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Pathwise Taylor expansions for random fields on multiple dimensional paths

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  • Buckdahn, Rainer
  • Ma, Jin
  • Zhang, Jianfeng

Abstract

In this paper we establish the pathwise Taylor expansions for random fields that are “regular” in terms of Dupire’s path-derivatives [6]. Using the language of pathwise calculus, we carry out the Taylor expansion naturally to any order and for any dimension, which extends the result of Buckdahn et al. (2011). More importantly, the expansion can be both “forward” and “backward”, and the remainder is estimated in a pathwise manner. This result will be the main building block for our new notion of viscosity solution to forward path-dependent PDEs corresponding to (forward) stochastic PDEs in our accompanying paper Buckdahn et al. [4].

Suggested Citation

  • Buckdahn, Rainer & Ma, Jin & Zhang, Jianfeng, 2015. "Pathwise Taylor expansions for random fields on multiple dimensional paths," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2820-2855.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:7:p:2820-2855
    DOI: 10.1016/j.spa.2015.02.004
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    References listed on IDEAS

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    1. Karandikar, Rajeeva L., 1995. "On pathwise stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 11-18, May.
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    Cited by:

    1. Ivan Guo & Gregoire Loeper, 2018. "Path Dependent Optimal Transport and Model Calibration on Exotic Derivatives," Papers 1812.03526, arXiv.org, revised Sep 2020.
    2. Keller, Christian & Zhang, Jianfeng, 2016. "Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 735-766.
    3. Bruno Dupire & Valentin Tissot-Daguette, 2022. "Functional Expansions," Papers 2212.13628, arXiv.org, revised Mar 2023.

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