NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

• URL: http://arxiv.org/abs/2607.14361v1
• Date: Wed, 15 Jul 2026 20:53:51 GMT
• Title: NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes
• Abstract summary: We introduce NeuralChaos, a neural operator architecture that produces elements of $mathcalH2_T(mathbbRd)$.<n>We prove that NeuralChaos is dense in $mathcalH2_T(mathbbRd)$ and achieves the best $N-term chaoslet rates for compressible and Malliavin--Sobolev processes.
• Score: 10.292476979020522
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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