2011/03/09 by Jacques Magnen, Jérémie Unterberger
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Constructive #Constructive proof #Fractional Brownian motion #Gaussian #Gaussian process #Hurst exponent #Iterated function #Malliavin calculus #Random Matrices and Applications #Stochastic calculus #Stochastic process #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60F05 #msc:60G15 #msc:60G18 #msc:60H05 #msc:81T08 #msc:81T18
paper · pdf · doi:10.1007/s00023-011-0119-y
arxiv created 2011/03/09 · openalex publication_date 2011/07/18 · arxiv updated 2015/05/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let B=(B1(t),...,Bd(t)) be a d-dimensional fractional Brownian motion with Hurst index α<1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \citeMagUnt1, this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as Lévy area.