Global Universal Approximation with Brownian Signatures

Battle Count: 65

The paper provides a rigorous theoretical foundation for signature-based models in quantitative finance. It directly justifies the universality of signature-based models driven by Brownian noise, which are used for option pricing, model calibration, portfolio optimization, and stochastic control. The result that linear functionals on Brownian signatures can approximate any p-integrable adapted process (including SDE solutions) is fundamental for the validity of signature-based trading strategies and derivative pricing models. However, the paper is purely theoretical and does not provide practical implementation guidance, numerical algorithms, or empirical validation, which limits its direct applicability to trading systems.

The theoretical framework requires deep expertise in rough path theory, functional analysis, and stochastic calculus. Key concepts include: geometric alpha-Hölder rough paths, Carnot-Carathéodory norms, iterated integrals (signatures), the Lyons extension theorem, weighted Stone-Weierstrass theorem, Fernique's theorem for Gaussian rough paths, and stopped rough path spaces. Practical implementation would require computing signatures of time-extended paths (iterated Stratonovich integrals), which is computationally expensive for high truncation levels. The paper does not provide code or numerical algorithms, making direct implementation from the paper alone very challenging.

As a purely theoretical mathematics paper, reproducibility is assessed by the completeness and self-containedness of proofs. The paper provides detailed proofs of all main theorems (Theorems 3.4, 3.13, Propositions 3.3, 3.11, 4.2, 4.6) and auxiliary lemmas (Appendix A). The approach builds on established results from [CST26] and [FV10]. All key technical steps are explicitly stated, though some proofs reference prior work (e.g., [CST26, Theorem 5.4]) rather than reproducing them in full. No computational experiments are needed to verify the results.