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Branched Signature Model

Munawar Ali, Qi Feng

TL;DR

This paper develops a universal approximation framework for branched rough paths by leveraging the branched signature and linking it to the classical geometric signature through the Hairer–Kelly extension map $\Psi$. It proves a branched-version universal approximation theorem and analyzes the limits of iterating lower-level signature models, advocating a layer-wise, two-step approach to reach higher levels with reduced computational burden. The authors provide an explicit construction of extended paths for Brownian and fractional Brownian inputs and also offer a data-driven route to learn extensions via neural networks, combined with physics-informed and shuffle-consistency losses. Numerical experiments on rough-volatility models demonstrate that incorporating the extended path via branched signatures improves calibration accuracy for both volatility and asset price paths, highlighting practical benefits for data-driven path-dependent modeling.

Abstract

In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [Gubinelli, Journal of Differential Equations, 248(4), 2010], which generalizes the classical geometric rough path. We establish a universal approximation theorem for the branched signature model and demonstrate that iterative compositions of lower-level signature maps can approximate higher-level signatures. Furthermore, building on the existence of the extension map proposed in [Hairer-Kelly. Annales de l'Institue Henri Poincaré, Probabilités et Statistiques 51, no. 1 (2015)], we show how to explicitly construct the extension of the original paths into higher-dimensional spaces via a map $Ψ$, so that the branched signature can be realized as the classical geometric signature of the extended path. This framework not only provides an efficient computational scheme for branched signatures but also opens new avenues for data-driven modeling and applications.

Branched Signature Model

TL;DR

This paper develops a universal approximation framework for branched rough paths by leveraging the branched signature and linking it to the classical geometric signature through the Hairer–Kelly extension map . It proves a branched-version universal approximation theorem and analyzes the limits of iterating lower-level signature models, advocating a layer-wise, two-step approach to reach higher levels with reduced computational burden. The authors provide an explicit construction of extended paths for Brownian and fractional Brownian inputs and also offer a data-driven route to learn extensions via neural networks, combined with physics-informed and shuffle-consistency losses. Numerical experiments on rough-volatility models demonstrate that incorporating the extended path via branched signatures improves calibration accuracy for both volatility and asset price paths, highlighting practical benefits for data-driven path-dependent modeling.

Abstract

In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [Gubinelli, Journal of Differential Equations, 248(4), 2010], which generalizes the classical geometric rough path. We establish a universal approximation theorem for the branched signature model and demonstrate that iterative compositions of lower-level signature maps can approximate higher-level signatures. Furthermore, building on the existence of the extension map proposed in [Hairer-Kelly. Annales de l'Institue Henri Poincaré, Probabilités et Statistiques 51, no. 1 (2015)], we show how to explicitly construct the extension of the original paths into higher-dimensional spaces via a map , so that the branched signature can be realized as the classical geometric signature of the extended path. This framework not only provides an efficient computational scheme for branched signatures but also opens new avenues for data-driven modeling and applications.

Paper Structure

This paper contains 19 sections, 15 theorems, 116 equations, 4 figures.

Table of Contents

  1. Preliminaries
  2. Signature and its properties
  3. Geometric Rough Path
  4. Branched Rough Path
  5. Hopf Algebra
  6. The Connes-Kreimer Hopf Algebra of rooted trees
  7. Geometric Realization of Branched Rough Paths
  8. Branched Signature Model
  9. Universal approximation theorem for branched signature
  10. Iterative application of signature model
  11. Construction of the extended path
  12. Explicit construction
  13. Multi-dimensional Brownian motion
  14. Multi-dimensional fractional Brownian motion
  15. Data-driven construction learned via a neural network
  16. ...and 4 more sections

Key Result

Proposition 1.11

Friz_Victoir_2010 Let $(\mathbf{X}_t)_{t \in [0,T]}$ be a continuous, $\mathbb{R}^d$-valued path of bounded variation. Then, the concatenated signature over intervals $[s, u]$ and $[u, t]$ satisfies for each $0 \leq s \leq u \leq t \leq T$. This identity can be equivalently expressed as follows: where $\mathbf{w}$ is an arbitrary word from $\mathbf{W}$.

Figures (4)

  • Figure 2.1: Application of Level 2 signature model on extended path $\Bar{\mathbf{X}}$
  • Figure 4.1: Left: calibration (path) loss per epoch. Right: shuffle-product residual loss per epoch. The right panel’s vertical axis is shown on a base-10 logarithmic scale.
  • Figure 4.2: Shuffle product residual matrix. Each cell $(i,j)$ displays $R_{ij}=\Delta X^{i}\Delta X^{j}-\!\left(\int_0^T X^{i}\,dX^{j}+\int_0^T X^{j}\,dX^{i}\right)$ on the plotted interval. Color encodes sign and magnitude red for positive, blue for negative and white near to $0$; rows/columns are coordinate indices $i$ and $j$. Near-white regions indicate good numerical adherence to the identity, while darker patches show where deviations are large. Final shuffle product MSE mean over all components is $6.1158 \times 10^{-3}$.
  • Figure :

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6: Shuffle Property
  • Remark 1.7
  • Example 1.8: Integration by parts
  • Example 1.9
  • Remark 1.10
  • ...and 42 more