用离散签名实现路径相关函数的全局近似，适用于布朗运动等过程。
Global universality via discrete-time signatures
- 基于分段线性路径的签名构造线性泛函，实现密度逼近。
- 证明了高斯过程离散签名在L^p范数下收敛，速率可量化。
- 适用于随机微分方程与路径依赖函数，理论严谨且具普适性。
我们建立了非前瞻性和一般路径依赖泛函在分段线性路径空间上的全局通用逼近定理，表明相应签名的线性泛函在$L^p$-和加权范数下是稠密的。验证了这些逼近结果适用于一类高斯过程的分段线性插值，包括布朗运动及赫斯特参数$H>1/4$的分数布朗运动。此外，我们推导出高斯过程离散签名逼近其连续时间对应物的定量收敛速率。由此得到高斯过程路径依赖泛函、以及由布朗运动驱动的随机常微分方程和随机微分方程的$L^p$-逼近结果。
原文摘要 · Abstract (English)
We establish global universal approximation theorems for non-anticipative and general path-dependent functionals on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms. We verify that these approximation results are applicable to piecewise linear interpolations of a class of Gaussian processes, including Brownian motion and fractional Brownian motion with Hurst parameter $H>1/4$. Moreover, we derive quantitative convergence rates for signatures of piecewise linear approximations of Gaussian processes towards their continuous-time counterparts. Consequently, we obtain $L^p$-approximation results for path-dependent functionals of Gaussian processes, as well as for random ordinary differential equations and stochastic differential equations driven by Brownian motion.
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