This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and their applications in areas such as stochastic differential equations (SDEs) and financial modeling are pointed out. In particular, exploiting the rough paths view, solutions of SDEs continuously depend on the lift of the driver. Such continuous mappings can be approximated using (truncated) signatures, which are solutions of high-dimensional linear systems. In order to lower the complexity of these models, this paper presents methods for reducing the order of high-dimensional truncated signature models while retaining essential characteristics. The derivation of reduced models and the universal approximation property of (truncated) signatures are treated in detail. Numerical examples, including applications to the (rough) Bergomi model in financial markets, illustrate the proposed reduction techniques and highlight their effectiveness.

翻译：本文聚焦于利用路径签名降低模型复杂度的数学框架。文中讨论了这些签名的结构（可将其解释为沿路径的迭代积分集合），并指出了它们在随机微分方程（SDEs）和金融建模等领域的应用。特别地，基于粗糙路径视角，随机微分方程的解连续依赖于驱动路径的提升映射。此类连续映射可通过（截断）签名进行逼近，而签名本身是高维线性系统的解。为降低此类模型的复杂度，本文提出了在保留本质特征的前提下，对高维截断签名模型进行降阶的方法。文中详细论述了降阶模型的推导过程以及（截断）签名的通用逼近性质。数值算例（包括金融市场中（粗糙）Bergomi模型的应用）展示了所提出的降维技术，并验证了其有效性。