The Annals of Probability

Stochastic calculus for fractional Brownian motion with Hurst exponent H>¼: A rough path method by analytic extension

Jérémie Unterberger

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The d-dimensional fractional Brownian motion (FBM for short) Bt=((Bt(1), …, Bt(d)), t∈ℝ) with Hurst exponent α, α∈(0, 1), is a d-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_{s}^{(i)}B_{t}^{(j)}]=\frac{1}{2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2\alpha})$. The long-standing problem of defining a stochastic integration with respect to FBM (and the related problem of solving stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either d or α. The case α=½ corresponds to the usual stochastic integration with respect to Brownian motion, while most computations become singular when α gets under various threshhold values, due to the growing irregularity of the trajectories as α→0.

We provide here a new method valid for any d and for α>¼ by constructing an approximation Γ(ɛ)t, ɛ→0, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process Γz on the cut plane z∈ℂ∖ℝ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see [5]) but as yet a little mysterious divergence of Lévy’s area for α→¼.

Article information

Ann. Probab., Volume 37, Number 2 (2009), 565-614.

First available in Project Euclid: 30 April 2009

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes 60G18: Self-similar processes 60H05: Stochastic integrals

Fractional Brownian motion stochastic integrals


Unterberger, Jérémie. Stochastic calculus for fractional Brownian motion with Hurst exponent H >¼: A rough path method by analytic extension. Ann. Probab. 37 (2009), no. 2, 565--614. doi:10.1214/08-AOP413.

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