Some results on the Signature and Cubature of the Fractional Brownian motion for $H>\frac{1}{2}$

TL;DR

This paper investigates the signature and cubature of fractional Brownian motion with Hurst parameter greater than 1/2, establishing convergence rates, bounds, and extending cubature methods beyond classical Brownian motion.

Contribution

It provides new convergence rate results, bounds for signature coefficients, and extends cubature techniques to fractional Brownian motion with H>1/2.

Findings

Expected signature convergence rate is 2H.

Uniform bounds for signature coefficients are established.

Sharp decay bounds for expected signature are derived.

Abstract

In this work we present different results concerning the signature and the cubature of fractional Brownian motion (fBm). The first result regards the rate of convergence of the expected signature of the linear piecewise approximation of the fBm to its exact value, for a value of the Hurst parameter $H\in(\frac{1}{2},1)$. We show that the rate of convergence is given by $2H$. We believe that this rate is sharp as it is consistent with the result of Ni and Xu, who showed that the sharp rate of convergence for the Brownian motion (i.e. fBm with $H=\frac{1}{2}$) is given by $1$. The second result regards the bound of the coefficient of the rate of convergence obtained in the first result. We obtain an uniform bound for the coefficient for the $2k$-th term of the signature of $\frac{\tilde{A}k(2k-1)}{(k-1)!2^{k}}$, where $\tilde{A}$ is a finite constant independent of $k$. The third result regards the sharp decay rate of the expected signature of the fBm. We obtain a sharp bound for the $2k$-th term of the expected signature of $\frac{1}{k!2^{k}}$. The last results concern the cubature method for the fBm for $H>\frac{1}{2}$. In particular, we develop the framework of the cubature method for fBm, provide a bound for the approximation error in the general case, and obtain the cubature formula for the fBm in a particular setting. These results extend the work of Lyons and Victoir, who focused on the Brownian motion case.