arXiv 2406.091892024-06math.AP

李对称网络：通过微分方程建模金融市场动态时保持守恒律

Lie Symmetry Net: Preserving Conservation Laws in Modelling Financial Market Dynamics via Differential Equations

Xuelian Jiang, Tongtian Zhu, Yingxiang Xu, Can Wang, Yeyu Zhang, Fengxiang He

AI总结：

本文提出李对称网络（LSN），通过引入源自守恒律的李对称风险，在结构风险最小化框架下训练模型以保持微分方程的李对称性，在金融市场动态建模中误差较现有方法降低一个数量级以上。

摘要：

本文采用一种新颖的基于李对称性的框架来建模金融市场中的内在对称性。具体而言，我们引入了李对称网络（LSN），它刻画了用于估计金融市场动态（如Black-Scholes方程）的微分方程（DE）的李对称性。为了以对称性感知的方式模拟这些微分方程，LSN引入了一种李对称风险，该风险源自与目标微分方程的李对称算子相关的守恒律。该风险度量了李对称性被实现的程度，并在结构风险最小化框架下指导LSN的训练。大量数值实验表明，LSN有效地实现了李对称性，并且与最先进的方法相比，误差降低了一个数量级以上。代码可在https://github.com/Jxl163/LSN_code获取。

英文摘要：

This paper employs a novel Lie symmetries-based framework to model the intrinsic symmetries within financial market. Specifically, we introduce Lie symmetry net (LSN), which characterises the Lie symmetries of the differential equations (DE) estimating financial market dynamics, such as the Black-Scholes equation. To simulate these differential equations in a symmetry-aware manner, LSN incorporates a Lie symmetry risk derived from the conservation laws associated with the Lie symmetry operators of the target differential equations. This risk measures how well the Lie symmetries are realised and guides the training of LSN under the structural risk minimisation framework. Extensive numerical experiments demonstrate that LSN effectively realises the Lie symmetries and achieves an error reduction of more than one order of magnitude compared to state-of-the-art methods. The code is available at https://github.com/Jxl163/LSN_code.

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