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This paper introduces NeuralChaos, a novel neural operator architecture designed to efficiently represent and compute $\mathbb{R}^{d}$-valued predictable square-integrable processes within the space $\mathcal{H}^2_T(\mathbb{R}^{d})$. By leveraging only finitely many evaluations of the driving Brownian motion, NeuralChaos maintains the essential properties of predictability and square-integrability while achieving optimal approximation rates for compressible processes. The findings demonstrate that traditional finite-dimensional Markovian neural SDE models are limited, while NeuralChaos provides a robust framework for stochastic control and mathematical finance applications.
NeuralChaos achieves optimal approximation rates for square-integrable processes using significantly fewer evaluations than traditional methods, reshaping the landscape of stochastic modeling.
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.
ETH