Brownian Signature Variables Overview

• Brownian signature variables are the tensor components of the iterated-integrals signature of Brownian motion, encoding rich path information via the Stratonovich formulation.
• The expected signature framework uses analytic and PDE methods to assess convergence properties and uniqueness, exemplified by finite-radius counterexamples in bounded domains.
• Time augmentation and orthogonalization techniques leverage these variables for efficient feature extraction in machine learning and robust numerical simulation.

Searching arXiv for recent and foundational papers on Brownian signatures, expected signatures, and related approximation/identifiability results. First, I’ll retrieve the core paper on the finite radius of convergence for the expected signature of stopped Brownian motion. Now I’ll gather the foundational expected-signature PDE paper and a few adjacent works on uniqueness, approximation, and time-augmented Brownian signatures. Brownian signature variables are the tensor components of the iterated-integrals signature of Brownian motion, taken coordinatewise in a fixed tensor basis and used as variables for rough path analysis, inference, classification, and learning (Boedihardjo et al., 2019). For a continuous path $X:[0,T]\to\mathbb R^d$, the signature is the formal series

$$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$

and for Brownian motion $B$ the same construction is taken in the Stratonovich sense, yielding random tensor coordinates $\pi_k S(B)_{0,T}$ at each level $k$ (Boedihardjo et al., 2019). These variables are central because signatures are group-like, satisfy Chen’s identity and shuffle relations, and provide a universal and faithful feature set up to reparametrization and tree-likeness, while their expectations, tails, and transforms expose delicate analytic structure that differs sharply between deterministic and random horizons (Lyons et al., 2011).

1. Signature coordinates and Brownian rough paths

For Brownian motion $B$ on $\mathbb R^d$, the stopped signature over a stopping time $\tau$ is the random element

$$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$

with level-$k$ projection

$$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$0

Brownian signature variables are precisely the coordinate components of these tensors in a chosen basis. In the tensor algebra $$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$1, signatures are group-like and multiplicative under concatenation, and coordinate iterated integrals obey shuffle identities; these algebraic properties are the basis for both probabilistic analysis and feature extraction (Lyons et al., 2011).

The Stratonovich formulation is structurally decisive. It preserves the geometric rough path interpretation, the shuffle algebra, and the pathwise relation between the signature and the driven differential equation viewpoint. The canonical Brownian rough path is the Stratonovich lift, and the signature path $$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$2 takes values in the group-like elements of the completed tensor algebra. This geometric viewpoint supports the interpretation of Brownian signature variables as noncommutative coordinates on path space rather than merely as collections of moments (Boedihardjo et al., 2016).

At low levels, the variables recover familiar stochastic quantities. Level $$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$3 is the increment, level $$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$4 contains symmetric quadratic variation contributions together with antisymmetric Lévy area, and higher levels encode nested temporal orderings of increments. In deterministic rough path theory, the full signature determines a tree-reduced path up to reparametrization; in the Brownian setting, the same principle underlies later uniqueness results for Brownian sample paths and Brownian rough paths (LeJan et al., 2011).

2. Expected signature and elliptic recursion

For a process $$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$5 with a suitable integration theory, the expected signature is defined level by level by

$$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$6

with the projective tensor norm implicit on each level. Its radius of convergence is the supremum of $$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$7 such that

$$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$8

For fixed deterministic time $$S(X)_{0,T}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<T} dX_{t_1}\otimes\cdots\otimes dX_{t_k},$$9, standard Brownian motion satisfies

$B$0

so the expected signature has infinite radius of convergence. The stopped case is more delicate because the horizon is controlled by exit geometry rather than by a fixed Gaussian covariance structure (Boedihardjo et al., 2019).

For Brownian motion stopped at the first exit time from a bounded domain $B$1, Lyons and Ni introduced the tensor-valued function

$B$2

and showed that it satisfies a recursive elliptic PDE system with Dirichlet-type boundary conditions. Writing $B$3 for the level-$B$4 component, one has for $B$5

$B$6

together with $B$7, $B$8, and $B$9 for $\pi_k S(B)_{0,T}$0. Under boundedness and regularity assumptions on $\pi_k S(B)_{0,T}$1, iterative Sobolev estimates imply geometric bounds of the form $\pi_k S(B)_{0,T}$2 and hence pointwise exponential bounds across levels (Lyons et al., 2011).

