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Perturbations of singular fractional SDEs

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  • Gassiat, Paul
  • Mądry, Łukasz

Abstract

We obtain well-posedness results for a class of ODE with a singular drift and additive fractional noise, whose right-hand-side involves some bounded variation terms depending on the solution. Examples of such equations are reflected equations, where the solution is constrained to remain in a rectangular domain, as well as so-called perturbed equations, where the dynamics depend on the running extrema of the solution. Our proof is based on combining the Catellier–Gubinelli approach based on Young nonlinear integration, with some Lipschitz estimates in p-variation for maps of Skorokhod type, due to Falkowski and Słominski. An important step requires to prove that fractional Brownian motion, when perturbed by sufficiently regular paths (in the sense of p-variation), retains its regularization properties. This is done by applying a variant of the stochastic sewing lemma.

Suggested Citation

  • Gassiat, Paul & Mądry, Łukasz, 2023. "Perturbations of singular fractional SDEs," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 137-172.
    • Handle: RePEc:eee:spapps:v:161:y:2023:i:c:p:137-172
    DOI: 10.1016/j.spa.2023.04.004
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    References listed on IDEAS

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    1. Zhang, Tu-Sheng, 1994. "On the strong solutions of one-dimensional stochastic differential equations with reflecting boundary," Stochastic Processes and their Applications, Elsevier, vol. 50(1), pages 135-147, March.
    2. Wen Yue & Tusheng Zhang, 2015. "Absolute Continuity of the Laws of Perturbed Diffusion Processes and Perturbed Reflected Diffusion Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 587-618, June.
    3. Nualart, David & Ouknine, Youssef, 2002. "Regularization of differential equations by fractional noise," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 103-116, November.
    4. Catellier, R. & Gubinelli, M., 2016. "Averaging along irregular curves and regularisation of ODEs," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2323-2366.
    5. Aida, Shigeki, 2015. "Reflected rough differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3570-3595.
    6. 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.
    7. Falkowski, Adrian & Słomiński, Leszek, 2022. "SDEs with two reflecting barriers driven by semimartingales and processes with bounded p-variation," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 164-186.
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