非几何粗糙路径的签名可逼近任意连续泛函，突破传统限制。
Universal approximation with signatures of non-geometric rough paths
- 通过扩展路径含时间与二次变差项，实现通用逼近
- 在紧集上，线性泛函均匀逼近连续泛函，误差可控
- 适用于金融建模中的期权定价与参数校准场景
我们建立了非弱几何粗糙路径签名的通用逼近定理。通过引入时间变量和粗糙路径括号项扩展路径，证明了相应粗糙路径签名的线性泛函可在紧集上一致逼近粗糙路径空间上的任意连续泛函。此外，基于Föllmer的路径积分框架，构造了包含路径逐点二次变差项的路径签名，支持路径式伊藤、斯特拉托诺维奇及反向伊藤积分。在概率设定下，对连续半鞅路径（经二次变差项扩展）的签名线性泛函，也获得了通用逼近结果，采用随机伊藤积分定义。数值实验展示了在模型校准与数学金融中的期权定价中，使用时间与二次变差扩展路径时签名的有效性。
原文摘要 · Abstract (English)
We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of the resulting rough paths approximate continuous functionals on rough path spaces uniformly on compact sets. Moreover, we construct the signature of a path extended by its pathwise quadratic variation terms based on general pathwise stochastic integration à la Föllmer, in particular, allowing for pathwise Itô, Stratonovich, and backward Itô integration. In a probabilistic setting, we obtain a universal approximation result for linear functionals of the signature of continuous semimartingales extended by the quadratic variation terms, defined via stochastic Itô integration. Numerical examples illustrate the use of signatures when the path is extended by time and quadratic variation in the context of model calibration and option pricing in mathematical finance.
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