These bounds show that the radius of convergence is strictly positive, but they do not imply infinite radius. The distinction is essential: a positive radius is compatible with rapid growth of $\pi_k S(B)_{0,T}$3 as $\pi_k S(B)_{0,T}$4, whereas the Chevyrev–Lyons uniqueness criterion for laws of group-like random variables requires the expected signature to have infinite radius. This gap motivated the unit-disc analysis that produced the first negative answer (Boedihardjo et al., 2019).

3. Finite-radius counterexample on the unit disc

For planar Brownian motion in the unit disc

$\pi_k S(B)_{0,T}$5

the expected stopped signature is

$\pi_k S(B)_{0,T}$6

The main theorem states that the expected signature $\pi_k S(B)_{0,T}$7 of two-dimensional Brownian motion stopped upon exiting the unit disk has a finite radius of convergence. Equivalently, $\pi_k S(B)_{0,T}$8, where $\pi_k S(B)_{0,T}$9 is the Chevyrev–Lyons topology built from continuity under all algebra developments (Boedihardjo et al., 2019).

The proof applies a specific $k$0 matrix-valued development map

$k$1

and studies

$k$2

For $k$3, this developed expected signature is smooth and satisfies the linear PDE

$k$4

Rotational invariance reduces the problem to three radial functions $k$5 satisfying a second-order ODE system with boundary conditions at $k$6 and $k$7 (Boedihardjo et al., 2019).

That ODE system admits an explicit solution in terms of Bessel functions $k$8 and $k$9. The decisive object is a denominator $B$0 arising from the boundary-value problem; Lemma 8 shows that $B$1 has a real zero $B$2, and the resulting formula for $B$3 has a pole at $B$4. Hence the series expansion in $B$5 of the developed expected signature has finite radius of convergence, and analytic continuation then forces the original expected signature $B$6 to have finite radius as well (Boedihardjo et al., 2019).

This example answers negatively the question posed by Lyons and Ni even in the simplest bounded smooth domain. The phenomenon is not caused by lack of existence or regularity of the expected signature; it is caused by a finite singularity in an analytic development. A plausible implication is that stopped Brownian signature variables remain analytically rich but no longer behave like fixed-time Gaussian signatures with infinite-radius exponential generating series.

4. Determinacy, tails, and the limits of expected-signature identification

Chevyrev and Lyons proved a sufficient criterion for determining the law of a group-like random variable from its expected signature: if two group-like random variables have the same expected signature and that expected signature has infinite radius of convergence, then the laws coincide. The unit-disc example shows that this criterion fails for stopped planar Brownian motion. The failure is specific to the sufficient condition: the paper explicitly notes that finite radius does not by itself produce a different law with the same expected signature, so uniqueness may still hold by other arguments in particular cases (Boedihardjo et al., 2019).

This limitation should be distinguished from results on the full signature, as opposed to its expectation. In dimension $B$7, the Stratonovich signatures of Brownian motion over a fixed interval generate the same completed $B$8-algebra as the Brownian path itself, so almost all Brownian sample paths are determined by their signatures (LeJan et al., 2011). At the rough-path level, the Brownian signature path is quasi-surely non-self-intersecting, the Brownian rough path is quasi-surely tree-reduced, and quasi-surely every Brownian sample path is uniquely determined by its signature up to reparametrization (Boedihardjo et al., 2016).

A complementary viewpoint comes from tail asymptotics. For the Stratonovich signature of $B$9-dimensional Brownian motion, Boedihardjo and Geng showed that

$\mathbb R^d$0

almost surely, with quadratic variation playing the role that length plays for $\mathbb R^d$1 paths (Boedihardjo et al., 2016). In one dimension, the Itô signature tail of a continuous semimartingale satisfies

$\mathbb R^d$2

and for one-dimensional fractional Brownian motion with the fWIS signature the corresponding tail is $\mathbb R^d$3, recovering the Hurst parameter from signature asymptotics (Gbúr, 2023).

A common misconception is that all forms of “signature determinacy” are equivalent. The literature shows three distinct regimes: almost-sure path recovery from the full signature, analytic recovery of a law from an expected signature under an infinite-radius condition, and reconstruction of roughness parameters from tail asymptotics. Brownian signature variables participate in all three, but the governing criteria differ sharply.

