Expected Signature in Stochastic Analysis

• Expected Signature is a non-commutative tensor series that aggregates all iterated integrals of a stochastic path, serving as a moment generating function.
• It bridges stochastic analysis and geometry by encoding dynamic path behavior through PDE recursions, cumulant transforms, and recursive differential equations.
• Its applications extend to statistical estimation, kernel methods, and parametric identification in models like Brownian motion, fractional Brownian motion, and McKean–Vlasov dynamics.

Searching arXiv for recent and foundational papers on expected signature to ground the article. Expected signature is the tensor-series expectation of path signatures, namely

$$\operatorname{ExpSig}(X)_T=\mathbb E\!\left[S(X)_{0,T}\right], \qquad S(X)_{0,T} = 1+\sum_{n=1}^{\infty}\int_{0<r_1<\cdots<r_n<T} dX_{r_1}\otimes\cdots\otimes dX_{r_n},$$

with stochastic iterated integrals interpreted in the Stratonovich sense in the semimartingale setting. It is described in the literature both as an analogue of the Laplace transform and as a path-space moment generating function, because it packages all expected iterated integrals of a random path into a single non-commutative object in the completed tensor algebra $T((\mathbb R^d))$ (Boedihardjo et al., 2019, Friz et al., 2021).

1. Algebraic definition and basic constructions

The signature of a sufficiently regular path takes values in the tensor algebra

$$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$

with level-$n$ projection denoted by $\operatorname{proj}_n$, $\pi_n$, or $T^n$ depending on context. For Brownian motion and more general semimartingales, the signature is defined through Stratonovich or Marcus integration, and for càdlàg semimartingales it is characterized by the stochastic equation $dS_t=S_t\circ dX_t$ in the appropriate geometric sense (Boedihardjo et al., 2019, Friz et al., 2024).

Conditional expected signature is typically written

$$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$

and its tensor logarithm

$\kappa_t(T)=\log \mu_t(T)\in T_0$

is called the signature cumulant. This logarithmic transform is the non-commutative analogue of passing from moments to cumulants; in one dimension, the expected signature reduces to the factorially normalized moment sequence, and the signature cumulant reduces to the ordinary cumulant sequence (Friz et al., 2024, Friz et al., 2021).

Several variants of the basic construction appear in the literature. On manifolds, one may define a $T((\mathbb R^d))$0-signature using an $T((\mathbb R^d))$1-valued one-form $T((\mathbb R^d))$2, with the geometrically important case $T((\mathbb R^d))$3 for an isometric embedding $T((\mathbb R^d))$4 (Geng et al., 2024). In learning-theoretic settings, one also encounters tensor-normalized expected signatures $T((\mathbb R^d))$5, where $T((\mathbb R^d))$6 is the time-augmented path and $T((\mathbb R^d))$7 rescales tensor levels to control growth (Romito et al., 2023).

2. Determinacy, analyticity, and radius of convergence

A central question is whether expected signature determines the law of a random signature. A standard sufficient criterion, quoted repeatedly in the literature, is infinite radius of convergence: $T((\mathbb R^d))$8 Under this condition, equal expected signatures imply equality in law for $T((\mathbb R^d))$9-valued random variables with the same projective expectation (Boedihardjo et al., 2019).

This criterion is satisfied in classical fixed-time Gaussian examples. For Brownian motion up to deterministic time $$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$0,

$$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$1

so the homogeneous levels have factorial decay and the radius of convergence is infinite (Boedihardjo et al., 2019). The same fixed-time explicit structure reappears in Euclidean expected-signature PDE calculations and in semimartingale cumulant formulas (Geng et al., 2024, Friz et al., 2024).

Random stopping can change this analytic behavior qualitatively. For planar Brownian motion stopped at the first exit time from the unit disc, the expected signature has finite radius of convergence; this gives a concrete negative answer to the question of whether Brownian motion stopped on exiting a bounded domain must satisfy the infinite-radius condition (Boedihardjo et al., 2019). On bounded $$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$2-domains $$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$3 with $$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$4, the expected signature of stopped Brownian motion has finite radius of convergence at every starting point (Li et al., 2020). Earlier PDE estimates for bounded domains yielded geometric bounds of the form

$$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$5

which imply only that the radius is strictly positive (Lyons et al., 2011).

These results also delimit a common misconception. Finite radius of convergence does not by itself show that expected signature fails to determine the law; it shows only that the standard sufficient criterion does not apply. This distinction is explicit in the stopped-disc example, where non-uniqueness is left open (Boedihardjo et al., 2019).

