NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

2026-07-15

AI Summary

This work addresses the long-standing challenge of efficiently representing and computing square-integrable predictable stochastic processes in $\mathcal{H}^2_T(\mathbb{R}^d)$, which has been hindered by the high-dimensional basis functions and intricate iterated integrals inherent in classical Wiener chaos expansions. We propose NeuralChaos, a novel neural operator architecture capable of generating processes that satisfy both predictability and square-integrability using only finitely many Brownian motion samples. Theoretically, we establish that NeuralChaos is dense in $\mathcal{H}^2_T(\mathbb{R}^d)$ and achieves the optimal $N$-term approximation rate with respect to chaoslet bases. Moreover, we show that compressible processes are generic, whereas finite-dimensional Markovian neural SDE models constitute a measure-zero, sparse subset. Integrating neural operators, Wiener chaos, Malliavin–Sobolev regularity, and non-degenerate sub-Gaussian sampling, our approach substantially enhances modeling expressivity and computational efficiency in stochastic dynamic control and dynamic hedging tasks.

Abstract

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.

Problem

• predictable processes
• square-integrable processes
• Wiener chaos expansion
• stochastic control
• mathematical finance

Innovation

• NeuralChaos
• predictable processes
• Wiener chaos expansion
• neural operators
• compressible processes