Concise $(\epsilon ,r)$-representations of a path

Abstract

Paths $X:[0,T]\to {\mathbb{R}}^d$ are traditionally stored in finite memory as time series. Recent research has underscored the benefits of instead representing them as collections of iterated integrals $\{{\int }_{0<u_1<\dots <u_n<T}\mathrm{d}{X}_{u_1}\otimes \cdots \otimes \mathrm{d}{X}_{u_n}{\}}_{n=0}^N$. These two encodings can be viewed as the extrema on a two-parameter spectrum of representations of the path as degree-$N$ signatures on $m$ intervals in a partition of $[0,T]$. We ask the question of which such representation takes up the least amount of memory, measured as number of real values needed to store the truncated log-signature, subject to the constraint of it being able to approximate solutions to linear controlled differential equations (CDEs) $\mathrm{d}Y=AY\mathrm{d}X$ with $|A|\le r$ at accuracy at least $\epsilon$. Estimating the error in terms of the length of $X$, we find that the optimal representation generally lies strictly in between the two naive choices $N=1$ or $m=1$, and derive its asymptotics as $r\to \infty$ and $\epsilon \to 0^{+}$. Similar considerations can be made when estimating the error in terms of the $p$-variation norm of $X$: in this regime we prove an error bound of the degree-$N$ Euler scheme for linear CDEs with decay in both $m$ and (factorially) in $N$ with the other arbitrarily fixed. We conclude by setting up the analogous problem for SDEs, with the error measured in $L^2$, and derive a similar $L^2$-Euler error estimate for Itô SDEs with drift. We include an empirical study of the optimisation problem, which we demonstrate for toy examples of $p$-rough paths and for fractional Brownian motion.

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