3.15 Path Signatures & Rough Paths: Iterated Integrals, Chen's Identity & the Signature of a Path

Basic Prerequisites: Itô Integral & Itô’s Lemma and Linear Algebra & Matrices (tensor algebra). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

A financial time series is a path: spot, vol, order flow, a yield curve - each is a curve through a state space, recorded only at sample times. Almost every quantitative problem is, at bottom, “summarize this path faithfully enough to decide.” The signature of a path is the principled answer to that: the complete, ordered inventory of iterated integrals of the path. It is a hierarchy - level 1 remembers the increment (where the path ended), level 2 remembers the area between coordinates (the shape of the loop, the covariances), level 3 the signed volumes, and so on. Collecting all levels recovers essentially the whole path, up to the genuinely irrelevant information (reparametrisation).

This folder is the topic-hub for two interlocking subjects:

1. The signature and its algebra - the tensor algebra of iterated integrals, Chen’s identity (the signature factorises under concatenation), the shuffle product identity (products of integrals are integrals over shuffles), the log-signature and the free Lie algebra, and Lyons’ uniqueness theorem (the signature determines the path up to tree-like equivalence).
2. Rough path theory - the reason the construction is stable: for paths rougher than finite variation (Brownian motion, realised volatility, order flow) the iterated integrals are not classically well defined, and rough path theory (Lyons 1998; Friz–Victoir 2010) supplies the analytic control - the $p$-variation norm, the rough-path lift, and the extension theorem - that makes signatures rigorous and computable in the wild.

The one-sentence essence:

The signature is a complete, reparametrisation-invariant, hierarchically-ordered summary of a path built from iterated integrals; its two algebraic laws - Chen’s identity and the shuffle product - are what make it computable, and rough path theory is what makes it rigorous for the irregular paths that finance actually produces; on top of that invariant summary one builds the log-signature (a compression into a Lie element), the expected signature and the signature kernel (universal features for machine learning on paths), and lead-lag / time augmentation (the fix that restores the shuffle law for raw, non-geometric data).

This folder is a hub: (a) the fast formula lookup below, and (b) six sub-pages that walk from raw intuition through the signature algebra, the log-signature and uniqueness, rough path theory, the practical failure modes (augmentation, lead-lag, hedging), and the frontier (expected signatures, signature kernels, signatures in machine learning, and lead-lag signatures for rough vol).

2. Mathematical Ground Truth & Derivations

Notation. $X=(X^1,\dots ,X^d)$ a path in ${\mathbb{R}}^d$ over $[0,T]$, $S(X{)}_{0,T}=(1,S^1,\dots ,S^w,\dots )$ its signature (a point in the tensor algebra $T(({\mathbb{R}}^d))$), $S^w$ the iterated integral indexed by the word $w=i_1\cdots i_k$, ${\Delta }_k=X_{t_{k+1}}-X_{t_k}$ the increments of a piecewise-linear discretisation, $u\bigsqcup v$ the shuffle of two words, $\ell =\log S(X)$ the log-signature, $[u,v]$ the tensor/Lie bracket, $B$ a Brownian motion, $H$ the Hurst exponent, $K(X,Y)$ the signature kernel.

Quick-Reference Lookup (job #1). Every formula below is stated in the verified form used throughout the folder, and every number in the check column was re-executed (§3 and the sub-pages) with stdlib-only, deterministic code.

Critical caveat (flagged in the corpus). The level-2 and level-3 formulas above are exact for a piecewise-linear path but are Riemann–Stieltjes / Stratonovich-type iterated integrals. For truly rough paths (Brownian vol) the same sums converge to the Stratonovich iterated integral only after a geometric rough-path lift has been chosen; the classical (Itô) integral has different, non-geometric terms that violate the shuffle identity. Every identity in this table is stated for the geometric (Stratonovich / rough-path) convention - the one the whole ML-on-signatures literature uses. §04 quantifies the difference.

