发表机构
Department of Mathematics, McMaster University; Vector Institute; The London School of Economics; ETH Zurich, Department of Mathematics（麦基尔大学数学系; 向量研究所; 伦敦政治经济学院; 苏黎世联邦理工学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究\(\mathbb{R}^{d}\)值可预测平方可积过程表示计算难题，提出神经混沌架构，仅用驱动布朗运动有限次评估生成\(\mathcal{H}^2_T(\mathbb{R}^{d})\)元素，证明其稠密性及逼近率，通过实验验证有效性，提升随机分析和数学金融建模效率与表现力。
AI 中文摘要
我们解决了在\([0,T]\)上表示和计算\(\mathbb{R}^{d}\)值可预测平方可积过程（收集在空间\(\mathcal{H}^2_T(\mathbb{R}^{d})\)中）的基本挑战。这些过程在连续时间随机控制、强化学习和数学金融中至关重要。尽管维纳混沌展开提供了强大的理论工具，但传统计算方法因需要大的混沌字典和高阶迭代积分而受阻。为克服这些障碍，我们引入神经混沌——一种神经算子架构，它仅使用驱动布朗运动的有限多次评估来生成\(\mathcal{H}^2_T(\mathbb{R}^{d})\)的元素，同时保持可预测性和平方可积性。我们证明神经混沌在\(\mathcal{H}^2_T(\mathbb{R}^{d})\)中是稠密的，并且对于可压缩和 Malliavin - Sobolev 正则过程实现了最佳的\(N\)项混沌小波逼近率。此外，在非退化次高斯采样下，可压缩性对于\(\mathcal{H}^2_T(\mathbb{R}^{d})\)中的过程是典型的。相比之下，我们表明有限维马尔可夫神经 SDE 模型在\(\mathcal{H}^2_T(\mathbb{R}^{d})\)中构成一个贫集且高斯零子集，无论离散化如何，而可压缩过程是通用的。在随机最优控制问题和动态套期保值上的数值实验突出了我们方法的实际有效性。我们的结果使随机分析和数学金融中的建模更高效且更具表现力。
英文摘要
We address fundamental challenges in representing and computing $\mathbb{R}^{d}$-valued predictable square-integrable processes over $[0,T]$, collected in the space $\mathcal{H}^2_T(\mathbb{R}^{d})$. These processes are central to continuous-time stochastic control, reinforcement learning, and mathematical finance. Although Wiener-chaos expansions offer strong theoretical tools, traditional computational methods are hindered by the need for large chaos dictionaries and high-order iterated integrals. To overcome these obstacles, we introduce NeuralChaos -- a neural operator architecture that produces elements of $\mathcal{H}^2_T(\mathbb{R}^{d})$ using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in $\mathcal{H}^2_T(\mathbb{R}^{d})$ and achieves the best $N$-term chaoslet approximation rates for compressible and Malliavin--Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from $\mathcal{H}^2_T(\mathbb{R}^{d})$ under non-degenerate sub-Gaussian sampling. In contrast, we show that finite-dimensional Markovian neural SDE models constitute a meagre and Gaussian-null subset in $\mathcal{H}^2_T(\mathbb{R}^{d})$, regardless of discretization, whereas compressible processes are generic. Numerical experiments on a stochastic optimal control problem and dynamic hedging highlight the practical effectiveness of our approach. Our results enable more efficient and expressive modelling in stochastic analysis and mathematical finance.
