Universal Approximation with Signatures of Non-Geometric Rough Paths

Battle Count: 69

The paper provides a rigorous theoretical foundation for using Itô signatures (as opposed to the more commonly used Stratonovich signatures) in quantitative finance applications. This is directly relevant because Itô integration is the natural choice in financial modeling (preserves martingale property, no-arbitrage principle). The numerical examples demonstrate practical advantages of Itô signatures for pricing derivatives that depend on quadratic variation (realized volatility options, covariance/correlation swaps). The signature-based calibration approach is directly applicable to model calibration tasks in quantitative trading. However, the paper is primarily theoretical and the numerical validation is limited to simulated data.

The theoretical framework requires deep understanding of rough path theory, tensor algebras, quasi-shuffle products, and stochastic integration. Implementation of signature computation is facilitated by existing packages (iisignature, esig). However, correctly implementing the path extension by quadratic variation, handling the quasi-shuffle algebra structure, and ensuring proper truncation requires significant expertise. The numerical experiments are relatively straightforward (linear regression on signature features), but the underlying theory is highly non-trivial.

Code for numerical experiments is publicly available on GitHub. The paper provides complete mathematical proofs for all theoretical results. However, the theoretical framework requires deep expertise in rough path theory. Specific model parameters, truncation levels (N=2), regularization parameters (α=10⁻⁵ for Lasso, α=10⁻⁶ for Ridge), and simulation settings are clearly stated. The use of standard packages (iisignature, esig) aids reproducibility.