反射倒向随机微分方程的深度学习：正则化与误差分析
Deep Learning for Reflected BSDEs: Regularization and Error Analysis
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中文总结 AI 辅助
本文提出两种深度学习方案（DFS和DBS）求解反射倒向随机微分方程，通过正则化约化并建立误差界，应用于高维美式期权定价，数值实验验证了高维下的准确性。
中文摘要 AI 辅助
反射倒向随机微分方程（RBSDEs）为障碍约束问题提供了概率论表述，但现有的针对其高维解的深度学习方法仍然有限。在本文中，我们提出了两种用于RBSDEs的深度学习方案，即深度前向方案（DFS）和深度后向方案（DBS），首先将反射问题约化为一族正则化的BSDEs。我们的主要理论贡献涉及DBS：我们建立了一个显式误差界，表明对于每个固定的正则化参数$\varepsilon>0$，DBS解与正则化BSDE解之间的逼近误差由相应的训练损失控制。我们证明了该训练损失可以通过神经网络的通用逼近能力来控制。综合这些结果，为基于深度学习的求解提供了理论基础，并补充了前向类型方法的现有分析。我们在高维美式期权定价中展示了该框架，其中反射表述使我们能够直接处理连续时间行权特征，而非通过Bermudan近似。数值实验表明，DFS和DBS在高维情况下均能提供精确的解。
英文摘要
Reflected backward stochastic differential equations (RBSDEs) provide a probabilistic formulation for obstacle constrained problems, but existing deep learning methods for their high dimensional solution remain limited. In this paper, we propose two deep learning schemes for RBSDEs, a deep forward scheme (DFS) and a deep backward scheme (DBS), by first reducing the reflected problem to a family of regularized BSDEs. Our main theoretical contribution concerns the DBS: we establish an explicit error bound showing that, for each fixed regularization parameter $\varepsilon>0$, the approximation error between the DBS solution and the solution to the regularized BSDE is controlled by the associated training loss. We prove that this training loss can be controlled by the universal approximation capability of neural networks. Together, these results yield a theoretical foundation for the deep learning-based solution and complement existing analysis for forward type methods. We illustrate the framework on high dimensional American option pricing, where the reflected formulation allows us to address the continuous time exercise feature directly rather than through a Bermudan approximation. Numerical experiments demonstrate that both DFS and DBS deliver accurate solutions in high dimensions.
发表机构
- University of California, Santa Barbara（加州大学圣塔芭芭拉分校）
- Adam Smith Business School, University of Glasgow（格拉斯哥大学亚当斯密商学院）
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