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Volterra equations driven by rough signals

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  • Harang, Fabian A.
  • Tindel, Samy

Abstract

This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon’s theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product behaving like a convolution extending the classical tensor product. We show that this convolution product is well defined for a large class of Volterra paths, and we provide an analogue of the extension theorem from the theory of rough paths (which guarantees in particular the existence of a Volterra signature). Moreover the concept of convolution product is essential in the construction of Volterra controlled paths, which is the natural class of processes to be integrated with respect to the driving noise in our situation. This leads to a rough integral given as a functional of the Volterra signature and the Volterra controlled paths, combined through the convolution product. The rough integral is then used in the construction of unique solutions to Volterra equations driven by Hölder noises with singular kernels. An example concerning Brownian noises and a singular kernel is treated.

Suggested Citation

  • Harang, Fabian A. & Tindel, Samy, 2021. "Volterra equations driven by rough signals," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 34-78.
    • Handle: RePEc:eee:spapps:v:142:y:2021:i:c:p:34-78
    DOI: 10.1016/j.spa.2021.08.001
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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. Christian Bayer & Peter K. Friz & Paul Gassiat & Jorg Martin & Benjamin Stemper, 2020. "A regularity structure for rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 782-832, July.
    3. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    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.
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    Cited by:

    1. Song, Jian & Tindel, Samy, 2022. "Skorohod and Stratonovich integrals for controlled processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 569-595.
    2. Qi Feng & Jianfeng Zhang, 2021. "Cubature Method for Stochastic Volterra Integral Equations," Papers 2110.12853, arXiv.org, revised Jul 2023.

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