baran, 2007. "Asymptotic pricing in large financial markets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(1), pages 1-20, August.
    4. Tomasz R. Bielecki & Igor Cialenco & Ismail Iyigunler & Rodrigo Rodriguez, 2012. "Dynamic Conic Finance: Pricing and Hedging in Market Models with Transaction Costs via Dynamic Coherent Acceptability Indices," Papers 1205.4790, arXiv.org, revised Jun 2013.
    5. Sergey Badikov & Mark H. A. Davis & Antoine Jacquier, 2018. "Perturbation analysis of sub/super hedging problems," Papers 1806.03543, arXiv.org, revised May 2021.
    6. Daniel Bartl & Michael Kupper & David J. Prömel & Ludovic Tangpi, 2019. "Duality for pathwise superhedging in continuous time," Finance and Stochastics, Springer, vol. 23(3), pages 697-728, July.
    7. Peter Christoffersen & Redouane Elkamhi & Bruno Feunou & Kris Jacobs, 2010. "Option Valuation with Conditional Heteroskedasticity and Nonnormality," The Review of Financial Studies, Society for Financial Studies, vol. 23(5), pages 2139-2183.
    8. Xiaotie Deng & Zhong Li & Shouyang Wang & Hailiang Yang, 2005. "Necessary and Sufficient Conditions for Weak No-Arbitrage in Securities Markets with Frictions," Annals of Operations Research, Springer, vol. 133(1), pages 265-276, January.
    9. Duffie, Darrell, 2003. "Intertemporal asset pricing theory," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, edition 1, volume 1, chapter 11, pages 639-742, Elsevier.
    10. Daniel Bartl, 2016. "Exponential utility maximization under model uncertainty for unbounded endowments," Papers 1610.00999, arXiv.org, revised Feb 2019.
    11. Eckhard Platen & Stefan Tappe, 2020. "The Fundamental Theorem of Asset Pricing for Self-Financing Portfolios," Research Paper Series 411, Quantitative Finance Research Centre, University of Technology, Sydney.
    12. Jos'e Manuel Corcuera, 2021. "The Golden Age of the Mathematical Finance," Papers 2102.06693, arXiv.org, revised Mar 2021.
    13. Balbás, Alejandro & Mirás, Miguel Ángel & Muñoz-Bouzo, María José, 2001. "Projective system approach to the martingale characterization of the absence of arbitrage," DEE - Working Papers. Business Economics. WB wb011505, Universidad Carlos III de Madrid. Departamento de Economía de la Empresa.
    14. Laurence Carassus & Massinissa Ferhoune, 2024. "Nonconcave Robust Utility Maximization under Projective Determinacy," Papers 2403.11824, arXiv.org.
    15. Micha{l} Barski, 2016. "Large losses - probability minimizing approach," Papers 1601.03388, arXiv.org.
    16. Miklos Rasonyi & Lukasz Stettner, 2005. "On utility maximization in discrete-time financial market models," Papers math/0505243, arXiv.org.
    17. Tomasz R. Bielecki & Igor Cialenco & Rodrigo Rodriguez, 2012. "No-Arbitrage Pricing for Dividend-Paying Securities in Discrete-Time Markets with Transaction Costs," Papers 1205.6254, arXiv.org, revised Jun 2013.
    18. Eckhard Platen & Stefan Tappe, 2020. "No arbitrage and multiplicative special semimartingales," Papers 2005.05575, arXiv.org, revised Sep 2022.
    19. Matteo Burzoni & Mario Sikic, 2018. "Robust martingale selection problem and its connections to the no-arbitrage theory," Papers 1801.03574, arXiv.org, revised Nov 2018.
    20. Romain Blanchard & Laurence Carassus & Miklos Rasonyi, 2018. "Optimal investment with possibly non-concave utilities and no-arbitrage: a measure theoretical approach," Post-Print hal-01883419, HAL.

    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:spr:jotpro:v:28:y:2015:i:2:d:10.1007_s10959-013-0522-z. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.