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Rough Volterra equations 2: Convolutional generalized integrals

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  • Deya, Aurélien
  • Tindel, Samy

Abstract

We define and solve Volterra equations driven by a non-differentiable signal, by means of a variant of the rough paths theory which allows us to handle generalized integrals weighted by an exponential coefficient. The results are applied to a standard rough path , with [gamma]>1/3, which includes the case of fractional Brownian motion with Hurst index H>1/3.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:8:p:1864-1899
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    References listed on IDEAS

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    1. Alòs, Elisa & Nualart, David, 1997. "Anticipating stochastic Volterra equations," Stochastic Processes and their Applications, Elsevier, vol. 72(1), pages 73-95, December.
    2. Wang, Zhidong, 2008. "Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1062-1071, July.
    3. Quer-Sardanyons, Lluís & Tindel, Samy, 2007. "The 1-d stochastic wave equation driven by a fractional Brownian sheet," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1448-1472, October.
    4. Cochran, W. George & Lee, Jung-Soon & Potthoff, Jürgen, 1995. "Stochastic Volterra equations with singular kernels," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 337-349, April.
    5. Yong, Jiongmin, 2006. "Backward stochastic Volterra integral equations and some related problems," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 779-795, May.
    6. David N. Keck & Mark A. McKibben, 2006. "Abstract semilinear stochastic Itó-Volterra integrodifferential equations," International Journal of Stochastic Analysis, Hindawi, vol. 2006, pages 1-22, July.
    7. Djehiche, Boualem & Eddahbi, M'hamed, 1999. "Large deviations for a stochastic Volterra-type equation in the Besov-Orlicz space," Stochastic Processes and their Applications, Elsevier, vol. 81(1), pages 39-72, May.
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    Cited by:

    1. 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.
    2. Harang, Fabian A. & Tindel, Samy, 2021. "Volterra equations driven by rough signals," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 34-78.
    3. Song, Jian & Tindel, Samy, 2022. "Skorohod and Stratonovich integrals for controlled processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 569-595.

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