NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

arXiv 2607.14361 ↗First posted 15 Jul 2026math.PR cs.LG q-fin.CPcited by 0

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Abstract (arXiv)

We address fundamental challenges in representing and computing ℝd-valued predictable square-integrable processes over [0,T], collected in the space H2T(ℝd). 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 H2T(ℝd) using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in H2T(ℝd) 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 H2T(ℝd) 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 H2T(ℝd), 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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References · 51

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