5. Time augmentation, approximation, and feature design

In machine learning and non-anticipative approximation, one often time-extends Brownian motion by adding a monotone time coordinate. For the time-extended Brownian rough path $\mathbb R^d$4, finite linear combinations

$\mathbb R^d$5

of signature coordinates are dense in $\mathbb R^d$6 both for terminal functionals $\mathbb R^d$7 and for progressively measurable processes $\mathbb R^d$8 adapted to the Brownian filtration, provided the exponential integrability assumptions of the weighted rough-path framework hold. In particular, any $\mathbb R^d$9-integrable adapted random variable or adapted process can be approximated in $\tau$0 by linear functionals of time-extended Brownian signature variables (Ceylan et al., 18 Dec 2025).

Time augmentation, however, introduces algebraic redundancy. For words over the alphabet $\tau$1, where $\tau$2 denotes time, the signature components of length $\tau$3 have the same linear span as the signature components of length at most $\tau$4. Two canonical minimal spanning families are the words of length at most $\tau$5 whose last letter is not time, and symmetrically those whose first letter is not time; both have cardinality $\tau$6, compared with the full truncated count $\tau$7. For additive Brownian-noise SDEs, these subfamilies are linearly independent, and the same independence persists for sufficiently fine piecewise-linear discretizations (Bourdon et al., 15 Jan 2026).

Brownian signature variables also appear in kernel methods and numerical simulation. General signature kernels reduce many computations to averages of PDE solutions, and for a factorial weighting one has the identity

$\tau$8

linking Brownian expected signatures to hyperbolic development (Cass et al., 2021). In high-order SDE simulation with time augmentation, Gaussian variables $\tau$9, $$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$0, and $$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$1 reconstruct low-level mixed time-space iterated integrals, while Lévy-area-type approximations provide tractable surrogates for non-Gaussian second- and third-level signature terms needed in strong and weak schemes (Foster, 2024).

These developments show that Brownian signature variables are not only abstract coordinates in tensor algebra. They form a structured family of features whose redundancy, approximation power, and numerical tractability depend strongly on whether time is included, which tensor levels are retained, and whether one works with raw coordinates, kernels, or carefully chosen spanning subfamilies.

6. Transform theory, orthogonalization, and geometric extensions

Recent work places Brownian signatures inside broader transform frameworks. Signature SDEs treat drift and diffusion coefficients as linear functionals of the current signature, and the prolonged signature process becomes affine and polynomial on the space of group-like elements. This yields explicit Fourier–Laplace and moment formulas via extended-tensor Riccati and linear ODEs, with Brownian signature variables entering as the graded coordinates on which these generators act (Cuchiero et al., 2023).

For the time-augmented one-dimensional Brownian signature, an infinite-dimensional affine transform theory gives a conditional Fourier–Laplace transform of the form

$$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$2

where $$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$3 solves a linear ODE on the extended tensor algebra. When the logarithm exists locally, one obtains

$$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$4

with $$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$5 solving a Riccati equation on the tensor algebra. Zeros of the Fourier–Laplace transform obstruct global logarithmic expansions, so the affine representation is intrinsically local unless one recenters and solves randomized Riccati equations with path-dependent terminal data (Jaber et al., 28 Jun 2026).

Orthogonalization gives another reorganization of Brownian signature variables. Under an infinite-radius hypothesis on expected signatures, linear signature functionals are dense in $$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$6 on grouplike elements, and one can construct orthogonal signature polynomials by block Gram–Schmidt on the shuffle algebra. For Brownian motion, a dimension-independent orthogonal signature exists with drift in the time-augmented Itô setting but not without drift in the pure Stratonovich setting; Hermite and Legendre families appear as special cases inside this orthogonalized path-space calculus (Chevyrev et al., 21 Feb 2026).

Brownian expected signatures also encode geometry beyond Euclidean space. On a compact Riemannian manifold, suitable asymptotics of the expected signature of Brownian bridge from $$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$7 to $$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$8 recover the Riemannian distance $$S(B)_{0,\tau}=1+\sum_{k=1}^{\infty}\int_{0<t_1<\cdots<t_k<\tau}\circ dB_{t_1}\otimes\cdots\otimes \circ dB_{t_k},$$9, and the fourth-level expected signature of the Brownian loop at $k$0 recovers the metric, Ricci curvature, and extrinsic second fundamental form through explicit small-time expansions (Geng et al., 2024). This suggests that Brownian signature variables are not merely stochastic path descriptors; they can serve as geometric probes whose tensorial asymptotics record both probabilistic and differential- geometric structure.