Stochastic calculus for fractional Brownian motion with Hurst exponent $H>1/4$: A rough path method by analytic extension

Abstract

The $d$-dimensional fractional Brownian motion (FBM for short) $B_t=((B_t^{(1)},...,B_t^{(d)}),t\in \mathbb{R})$ with Hurst exponent $\alpha$, $\alpha \in (0,1)$, is a $d$-dimensional centered, self-similar Gaussian process with covariance $\mathbb{E}[B_s^{(i)}{B}_t^{(j)}]=1/2{\delta }_{i,j}(|s{|}^{2\alpha }+|t{|}^{2\alpha }-|t-s{|}^{2\alpha }).$ 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 $\alpha$. The case $\alpha =1/2$ corresponds to the usual stochastic integration with respect to Brownian motion, while most computations become singular when $\alpha$ gets under various threshhold values, due to the growing irregularity of the trajectories as $\alpha \to 0$. We provide here a new method valid for any $d$ and for $\alpha >1/4$ by constructing an approximation $\Gamma (\epsilon {)}_t$, $\epsilon \to 0$, 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 ${\Gamma }_z$ on the cut plane $z\in \mathbb{C}\setminus \mathbb{R}$ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see \citeCQ02) but as yet a little mysterious divergence of Lévy's area for $\alpha \to 1/4$.

arXiv commentPublished in at http://dx.doi.org/10.1214/08-AOP413 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (14)

• Quantitative analysis of single particle trajectories: mean maximal excursion method
• Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble
• A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion
• First passage behaviour of fractional Brownian motion in two-dimensional wedge domains
• Regularized fractional Ornstein-Uhlenbeck processes, and their relevance to the modeling of fluid turbulence
• A construction of the rough path above fractional Brownian motion using Volterra's representation
• Mesoscopic linear statistics of Wigner matrices
• Hölder-continuous rough paths by Fourier normal ordering
• Parameter estimation for rough differential equations
• A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index $H<1/4$
• From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics
• Pathwise solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in $(1/3,1/2]$
• From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the Lévy area of fractional Brownian motion with Hurst index $\alpha \in (1/8,1/4)$
• The curious spectra and dynamics of non-locally finite crystals