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 $\alpha\in(1/8,1/4)$

TL;DR

This paper provides a rigorous constructive proof of convergence for the Lévy area of fractional Brownian motion with low Hurst index, using advanced field theory techniques to handle low regularity paths.

Contribution

It introduces a novel constructive approach employing cluster expansions and renormalization to prove convergence of second-order iterated integrals for fractional Brownian motion with Hurst index between 1/8 and 1/4.

Findings

Proves convergence of Lévy area for fractional Brownian motion with Hurst index in (1/8, 1/4)

Uses field theory tools like cluster expansions and renormalization for stochastic calculus

Extends Gaussian tools such as Malliavin calculus to low regularity processes

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.