From constructive field theory to fractional stochastic calculus. (I) The Lévy area of fractional Brownian motion with Hurst index $\alpha \in (1/8,1/4)$

Abstract

Let $B=(B_1(t),\dots ,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha <1/4$. 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 show in this paper how to obtain second-order iterated integrals as the limit when the ultra-violet cut-off goes to infinity of iterated integrals of weakly interacting fields defined using the tools of constructive field theory, in particular, cluster expansion and renormalization. The construction extends to a large class of Gaussian fields with the same short-distance behaviour, called multi-scale Gaussian fields. Previous constructions \cite{Unt-Holder,Unt-fBm} were of algebraic nature and did not provide such a limiting procedure.

86 pages, 5 figures

References in corpus (2)

• A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index
• A renormalized rough path over fractional Brownian motion