A different determinacy statement appears in the bounded-variation framework with tensor normalization. There the law of a process with continuous BV trajectories is characterized by normalized expected signature: $$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$6 The same paper also notes that raw expected signature does not in general determine the law in that framework, which is why normalization is built into the theorem (Romito et al., 2023).

3. Differential equations, recursive structure, and cumulant dynamics

Expected signatures are often computable because they satisfy recursive PDEs or functional equations. For Brownian motion started at $$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$7 and stopped at the first exit time $$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$8, the tensor-valued expected signature

$$T((\mathbb R^d))=\prod_{n=0}^{\infty}(\mathbb R^d)^{\otimes n},$$9

satisfies the elliptic system

$n$0

together with $n$1, $n$2, and the probabilistic boundary condition $n$3 as $n$4 (Lyons et al., 2011). Projecting onto tensor level $n$5 yields a triangular recursion in which the $n$6-th level depends only on levels $n$7 and $n$8.

For continuous semimartingales, the conditional expected signature admits a backward functional equation: $n$9 For càdlàg semimartingales the corresponding formula acquires jump-compensation terms involving $\operatorname{proj}_n$0 (Friz et al., 2024). These identities provide level-by-level recursions for $\operatorname{proj}_n$1 and are the semimartingale counterpart of the elliptic recursion for stopped Brownian motion.

Passing from $\operatorname{proj}_n$2 to the signature cumulant $\operatorname{proj}_n$3 yields a Magnus-type structure. The logarithmic equations are expressed through operator series

$\operatorname{proj}_n$4

and produce a universal functional relation for semimartingales. In the deterministic bounded-variation case, all bracket terms vanish and the cumulant equation reduces to the backward Hausdorff differential equation

$\operatorname{proj}_n$5

whose recursive solution is the Magnus expansion (Friz et al., 2021, Friz et al., 2024).

For time-inhomogeneous Lévy processes, the expected signature becomes especially transparent: it solves a deterministic backward ODE

$\operatorname{proj}_n$6

with generalized Lévy–Khintchine exponent

$\operatorname{proj}_n$7

In this sense, the expected signature of the stochastic process is the signature of a deterministic path driven by $\operatorname{proj}_n$8 (Friz et al., 2024).

4. Geometric information encoded by expected signatures

One of the strongest recent developments is the use of expected signatures as geometric probes. On a compact $\operatorname{proj}_n$9-dimensional oriented boundaryless Riemannian manifold $\pi_n$0, the expected $\pi_n$1-signature of Brownian motion satisfies an intrinsic parabolic equation. For $\pi_n$2 arising from an isometric embedding $\pi_n$3, the embedded signature records both intrinsic and extrinsic geometry (Geng et al., 2024).

For Brownian bridges, the high-level/small-time asymptotics recover distance. If $\pi_n$4 is isometric and $\pi_n$5, then there exist constants $\pi_n$6 such that

$\pi_n$7

Equivalently, for any $\pi_n$8,

$\pi_n$9

This is a signature-theoretic analogue of recovering distance from heat-kernel asymptotics, but the recovery is linear in $T^n$0 rather than quadratic in $T^n$1 (Geng et al., 2024).

For Brownian loops, the first three expected signature levels vanish by time-reversal symmetry, and the first nontrivial information appears at level four. Writing

$T^n$2

the contracted fourth level has an expansion

$T^n$3

The coefficient $T^n$4 determines the tangent space $T^n$5 and the metric $T^n$6, while $T^n$7 restricted to tangent directions yields a linear combination of scalar curvature $T^n$8, Ricci curvature $T^n$9, the pairing $dS_t=S_t\circ dX_t$0 between the second fundamental form and mean curvature vector, and $dS_t=S_t\circ dX_t$1 (Geng et al., 2024). The distinction between intrinsic and extrinsic quantities is explicit: $dS_t=S_t\circ dX_t$2, $dS_t=S_t\circ dX_t$3, and $dS_t=S_t\circ dX_t$4 are embedding-independent, whereas the terms involving $dS_t=S_t\circ dX_t$5 and $dS_t=S_t\circ dX_t$6 depend on the chosen isometric embedding.

This geometric use of expected signature also clarifies a limitation. Distance recovery is proved only in the regime $dS_t=S_t\circ dX_t$7, and curvature recovery is established only up to fourth signature level and order $dS_t=S_t\circ dX_t$8; the full Riemann curvature tensor and full second fundamental form are not reconstructed in the paper (Geng et al., 2024).

