Convergence of trapezoid rule to rough integrals

TL;DR

This paper develops a trapezoid rule for rough integrals driven by Gaussian processes, simplifying the approximation of such integrals without correction terms, applicable to fractional Brownian motion with Hurst parameter greater than 1/4.

Contribution

It introduces a natural trapezoid rule for rough integrals driven by Gaussian processes, avoiding complex correction terms and broadening applicability.

Findings

Trapezoid rule converges for rough integrals driven by Gaussian processes.

Applicable to fractional Brownian motion with Hurst parameter > 1/4.

Midpoint rule convergence for integrals of the form ∫f(X)dX.

Abstract

Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough integrals are usually Riemann-Stieltjes integrals with correction terms that are sometimes seen as unnatural. As opposed to those somewhat artificial correction terms, our endeavor in this note is to produce a trapezoid rule for rough integrals driven by general $d$-dimensional Gaussian processes. Namely we shall approximate a generic rough integral $\int y \, dX$ by Riemann sums avoiding the usual higher order correction terms, making the expression easier to work with and more natural. Our approximations apply to all controlled processes $y$ and to a wide range of Gaussian processes $X$ including fractional Brownian motion with a Hurst parameter $H>1/4$. As a corollary of the trapezoid rule, we also consider the convergence of a midpoint rule for integrals of the form $\int f(X) dX$.