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.

翻译：本文建立了一个适用于非弱几何粗糙路径签名的通用逼近定理。通过将路径扩展至包含时间及其粗糙路径括号项，我们证明了所得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连续泛函。此外，我们基于Föllmer提出的通用路径随机积分方法，构造了通过路径二次变差项扩展的路径签名，该方法特别适用于路径Itô积分、Stratonovich积分和后向Itô积分。在概率论框架下，我们获得了通过随机Itô积分定义的二次变差项扩展的连续半鞅签名的线性泛函的通用逼近结果。数值算例展示了在数理金融的模型校准与期权定价场景中，当路径通过时间与二次变差扩展时签名的应用价值。