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One-dimensional reflected rough differential equations

Author

Listed:
  • Deya, Aurélien
  • Gubinelli, Massimiliano
  • Hofmanová, Martina
  • Tindel, Samy

Abstract

We prove existence and uniqueness of the solution of a one-dimensional rough differential equation driven by a step-2 rough path and reflected at zero. The whole difficulty of the problem (at least as far as uniqueness is concerned) lies in the non-continuity of the Skorohod map with respect to the topologies under consideration in the rough case. Our argument to overcome this obstacle is inspired by some ideas we introduced in a previous work dealing with rough kinetic PDEs arXiv:1604.00437.

Suggested Citation

  • Deya, Aurélien & Gubinelli, Massimiliano & Hofmanová, Martina & Tindel, Samy, 2019. "One-dimensional reflected rough differential equations," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3261-3281.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:9:p:3261-3281
    DOI: 10.1016/j.spa.2018.09.007
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    References listed on IDEAS

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    1. Deya, Aurélien & Tindel, Samy, 2011. "Rough Volterra equations 2: Convolutional generalized integrals," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1864-1899, August.
    2. Aida, Shigeki, 2015. "Reflected rough differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3570-3595.
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    Cited by:

    1. Coghi, Michele & Nilssen, Torstein, 2021. "Rough nonlocal diffusions," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 1-56.
    2. Allan, Andrew L. & Liu, Chong & Prömel, David J., 2021. "Càdlàg rough differential equations with reflecting barriers," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 79-104.
    3. Gassiat, Paul & Mądry, Łukasz, 2023. "Perturbations of singular fractional SDEs," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 137-172.

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