Weak and strong discretization errors for signatures of Gaussian processes

Abstract

Let $X$ be a continuous Gaussian process in $\mathbb{R}^d$, $d\ge2$, with independent components of roughness $H\in(\tfrac14,1)$, possibly with a linear drift, and let $X^n$ be its piecewise-linear interpolant on a mesh of width $δ$. We study how well the truncated signature of $X^n$ approximates that of the natural lift of $X$, in mean (the weak error) and in $L^2$ (the strong error). Over a class of drivers defined by quantitative regularity conditions, the weak error at every fixed truncation level is $O(δ^{γ(H)})$ with $γ(H)=\min(2H,4H-1)$, uniformly over the class, and for truncation levels $M\ge4$ fractional Brownian motion attains this rate. For centred drivers the weak error vanishes at levels $1$ to $3$, Gaussian or not. It first appears at level~$4$, where we compute its leading constant for independent fractional Brownian motion in $d=2$: on $(\tfrac12,1)$ it is $1/(2\sqrt3)$, independent of $H$, and on $(\tfrac14,\tfrac12)$ it is given by a $ζ$-series. For fractional Brownian motion with drift, a level-$3$ error of the same order appears for $H\ge\tfrac12$. Below $\tfrac12$ the drift reaches the rate only at levels $\ge5$ (proved for $d\ge3$). The strong error is of exact order $δ^{2H-1/2}$ for fractional Brownian motion with $H\in(\tfrac12,1)$, so in mean square the weak error lies a full power of $δ$ below it. For $H\le\tfrac12$ the strong error is $O(δ^{\vartheta})$ for every $\vartheta<2H-\tfrac12$, uniformly over the class, and the exponent $2H-\tfrac12$ cannot be improved. Finally, we construct pairs of laws in the class whose mesh data have the same law but whose expected signatures differ by order $δ^{γ(H)}$. Hence no quantity computed from the mesh data has mean within $o(δ^{γ(H)})$ of the expected signature uniformly over the class.