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Skorohod and Stratonovich integrals for controlled processes

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  • Song, Jian
  • Tindel, Samy

Abstract

Given a continuous Gaussian process x which gives rise to a p-geometric rough path for p∈(2,3), and a general continuous process y controlled by x, under proper conditions we establish the relationship between the Skorohod integral ∫0tysd♢xs and the Stratonovich integral ∫0tysdxs. Our strategy is to employ the tools from rough paths theory and Malliavin calculus to analyze discrete sums of the integrals.

Suggested Citation

  • Song, Jian & Tindel, Samy, 2022. "Skorohod and Stratonovich integrals for controlled processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 569-595.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:569-595
    DOI: 10.1016/j.spa.2022.05.002
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    References listed on IDEAS

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    1. Benjamin Gess & Cheng Ouyang & Samy Tindel, 2020. "Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths," Journal of Theoretical Probability, Springer, vol. 33(2), pages 611-648, June.
    2. 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.
    3. Baudoin, Fabrice & Coutin, Laure, 2007. "Operators associated with a stochastic differential equation driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 117(5), pages 550-574, May.
    4. Ivan Nourdin & David Nualart, 2010. "Central Limit Theorems for Multiple Skorokhod Integrals," Journal of Theoretical Probability, Springer, vol. 23(1), pages 39-64, March.
    5. Harang, Fabian A. & Tindel, Samy, 2021. "Volterra equations driven by rough signals," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 34-78.
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    Cited by:

    1. Akihiko Takahashi & Toshihiro Yamada, 2023. "New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion," CARF F-Series CARF-F-563, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Feb 2024.
    2. Huang, Chuying & Wang, Xu, 2023. "Strong convergence rate of the Euler scheme for SDEs driven by additive rough fractional noises," Statistics & Probability Letters, Elsevier, vol. 194(C).
    3. Akihiko Takahashi & Toshihiro Yamada, 2023. "New Asymptotic Expansion Formula via Malliavin Calculus and Its Application to Rough Differential Equation Driven by Fractional," CIRJE F-Series CIRJE-F-1215, CIRJE, Faculty of Economics, University of Tokyo.

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