# Exponentially Fading Memory Signature

Eduardo Abi Jaber*¹ and Dimitri Sotnikov†¹,²

¹École Polytechnique, CMAP

²Engie Global Markets

July 4, 2025

## Abstract

We introduce the exponentially fading memory (EFM) signature, a time-invariant transformation of an infinite (possibly rough) path that serves as a mean-reverting analogue of the classical path signature. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity. The EFM-signature retains many of the key algebraic and analytical properties of classical signatures, including a suitably modified Chen identity, the linearization property, path-determinacy, and the universal approximation property. From the probabilistic perspective, the EFM-signature provides a “stationarized” representation, making it particularly well-suited for time-series analysis and signal processing overcoming the shortcomings of the standard signature. In particular, the EFM-signature of time-augmented Brownian motion evolves as a group-valued Ornstein-Uhlenbeck process. We establish its stationarity, Markov property, and exponential ergodicity in the Wasserstein distance, and we derive an explicit formula à la Fawcett for its expected value in terms of Magnus expansions. We also study linear combinations of EFM-signature elements and the computation of associated characteristic functions in terms of a mean-reverting infinite dimensional Riccati equation.

## 1 Introduction

Fading memory and time invariance are important properties when modeling “memory” effects of dynamical systems. *Time invariance* means that the output signal $Y_t$ at time $t$ depends only on the continuous input signal $(X_s)_{-\infty<s\le t}$ up to time $t$, but not on the absolute time $t$. In other words, a parallel shift in time of the input signal results in an identical shift of the output signal. More precisely, we suppose that there exists a functional $F: C(\mathbb{R}_+, \mathbb{R}^d) \to \mathbb{R}$ such that, for all $t \in \mathbb{R}$,

$$Y_t = F((X_{t-s})_{s \ge 0}). \qquad (1.1)$$

Time invariance postulates, in some sense, a stationarity in the relationship between input and output. This is a much weaker condition than the stationarity of the output itself, as it allows for a nonstationary input. Nevertheless, the assumption of stationary dependence is crucial for the application of machine learning and econometric methods in time-series analysis. It is worth noting that, in practice, often only a single realization of a time series or a process trajectory—referred to more generally here as a “signal”—is available, for instance, in financial data. Within this framework, it is natural to assume that, at any given time $t$, the input signal $X$ is observable from the infinitely distant past $-\infty$ up to $t$. Without this assumption, the notion of time invariance would not make sense.

*Fading memory* is an additional assumption on the functional $F$, introduced to handle the infinite length of the past in a tractable way. Informally, it means that the influence of the distant past diminishes over time, i.e., that $Y_t$ gradually “forgets” the remote history of $X$. This idea dates back to the works of Volterra (2005) and Wiener (1958). Boyd and Chua (1985) formalize this concept by requiring the functional $F$ to be

<footnote>*eduardo.abi-jaber@polytechnique.edu. EAJ is grateful for the financial support from the Chaires FiME-FDD, Financial Risks and Deep Finance & Statistics at Ecole Polytechnique.</footnote>

<footnote>†dmitrii.sotnikov@polytechnique.edu. DS is grateful for the financial support provided by Engie Global Markets. Both authors thank Paul Hager, Fabian Harang, Luca Pelizzari, Josef Teichmann, and William Turner for fruitful discussions and insightful comments.</footnote>
continuous not with respect to the uniform topology, but with respect to a weighted uniform topology, thereby strengthening the notion of continuity. That is, F has fading memory if its output remains close for input paths that are close in the recent past, even if they differ in the distant past.

Moreover, Boyd and Chua (1985, Theorem 5) prove a folk theorem stating that any linear time-invariant operator with fading memory admits a convolution representation. In discrete time, this implies that the output can be written as a weighted moving average:

$$y_n = F((x_{n-k})_{k \ge 0}) = \sum_{k \ge 0} w_k x_{n-k},$$

for some sequence of weights $w = (w_k)_{k \ge 0} \in \ell^1$. A particularly popular example of such a weighted moving average is the exponential moving average (EWMA), defined as

$$y_n = \sum_{k \ge 0} \rho^k x_{n-k}, \quad \rho \in (0, 1).$$

Unlike the general case, the EWMA yields a Markovian sequence, as it satisfies $y_{n+1} = \rho y_n + x_{n+1}$, which significantly simplifies both analysis and computation.