3. Computational Implementation - the signature engine

Stdlib only (math, itertools, random) - no numpy/scipy. The whole folder runs on two primitives: the explicit iterated-integral sums for a piecewise-linear path (levels 1–3, exact), and the segments composition method (S_word) that evaluates any word’s integral exactly via Chen’s identity across the breakpoints. Every identity in §2 is verified to machine precision by the engine:

The two engines - the direct explicit sums and the Chen-composition S_word - agree to $10^{-15}$, which is itself a numerical proof that the explicit sums and Chen’s identity describe the same object. Chen’s identity then holds to machine zero at level 2 and $10^{-15}$ at level 3; the shuffle identity holds to the same order; and the level-2 log-signature (the Lévy area) is exactly antisymmetric - the first hint that the log-signature is a Lie element, not a general tensor (§03).

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. The signature of a 1D path is just its endpoint. A single coordinate has no area, no loop, no shape - $S^{11}=\frac{1}{2}(S^1{)}^2$ is determined by $S^1$. Using raw univariate signatures is throwing the path away. The fix is time augmentation or lead-lag (§05).
2. Non-geometric (Itô) data breaks the shuffle law. The identities in §2 hold for geometric (Stratonovich / rough-path) integrals; raw increments are Itô-type and the shuffle relation fails at the diagonal. Augmenting is not optional - it is the fix (§05).
3. Truncation is a modelling choice with a price. The signature is infinite; any computation truncates at level $N$. The price is the reparametrisation-scale information in the high levels - small, fast oscillations live at high level, so truncation literally throws away the short-time structure (the one that matters for rough vol, §06).
4. The signature is not a probability distribution. Reparametrisation invariance means the signature is insensitive to the speed of the path - perfect for geometric features, wrong if the timing (e.g. the arrival structure of order flow) is itself the signal. Lead-lag reintroduces time, carefully (§05).
5. Expected signatures are not moments to be read naively. The expected signature is a universal characterising object (Chevyrev–Lyons), but Monte Carlo estimates converge slowly at high level - the weights decay only factorially - so truncation error and estimator bias interact (§06).
6. Roughness is a regularity statement, not a style choice. A $p$-variation norm below what the data actually supports gives uncontrolled iterated integrals; the whole rough-path apparatus exists because you cannot ignore this (§04).

5. References

• Lyons, Terry J. (1998), Differential equations driven by rough signals, Revista Matemática Iberoamericana 14(2), 215–310
• Lyons, T., Caruana, M., Lévy, T. (2007), Differential Equations Driven by Rough Paths (Springer Lecture Notes 1908)
• Lyons, T. J., Ni, Hao, Zhang, Hao (2019), Machine Learning Models of Financial Time Series, arXiv:1905.11666
• Hambly, B., Lyons, T. (2010), Uniqueness for the signature of a path of bounded variation and the reduced path group, Annals of Mathematics 171(1), 109–167
• Chevyrev, Ilya & Kormilitzin, Andrey (2016), A Primer on the Signature Method in Machine Learning, arXiv:1603.03788
• Friz, Peter K. & Victoir, Nicolas (2010), Multidimensional Stochastic Processes as Rough Paths (Cambridge Studies in Advanced Mathematics 120)
• Boedihardjo, H., Geng, X., Lyons, T., Yang, D. (2016), The signature of a rough path: uniqueness, Advances in Mathematics 293, 720–737
• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough

6. Connected Graph Bridges

• Foundational base: Stochastic Calculus & Itô’s Lemma · Brownian Motion & Martingales · Itô Integral & Itô’s Lemma · Linear Algebra & Matrices (tensor algebra) · Probability & Measure Theory
• Sibling topics: Advanced Volatility - Heston & SABR (rough vol lives here) · Exotic & Path-Dependent Options (the pricing objects that live on paths) · Numerical Methods (simulation of the paths)
• Related flat notes: Rough Volatility (from the SV side) · Lookbacks & Asians · Volatility Modeling · Deep Learning for Sequences
• Sub-pages (in-folder): 01 From Zero · 02 The Signature Algebra · 03 Log-Signature & Uniqueness · 04 Rough Path Theory · 05 Failure Modes & Practice · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05