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A C^{0,1}-functional Itô's formula and its applications in mathematical finance

Author

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  • Bruno Bouchard

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique, PSL - Université Paris Sciences et Lettres)

  • Grégoire Loeper

    (Monash University [Melbourne])

  • Xiaolu Tan

    (Department of mathematics, Chinese University of Hong Kong - CUHK - The Chinese University of Hong Kong [Hong Kong])

Abstract

Using Dupire's notion of vertical derivative, we provide a functional (path-dependent) extension of the Itô's formula of Gozzi and Russo (2006) that applies to C^{0,1}-functions of continuous weak Dirichlet processes. It is motivated and illustrated by its applications to the hedging or superhedging problems of path-dependent options in mathematical finance, in particular in the case of model uncertainty.

Suggested Citation

  • Bruno Bouchard & Grégoire Loeper & Xiaolu Tan, 2022. "A C^{0,1}-functional Itô's formula and its applications in mathematical finance," Post-Print hal-03105342, HAL.
    • Handle: RePEc:hal:journl:hal-03105342
    DOI: 10.1016/j.spa.2022.02.010
    Note: View the original document on HAL open archive server: https://hal.science/hal-03105342
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    References listed on IDEAS

    as
    1. Gozzi, Fausto & Russo, Francesco, 2006. "Weak Dirichlet processes with a stochastic control perspective," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1563-1583, November.
    2. Bruno Bouchard & Xiaolu Tan, 2019. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Working Papers hal-02398881, HAL.
    3. Ren, Zhenjie & Tan, Xiaolu, 2017. "On the convergence of monotone schemes for path-dependent PDEs," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1738-1762.
    4. Bruno Bouchard & Jean-François Chassagneux, 2016. "Fundamentals and Advanced Techniques in Derivatives Hedging," Post-Print hal-01348864, HAL.
    5. Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    6. Jianfeng Zhang & Jia Zhuo, 2014. "Monotone schemes for fully nonlinear parabolic path dependent PDEs," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 1-23.
    7. Ariel Neufeld & Marcel Nutz, 2012. "Superreplication under Volatility Uncertainty for Measurable Claims," Papers 1208.6486, arXiv.org, revised Apr 2013.
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