However, linear functionals are quite limited in their ability to capture complex memory effects. Boyd and Chua (1985, Theorem 1) address the nonlinear case and provide an approximation result for time-invariant fading memory operators using Volterra series. This result is a consequence of the Stone-Weierstrass theorem and thus relies on the algebraic structure of the Volterra series. Nevertheless, as the authors point out, identifying an explicit approximating Volterra series remains a challenging task:

> "While the approximations are certainly strong enough to be useful in applications like macro-modeling of complicated systems or in universal nonlinear system identifiers, we know of no general procedure, based only on input/output measurements, by which an approximation can be found."

For a review of the Volterra series method and the different approaches to estimating Volterra kernels, we refer to Cheng, Peng, Zhang, and Meng (2017).

Path signatures offer an alternative to Volterra series. Originally introduced by Chen (1958), signatures have played an important role in stochastic Taylor expansions (see Kloeden and Platen (1992); Ben Arous (1989)) and have attracted significant attention through the work of Terry Lyons and the development of rough path theory (see Lyons, Caruana, and Lévy (2007)), which has led to a wide range of applications, particularly in machine learning. We refer the interested reader to the comprehensive surveys by Chevyrev and Kormilitzin (2016) and Lyons and McLeod (2022). Similar to Volterra series, path signatures possess a rich algebraic structure and enjoy a linearization property, making them powerful tools for functional approximation. A key advantage of path signatures, compared to Volterra series, lies in their computational tractability, made possible by the celebrated Chen (1958) identity.

When applied to time series or stochastic processes, the standard approach treats path signatures as features that efficiently summarize the information contained in the signal; see, e.g., Levin, Lyons, and Ni (2016); Morrill, Fermanian, Kidger, and Lyons (2020). Recent works by Cuchiero, Gazzani, and Svaluto-Ferro (2023a) and Abi Jaber and Gérard (2025) propose an alternative perspective, where linear functionals on path signatures are studied as stochastic processes themselves. This viewpoint allows for explicit representations of path-dependent processes as linear functionals of the time-augmented Brownian motion signature, as shown in Abi Jaber, Gérard, and Huang (2024a).

However, this framework is not naturally suited for modeling time-invariant dependencies as defined in (1.1). In particular, one cannot directly compute the signature of an infinite path. To incorporate information from the entire past, a natural model would be

$$Y_t = \langle \ell, \hat{\mathbb{X}}_{0,t} \rangle, \quad t \ge 0, \qquad (1.2)$$

where $\hat{\mathbb{X}}_{0,t}$ denotes the signature of the time-augmented path $\hat{X}_t = (t, X_t)$ over $[0, t]$.

This model is clearly not time-invariant and can easily lead to overfitting, as illustrated in the examples below. Furthermore, since signatures play a role analogous to polynomials in path space, (1.2) can be interpreted as a
kind of Taylor expansion of the output process Y; see, e.g., Litterer and Oberhauser (2014); Kloeden and Platen (1992). While such expansions offer excellent local approximation properties, they may fail globally: a model trained on an interval $[0, T_1]$ may diverge when applied on a longer interval $[0, T_2]$ with $T_2 > T_1$, as reported in Abi Jaber and Gérard (2025). This behavior is undesirable in the context of modeling time-invariant systems (1.1).

One could “stationarize” the signature model (1.2) to make it time-invariant by computing the path signature over a rolling window of fixed length $\Delta > 0$, that is, by considering

$$Y_t = \langle \ell, \hat{X}_{t-\Delta,t} \rangle, \quad t \in \mathbb{R}.$$

This approach serves as a continuous-time analogue of the expected signature (ES) model introduced by Levin, Lyons, and Ni (2016). However, this modification introduces several limitations. First, it assumes that the future depends only on a fixed part of the past, which makes it impossible to approximate more general fading-memory time-invariant functionals where the whole past can matter. Second, although this model enforces time invariance, it does so at the cost of significant mathematical intractability: the moving lower limit $t - \Delta$ in the signature computation leads to a delayed equation in $t$ and prevents one from deriving a simple dynamics of $(\hat{X}_{t-\Delta,t})_{t \in \mathbb{R}}$ or from obtaining a semimartingale decomposition for linear functionals applied to it.

