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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819855

Digital Object Identifier
doi:10.1214/EJP.v15-843

Mathematical Reviews number (MathSciNet)
MR2745728

Zentralblatt MATH identifier
1225.60089

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

Keywords
Stochastic integration Stratonovich integral fractional Brownian motion weak convergence Malliavin calculus

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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