AI 中文总结
本文提出路径组合优化理论，基于路径优先框架将组合转化为签名的线性系统，发现提升的几何性缺陷等结构结果，实证表明二次路径泛函可显著提升收益，且样本量下限与非结构化估计相关。
AI 中文摘要
本文构建了路径组合优化（Path Portfolio Optimization）：一种基于路径优先框架的组合理论，其中签名（signature）是价格路径的通用坐标，并探究其是否可被估计。组合是签名的线性泛函，因此控制量存在于截断张量代数中；签名坐标的协方差是期望签名的非群状部分，即缺陷形式（defect form）；整个均值-方差问题成为单一张量中的线性系统。由此得出两个结构性结果：提升（lift）是执行约定，Marcus提升与前向提升之间的差距与组合权重的乘积恰好是Fernholz的超额增长率，因此超额增长率是组合映射的几何性缺陷；二阶反对称块是路径意义上的提升不变量，因此方向信号是约定无关的，而方差信号与破产则并非如此。实证发现存在维度权衡：当期望签名已知时，二次路径泛函使一对资产的确定等价收益提升11倍，20个资产横截面的确定等价收益提升60倍；当期望签名被估计时，未正则化策略在样本量超过每个参数约6个观测值前表现严重为负，而收缩（shrinkage）从一对资产中的有害因素转变为横截面中的必要因素。全部收益都来自对称块，它是终端增量的凸性而非路径依赖；当驱动项无期望面积时，路径依赖的反对称块无收益。样本量下限属于非结构化估计而非路径复杂度：一种仅拟合驱动项生成元并重建期望签名的估计器，在每个参数仅1个观测值时就能恢复几乎所有可实现价值。
英文摘要
This paper builds Path Portfolio Optimization: portfolio theory on a path-first framework in which the signature is the universal coordinate of the price path, and asks whether it survives estimation. A portfolio is a linear functional of the signature, so the control lives in a truncated tensor algebra, the covariance of signature coordinates is the non-group-like part of the expected signature --- a defect form --- and the whole mean--variance problem becomes a linear system in one tensor. Two structural results follow. The lift is the execution convention: the gap between the Marcus and forward lifts, contracted with portfolio weights, is Fernholz's excess growth rate exactly, so excess growth is the geometricity defect of the portfolio map. And the antisymmetric block at level two is lift-invariant pathwise, so directional signals are convention-free while variance signals and ruin are not. The empirical finding is a dimensional trade-off. With the expected signature known, quadratic path functionals raise the certainty equivalent elevenfold for a pair of assets and sixtyfold for a cross section of twenty; with it estimated, the unregularized policy is severely negative until the sample exceeds roughly six observations per parameter, and shrinkage flips from harmful in the pair to indispensable in the cross section. The entire gain sits in the symmetric block, which is convexity in the terminal increment rather than path-dependence; the path-dependent antisymmetric block earns nothing when the driver has no expected area. And the sample-size floor belongs to unstructured estimation rather than to path complexity: an estimator that fits only the generator of the driver and rebuilds the expected signature recovers almost all of the attainable value at barely one observation per parameter
