STOCHASTIC CALCULUS FOR FRACTIONAL BROWNIAN MOTION WITH HURST EXPONENT H > 1/4: A ROUGH PATH METHOD BY ANALYTIC EXTENSION

• 成果类型: Article
• 署名作者: Unterberger, Jeremie
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lorraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP413
• 发表日期: 2009
• 页码: 565-614
• 关键词: integration; driven

摘要

The d-dimensional fractional Brownian motion (FBM for short) B(t) = ((B(t)((1)),...,B(t)((d))), t is an element of R) with Hurst exponent alpha, alpha is an element of (0, 1), is a d-dimensional centered, self-similar Gaussian process with covariance E[B(s)((i)) B(t)((j))] = 1/2 delta(i), j (vertical bar s vertical bar(2 alpha) + vertical bar t vertical bar(2 alpha) - vertical bar t -s vertical bar(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 threshold values, due to the growing irregularity of the trajectories as alpha -> 0. We provide here a new method valid for any d and for alpha > 1/4 by constructing an approximation Gamma(epsilon)(t), epsilon -> 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 is an element of 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 Levy's area for alpha -> 1/4.