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On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands

Author

Listed:
  • Ehsan Azmoodeh

    (University of Liverpool)

  • Pauliina Ilmonen

    (Aalto University School of Science)

  • Nourhan Shafik

    (Aalto University School of Science)

  • Tommi Sottinen

    (University of Vaasa)

  • Lauri Viitasaari

    (Uppsala University)

Abstract

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order $$H>\frac{1}{2}$$ H > 1 2 with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $$L^1$$ L 1 -distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $$n^{1-2H}$$ n 1 - 2 H , which is twice as good as the best known results in the case of discontinuous integrands and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply a change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

Suggested Citation

  • Ehsan Azmoodeh & Pauliina Ilmonen & Nourhan Shafik & Tommi Sottinen & Lauri Viitasaari, 2024. "On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands," Journal of Theoretical Probability, Springer, vol. 37(1), pages 721-743, March.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-023-01272-7
    DOI: 10.1007/s10959-023-01272-7
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    References listed on IDEAS

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    1. Chen, Zhe & Leskelä, Lasse & Viitasaari, Lauri, 2019. "Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2723-2757.
    2. Ehsan Azmoodeh & Yuliya Mishura & Farzad Sabzikar, 2022. "How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?," Journal of Theoretical Probability, Springer, vol. 35(1), pages 484-527, March.
    3. Ehsan Azmoodeh & Lauri Viitasaari, 2015. "Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 28(1), pages 396-422, March.
    4. Tommi Sottinen & Lauri Viitasaari, 2016. "Pathwise Integrals and Itô–Tanaka Formula for Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 29(2), pages 590-616, June.
    5. Yaskov, Pavel, 2019. "On pathwise Riemann–Stieltjes integrals," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 101-107.
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