[Paper Review] Linear independence properties of the signature components of time-augmented stochastic processes

Arthur Bourdon, Benjamin Jourdain+1 more|arXiv (Cornell University)|Jan 15, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper studies linear independence among time-augmented signature components, identifies basis-of-words structures that span the same space as full N-step signatures, and proves independence results for SDEs with additive Brownian noise, including discretized paths.

ABSTRACT

The addition of the running time as a component of a path before computing its signature is a widespread approach to ensure the one-to-one property between them and leads to universal approximation theorems (Cuchiero, Primavera and Svaluto-Ferro, 2023). However, this also leads to the linear dependence of the components of the terminal value of the signature of the time-augmented path. More precisely, for a given natural number $N$, the signature components associated with words of length $N$ have the same linear span as the signature components associated with words of length not greater than $N$. We generalize this result by exhibiting other subfamilies of signature components with the same spanning properties. In particular we recover the result of Dupire and Tissot-Daguette which states that the spanning of the iterated integrals with the last integrator different from the time variable is the same as the spanning of all iterated integrals. We check that this choice leads to the minimal computation time when the terms of the signature are calculated using Chen's relation in a backward way. The same optimal computation time is symmetrically achieved in a forward way for the iterated integrals with the first integrator different from the time variable. Building on these results, we derive several results regarding the linear independence of the signature components of a time-augmented stochastic process. We show that if the stochastic process we consider is solution to some SDE with additive Brownian noise then any subfamily of components proposed previously is linearly independent. We also prove that the linear independence of these subfamilies of components is still true when we consider the discretization of the sample paths of this stochastic process on a grid with a sufficiently small discretization time step.

Motivation & Objective

• Motivate the use of time augmentation to achieve a one-to-one correspondence between paths and their signatures and to enable universal approximation properties.
• Characterize which subfamilies of signature components span the same space as the full N-step truncated time-augmented signature.
• Develop a notion of a basis of words that yields identical spans and analyze its implications for identifiability, storage, and computation.
• Prove linear independence results for time-augmented signatures in L2 under additive Brownian noise and extend to discretized paths.
• Provide guidance on optimal computation of signature components via Chen’s relation in backward and forward schemes.

Proposed method

• Define the time-augmented signature and its N-step truncation and describe the corresponding feature space.
• Introduce the shuffle product and dual-bracket formalism to study spans of signature components.
• Define bases of words for W_N and establish necessary and sufficient conditions for being a basis (Theorems 1–3).
• Prove linear independence results for subfamilies associated with bases of words under SDEs with additive Brownian noise (Theorem 4) and corollaries (Corollary 3).
• Show that discretized paths preserve independence for small time steps (Theorem 5).
• Discuss optimal computation of signature components using Chen’s relations in backward and forward schemes.

Experimental results

Research questions

• RQ1 Which subfamilies of time-augmented signature components share the same linear span as the full N-step truncated signature?
• RQ2 Under what conditions are these subfamilies linearly independent in L2, particularly for SDEs with additive Brownian noise?
• RQ3 How can one construct a basis of words for W_N that minimizes computation while preserving span and identifiability?
• RQ4 Does discretization of sample paths affect the linear independence of these signature components, and under what discretization step size is independence preserved?
• RQ5 What are the practical implications for regression and feature selection when using bases of words in time-augmented signatures?

Key findings

• Any subfamily corresponding to a basis of words for W_N has the same span as the full N-step truncated signature.
• A basis can be decomposed by pure words, enabling modular construction across word classes (gamma-prefixes and gamma-suffixes).
• For time-augmented paths, the span of signatures with words of length N equals the span of all words up to length N, implying linear dependence across lengths.
• In SDEs with additive Brownian noise, subfamilies arising from bases of words are linearly independent in L2, and this independence persists under sufficiently fine discretization.
• Optimal computation times are achieved by using words that avoid unnecessary time-letter endings or beginnings, enabling backward or forward Chen-based calculations.
• For piecewise linear interpolations of such SDE solutions and appropriate integrability, the chosen basis retains independence in the discretized setting.