5. Estimation, kernels, and learning from path data

Recent work has moved expected signature from purely analytic questions to statistical estimation. For a stationary rough-path-valued process observed along a single long dependent trajectory, the target is the expected truncated block signature

$dS_t=S_t\circ dX_t$9

and the natural estimator is the block average

$$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$0

where $$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$1 is the signature of the piecewise-linearly interpolated block. Under moment control, strict stationarity, a blockwise discretization bound, and either mixing or direct covariance decay of block signatures, the mean-squared error satisfies

$$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$2

The first term is a discretization contribution and the second is a dependence-driven fluctuation term. In fractional Ornstein–Uhlenbeck examples, the paper verifies these assumptions in the rough, semimartingale, and long-range regimes $$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$3, $$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$4, and $$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$5 (Schenck, 19 May 2026).

In supervised learning, expected signature can be constructed from a single observed time series by generating an ensemble of augmented paths. A Gaussian-process-based data-augmentation module produces $$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$6 sampled trajectories, time augmentation is added, each path is mapped to a truncated signature, a tensor normalization $$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$7 is applied, and the empirical expected normalized signature

$$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$8

is fed to a classifier. The theoretical role of normalization is explicit: for continuous BV trajectories,

$$\mu_t(T)=\mathbb E_t\!\big(\operatorname{Sig}(X)_{t,T}\big)\in T_1,$$9

while a concentration bound based on Hoeffding’s inequality controls Monte Carlo approximation error in $\kappa_t(T)=\log \mu_t(T)\in T_0$0 (Romito et al., 2023).

The expected signature also induces kernels on laws of paths. If $\kappa_t(T)=\log \mu_t(T)\in T_0$1, then for probability measures $\kappa_t(T)=\log \mu_t(T)\in T_0$2,

$\kappa_t(T)=\log \mu_t(T)\in T_0$3

For inhomogeneous Lévy rough paths this expected signature kernel is characterized by a coupled PDE–ODE system, infinite-dimensional in the presence of jumps and finite-dimensional in continuous cases. In particular, the expected signature kernel of Gaussian martingales satisfies a scalar Goursat PDE (Friz et al., 9 Sep 2025).

6. Specialized models, parametric identification, and mean-field extensions

Expected signature has also become a model-dependent observable in several specialized stochastic systems. For the generalized physical Brownian motion

$\kappa_t(T)=\log \mu_t(T)\in T_0$4

the expected signature $\kappa_t(T)=\log \mu_t(T)\in T_0$5 admits a small-mass asymptotic expansion at each tensor level. As $\kappa_t(T)=\log \mu_t(T)\in T_0$6,

$\kappa_t(T)=\log \mu_t(T)\in T_0$7

The limit is nontrivial: higher levels survive through deterministic drift-decay tensors $\kappa_t(T)=\log \mu_t(T)\in T_0$8 and the second-level anomaly tensor $\kappa_t(T)=\log \mu_t(T)\in T_0$9 (Li et al., 2023).

For mixed fractional Brownian motion

$T((\mathbb R^d))$00

the expected signature up to level three is completely controlled by two scale transforms,

$T((\mathbb R^d))$01

At level two, the only parameter-dependent coordinate is $T((\mathbb R^d))$02; at level three, the only new parameter-dependent information is $T((\mathbb R^d))$03 together with $T((\mathbb R^d))$04. This leads to a sharp scale tradeoff: $T((\mathbb R^d))$05 distinct scales suffice for injectivity from level-two data alone, whereas $T((\mathbb R^d))$06 scales suffice when one uses the selected level-two and level-three information (Cai, 8 Jun 2026).

A different extension uses expected signature as the law variable in McKean–Vlasov dynamics. In signature McKean–Vlasov SDEs, coefficients depend on

$T((\mathbb R^d))$07

where $T((\mathbb R^d))$08 is the signature of the time-augmented geometric $T((\mathbb R^d))$09-rough path lift of the solution. Under boundedness and Lipschitz assumptions on the coefficients, the model admits strong existence and uniqueness; the paper also proves approximation of a class of path-dependent McKean–Vlasov SDEs by signature-based ones and establishes propagation of chaos for the associated particle system (Benth et al., 22 Jun 2026).

Across these examples, expected signature appears in three distinct roles: as an analytic transform with delicate radius-of-convergence behavior, as a geometric probe that can reconstruct metric and curvature data, and as a statistical or dynamical state variable usable for estimation, kernels, and mean-field interactions. The literature also makes clear that these roles come with model-specific boundaries: stopped Brownian motion can have finite radius of convergence (Boedihardjo et al., 2019, Li et al., 2020), geometric reconstruction on manifolds is currently local and low-order (Geng et al., 2024), and several estimation results are presently formulated only for fixed truncation level $T((\mathbb R^d))$10 rather than the full infinite expected signature (Schenck, 19 May 2026).