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A ℂ0,1-functional Itô’s formula and its applications in mathematical finance

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  • Bouchard, Bruno
  • Loeper, Grégoire
  • Tan, Xiaolu

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 C0,1-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. In this context, we also prove a new regularity result for the vertical derivative of candidate solutions to a class of path-depend PDEs, using an approximation argument which seems to be original and of own interest.

Suggested Citation

  • Bouchard, Bruno & Loeper, Grégoire & Tan, Xiaolu, 2022. "A ℂ0,1-functional Itô’s formula and its applications in mathematical finance," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 299-323.
    • Handle: RePEc:eee:spapps:v:148:y:2022:i:c:p:299-323
    DOI: 10.1016/j.spa.2022.02.010
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    References listed on IDEAS

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    1. Sara Biagini & Bruno Bouchard & Constantinos Kardaras & Marcel Nutz, 2017. "Robust Fundamental Theorem For Continuous Processes," Mathematical Finance, Wiley Blackwell, vol. 27(4), pages 963-987, October.
    2. 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.
    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. Sara Biagini & Bruno Bouchard & Constantinos Kardaras & Marcel Nutz, 2017. "Robust Fundamental Theorem for Continuous Processes," Post-Print hal-01076062, HAL.
    5. Bruno Bouchard & Jean-François Chassagneux, 2016. "Fundamentals and Advanced Techniques in Derivatives Hedging," Post-Print hal-01348864, HAL.
    6. Russo, Francesco & Vallois, Pierre, 1995. "The generalized covariation process and Ito formula," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 81-104, September.
    7. Errami, Mohammed & Russo, Francesco, 2003. "n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes," Stochastic Processes and their Applications, Elsevier, vol. 104(2), pages 259-299, April.
    8. Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    9. 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.
    10. Bruno Bouchard & Xiaolu Tan, 2021. "A quasi-sure optional decomposition and super-hedging result on the Skorokhod space," Finance and Stochastics, Springer, vol. 25(3), pages 505-528, July.
    11. Ariel Neufeld & Marcel Nutz, 2012. "Superreplication under Volatility Uncertainty for Measurable Claims," Papers 1208.6486, arXiv.org, revised Apr 2013.
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