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From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L'evy area of fractional Brownian motion with Hurst index $alphain(1/8,1/4)$ | | Jacques Magnen
; Jérémie Unterberger
; | | Date: |
9 Mar 2011 | | Abstract: | {Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion
with Hurst index $alpha<1/4$, 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 H"older 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 cite{MagUnt1}, this one concentrates on the
details of the constructive proof of convergence for second-order iterated
integrals, also known as L’evy area. | | Source: | arXiv, 1103.1750 | | Services: | Forum | Review | PDF | Favorites | |
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