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On Simpson’s Rule and Fractional Brownian Motion with $$H = 1/10$$ H = 1 / 10

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  • Daniel Harnett

    (University of Wisconsin-Stevens Point
    University of Kansas)

  • David Nualart

    (University of Wisconsin-Stevens Point
    University of Kansas)

Abstract

We consider stochastic integration with respect to fractional Brownian motion (fBm) with $$H 1/10$$ H > 1 / 10 . For the case $$H = 1/10$$ H = 1 / 10 , we prove that the sequence of sums converges in distribution. Consequently, we have an Itô-like formula for the resulting stochastic integral. The convergence in distribution follows from a Malliavin calculus theorem that first appeared in Nourdin and Nualart (J Theor Probab 23:39–64, 2010).

Suggested Citation

  • Daniel Harnett & David Nualart, 2015. "On Simpson’s Rule and Fractional Brownian Motion with $$H = 1/10$$ H = 1 / 10," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1651-1688, December.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0552-1
    DOI: 10.1007/s10959-014-0552-1
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    References listed on IDEAS

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    1. Ivan Nourdin & David Nualart, 2010. "Central Limit Theorems for Multiple Skorokhod Integrals," Journal of Theoretical Probability, Springer, vol. 23(1), pages 39-64, March.
    2. Harnett, Daniel & Nualart, David, 2012. "Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3460-3505.
    3. Nualart, D. & Ortiz-Latorre, S., 2008. "Central limit theorems for multiple stochastic integrals and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 614-628, April.
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    Cited by:

    1. Daniel Harnett & Arturo Jaramillo & David Nualart, 2019. "Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1105-1144, September.

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