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On the support of solutions to stochastic differential equations with path-dependent coefficients

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  • Cont, Rama
  • Kalinin, Alexander

Abstract

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron–Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock–Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on functional Itô calculus.

Suggested Citation

  • Cont, Rama & Kalinin, Alexander, 2020. "On the support of solutions to stochastic differential equations with path-dependent coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2639-2674.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:2639-2674
    DOI: 10.1016/j.spa.2019.07.015
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    References listed on IDEAS

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    1. Ledoux, M. & Qian, Z. & Zhang, T., 2002. "Large deviations and support theorem for diffusion processes via rough paths," Stochastic Processes and their Applications, Elsevier, vol. 102(2), pages 265-283, December.
    2. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
    3. Cont, Rama & Lu, Yi, 2016. "Weak approximation of martingale representations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 857-882.
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    Cited by:

    1. Damiano Brigo & Federico Graceffa & Alexander Kalinin, 2021. "Mild to classical solutions for XVA equations under stochastic volatility," Papers 2112.11808, arXiv.org.
    2. Giulia Nunno & Michele Giordano, 2024. "Stochastic Volterra equations with time-changed Lévy noise and maximum principles," Annals of Operations Research, Springer, vol. 336(1), pages 1265-1287, May.

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