In this paper, we aim to construct an object that combines the best features from three different approaches:

1. Time-invariance of Volterra series,

2. Algebraic properties, approximation power, and numerical tractability of signatures,

3. Markovian structure of EWMA.

To achieve this, we introduce the exponentially fading memory signature (EFM-signature for short), defined as follows: for an $\mathbb{R}^d$-valued continuous path $X = (X_t)_{t \in \mathbb{R}}$ and a vector $\lambda \in \mathbb{R}^d$ with positive entries, the fading-memory signature is defined as a collection of iterated integrals of different orders $n$

$$X_t^{\lambda, i_1 \cdots i_n} := \int_{-\infty < u_1 < \cdots < u_n < t} e^{-\lambda^{i_1} (t-u_1)} dX_{u_1}^{i_1} \cdots e^{-\lambda^{i_n} (t-u_n)} dX_{u_n}^{i_n}, \quad t \in \mathbb{R},$$

where $i_1, \dots, i_n \in \{1, \dots, d\}$.

In particular, we will show that when the input signal is a Brownian motion, its EFM-signature becomes a stationary $T((\mathbb{R}^d))$-valued Markov process, which can be treated with standard tools of stochastic calculus.

The well-posedness of the iterated integrals is ensured by the exponential weights. These weights maintain the same order of magnitude for each component of the EFM-signature as $t$ evolves, which is crucial for many applications. Moreover, the exponential form is crucial for preserving the algebraic structure of signatures, in particular, the Chen’s identity and the shuffle property. The same reason explains the choice of the argument $(t-u_j)$ in the terms $e^{-\lambda(t-u_j)} \odot dX_{u_j}$, rather than the choice $(u_{j+1}-u_j)$, which would correspond to a particular case of the Volterra signature introduced by Harang and Tindel (2021). Indeed, $(t-u_j)$ guarantees that the mean-reversion rate increases with the signature level, which is necessary for the multiplicative structure. In contrast, in the Volterra case with $(u_{j+1}-u_j)$, neither Chen’s identity nor the shuffle property hold. For this reason, Harang and Tindel (2021) propose replacing the tensor product with a convolution product to preserve multiplicativity, which makes the computation of the Volterra signature more intricate.

A discretized version of the EFM-signature can be seen as a particular case of the general weighted iterated sum introduced by Diehl and Krieg (2024) for an appropriately chosen weight function. This work examines the application of the signature approach to time series, where it has been demonstrated to perform well in classification tasks. However, the authors do not introduce this particular form and do not leverage its algebraic and analytical properties.

We develop a framework for the EFM-signature, establishing its principal algebraic, analytic, and probabilistic properties. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity (Theorem 3.13) and we prove the fading memory property (Theorem 3.14). We show that the EFM-signature preserves many of the essential features of classical signatures—such as a modified Chen identity (Proposition 4.1), the linearization property (Proposition 4.8), uniqueness (Theorem 4.7), the universal approximation theorem (Theorem 4.10), and moment determinacy
(Proposition 4.11)—while naturally adapting them to a mean-reverting, time-invariant setting. The EFM-signature of piece-wise linear path can be computed explicitly (Theorem 4.4), which allows for an efficient numerical computation and simulation. We pay particular attention to the case of time-augmented Brownian motion, for which the EFM-signature evolves as a group-valued Ornstein–Uhlenbeck process. In this context, we prove its Markov property (Theorem 5.2), stationarity (Theorem 3.16), and exponential ergodicity under the Wasserstein distance (Theorem 5.3), and we derive explicit formulas for its expected EFM-signature in the spirit of Fawcett’s formula via Magnus expansions (Theorem 5.5). Finally, we investigate linear functionals of the EFM-signature and establish an Itô-type decomposition (Proposition 6.2) along with a representation of the associated characteristic function via an infinite-dimensional, mean-reverting Riccati equation (Theorem 6.4).

**Outline.** Section 2 illustrates the motivation for introducing the EFM-signature with two simple examples. In Section 3, we introduce the notation, provide a definition of the fading-memory signature, and discuss its first properties. Section 4 establishes the algebraic and analytical properties of the EFM-signature for the general semimartingale path. In Section 5, we treat the case of the Brownian motion path and study its probabilistic properties. Section 6 is dedicated to the linear functionals of the EFM-signature of Brownian motion. Finally, in Section 7, we develop the rough path tools necessary to treat the EFM-signature and we establish the integration result and the fading memory property.

# 2 Motivating examples

To illustrate shortcomings of the standard signature model (1.2) and the relevance of our approach, we consider two toy examples.

