AI 中文总结
本文研究路径的最优简洁(ε,r)-表示，推导其渐近行为与误差界，建立SDEs的类似问题并开展实证研究，为路径存储与CDE/SDE求解提供了优化方案。
AI 中文摘要
路径X:[0,T]→ℝᵈ传统上以时间序列形式存储，占用有限内存。近期研究强调，将路径表示为迭代积分集合{∫_{0<u₁<…<uₙ<T} dX_{u₁}⊗…⊗dX_{uₙ}}_{n=0}^N具有诸多优势。这两种编码可视为路径表示的双参数谱的极值，该表示是[0,T]划分中m个区间上的N阶签名。本文研究问题：在满足能以至少ε的精度逼近线性受控微分方程(CDEs)dY=AYdX（|A|≤r）解的约束下，哪种表示所需存储的实数值最少（以截断对数签名的存储量衡量）。通过以X的长度估计误差，发现最优表示通常严格介于N=1或m=1这两种简单选择之间，并推导了r→∞和ε→0⁺时的渐近行为。当以X的p-变差范数估计误差时，本文证明了线性CDEs的N阶欧拉格式的误差界，其随m衰减、随N呈阶乘衰减，其余参数可任意固定。最后，本文建立了SDEs的类似问题，以L²衡量误差，并推导了带漂移项的伊藤SDEs的类似L²-欧拉误差估计。本文还对该优化问题进行了实证研究，针对p-粗糙路径和分数布朗运动的示例进行了验证。
英文摘要
Paths $X \colon [0,T] \to \mathbb R^d$ are traditionally stored in finite memory as time series. Recent research has underscored the benefits of instead representing them as collections of iterated integrals $\{\int_{0 < u_1 < \ldots < u_n < T} \mathrm{d} X_{u_1} \otimes \cdots \otimes \mathrm{d} X_{u_n}\}_{n = 0}^N$. 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 $|A| \leq r$ at accuracy at least $\varepsilon$. 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 $\varepsilon \to 0^+$. 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.
