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Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients

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  • Keller, Christian
  • Zhang, Jianfeng

Abstract

This paper introduces path derivatives, in the spirit of Dupire’s functional Itô calculus, for controlled rough paths in rough path theory with possibly non-geometric rough paths. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients are less regular in the time variable, which is not covered in the existing literature. The results are useful for studying viscosity solutions of stochastic PDEs.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:3:p:735-766
    DOI: 10.1016/j.spa.2015.09.018
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    References listed on IDEAS

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    1. Buckdahn, Rainer & Ma, Jin, 2001. "Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 205-228, June.
    2. Buckdahn, Rainer & Ma, Jin, 2001. "Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 181-204, June.
    3. 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.
    4. Karandikar, Rajeeva L., 1995. "On pathwise stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 11-18, May.
    5. Terry Lyons, 2014. "Rough paths, Signatures and the modelling of functions on streams," Papers 1405.4537, arXiv.org.
    6. 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.
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