English

NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

Probability 2026-07-15 v1 Machine Learning Computational Finance Machine Learning

Abstract

We address fundamental challenges in representing and computing Rd\mathbb{R}^{d}-valued predictable square-integrable processes over [0,T][0,T], collected in the space HT2(Rd)\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 HT2(Rd)\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 HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}) and achieves the best NN-term chaoslet approximation rates for compressible and Malliavin--Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from HT2(Rd)\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 HT2(Rd)\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.

Cite

@article{arxiv.2607.14361,
  title  = {NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes},
  author = {Anastasis Kratsios and Giulia Livieri and Philipp Schmocker},
  journal= {arXiv preprint arXiv:2607.14361},
  year   = {2026}
}