Electronic Journal of Probability

The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

Ivan Nourdin, Anthony Réveillac, and Jason Swanson

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Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int\!g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of $B$.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 70, 2117-2162.

Accepted: 14 December 2010
First available in Project Euclid: 1 June 2016

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Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals
Secondary: 60G15: Gaussian processes 60G18: Self-similar processes 60J05: Discrete-time Markov processes on general state spaces

Stochastic integration Stratonovich integral fractional Brownian motion weak convergence Malliavin calculus

This work is licensed under aCreative Commons Attribution 3.0 License.


Nourdin, Ivan; Réveillac, Anthony; Swanson, Jason. The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6. Electron. J. Probab. 15 (2010), paper no. 70, 2117--2162. doi:10.1214/EJP.v15-843. https://projecteuclid.org/euclid.ejp/1464819855

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