Approximation of the finite dimensional distributions of multiple fractional integrals

TL;DR

This paper develops an approximation method for finite-dimensional distributions of multiple fractional integrals, specifically for processes converging to multiple Wiener-Itô integrals with respect to fractional Brownian motion for Hurst parameter H > 1/2.

Contribution

It introduces a new family of continuous stochastic processes that approximate multiple fractional integrals in finite-dimensional distribution sense for H > 1/2.

Findings

Convergence of the constructed processes to the target integrals.

Applicability to a broad class of integrands.

Extension of approximation techniques to fractional Brownian motion.

Abstract

We construct a family $I_{n_{\eps}}(f)_{t}$ of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-It\^o integral $I_{n}^{H}(f1^{\otimes n}_{[0,t]})$ with respect to the fractional Brownian motion. We assume that $H>{1/2}$ and we prove our approximation result for the integrands $f$ in a rather general class.