Convergence of trapezoid rule to rough integrals

Yanghui Liu; Zachary Selk; Samy Tindel

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arXiv · 2005.06500

Convergence of trapezoid rule to rough integrals

Yanghui Liu·Zachary Selk·Samy Tindel

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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 $\int f(X) dX$.

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BibTeX RIS

Yanghui Liu, Zachary Selk, Samy Tindel. 2020-05-13. Convergence of trapezoid rule to rough integrals. https://arxiv.org/abs/2005.06500

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