## 2.1 Signature representation of an Ornstein–Uhlenbeck process

Consider an Ornstein–Uhlenbeck signal $Y$ given by

$$dY_t = -\mu Y_t dt + dW_t, \quad t \in \mathbb{R}, \quad Y_0 \in \mathbb{R}.$$

It can be written as an infinite linear combination of signature elements of the time extended Brownian motion $(t, W_t)$:

$$Y_t = Y_0 + Y_0 \sum_{k \ge 1} (-\mu)^k \int_{0<s_1<\dots<s_k<t} ds_1 \dots ds_k + \sum_{k \ge 1} (-\mu)^{k-1} \int_{0<s_1<\dots<s_k<t} dW_{s_1} \dots dW_{s_k}, \quad (2.1)$$

as shown by Abi Jaber and Gérard (2025). In practice, this series should be truncated at some order $N$. For the numerical example, we chose $N = 10$. Similarly, we will show in Example 6.3.1 that $Y$ can be approximated by a linear functional (6.4) of the EFM-signature, also truncated at order $N = 10$.

The trajectories of these linear combinations are shown in Figure 1, the representation (2.1) using the standard signature on the left, and (6.4) using the EFM-signature on the right. We clearly observe the explosive behavior of the standard signature when $T$ is far from zero on the left graph, exactly as in the case of the Taylor expansion of real-valued functions, which is accurate in a neighborhood of the expansion point. On the contrary, the EFM-signature on the right-hand side provides a uniform global approximation independent of the time interval length overcoming the explosive problem of the standard signature.
<figure><img src="image_1.png" /><figcaption>Figure 1: Representation of an OU process as a linear functional of the signature (on the left) and the EFM-signature (on the right) of the time-augmented Brownian motion. The parameter of the EFM-signature is $\lambda = 3$, and the mean-reversion parameter of the OU process is $\mu = 5$. Both signatures were truncated at order $N = 10$.</figcaption></figure>

## 2.2 Learning stochastic dynamics with signatures

In the second example, we compare regression models based on the signature and the EFM-signature to fit the dynamics of a stationary SDE solution.

Consider a signal $Y$ following the dynamics:

$$dY_t = -\mu Y_t^p dt + dW_t, \quad t \in \mathbb{R}, \qquad (2.2)$$

for some $\mu > 0$ and an odd $p \in \mathbb{N}$. Our goal is to learn the dynamics (2.2) by observing the signal $Y$ and the Brownian motion $W$, using only a single trajectory of the process. Given a time grid $0 = t_0 < t_1 < \dots < t_N = T$, we formulate the following regression problem:

$$\min_{\ell \in T^N(\mathbb{R}^2)} \left\{ \frac{1}{2(N+1)} \sum_{0 \le t_i \le T} |Y_{t_i} - \langle \ell, \mathbb{X}_{t_i} \rangle|^2 + \alpha \omega \|\ell\|_1 + \frac{1}{2} \alpha (1-\omega) \|\ell\|_2^2 \right\},$$

where $\alpha > 0$ is a regularization parameter, $\omega \in [0, 1]$ is a mixing parameter, and the $\ell^q$-norm for $q \ge 1$ is defined by

$$\|\ell\|_q := \left( \sum_{n=0}^{N} \sum_{|v|=n} |\ell^v|^q \right)^{1/q}, \quad \ell \in T^N(\mathbb{R}^2),$$

we refer to Section 3.1 for the precise definitions and notations. Three options for the signature $\mathbb{X}_t$ are considered:

1. **Signature of time-augmented Brownian motion $\hat{W}_t = (t, W_t)$: $\mathbb{X}_t = \hat{\mathbb{W}}_t$**. An approach implemented by Cuchiero, Gazzani, and Svaluto-Ferro (2023a) and Abi Jaber and Gérard (2025) seems to be the most natural from a theoretical point of view and consists of regressing the signal against the (extended) signature of the underlying Brownian motion. Although this method is able to fit nonstationary dynamics given many trajectory samples, as indicated in the mentioned works, the model quickly overfits when the dynamics is stationary and can hardly reproduce a reasonable behavior on out-of-sample intervals, as shown in Figure 2 (brown trajectory).

2. **Signature of time-augmented stationary OU process $\hat{U}_t = (t, U_t)$: $\mathbb{X}_t = \hat{\mathbb{U}}_t$, where $U$ is an Ornstein–Uhlenbeck process satisfying**

$$dU_t = -\lambda U_t dt + dW_t, \quad t \in \mathbb{R},$$

for some $\lambda > 0$. At first glance, this approach seems to be better suited for dealing with stationary processes. For instance, Cuchiero, Gazzani, Möller, and Svaluto-Ferro (2025) use the signatures of
correlated OU processes for stochastic volatility modeling. However, due to the presence of nonstationary terms corresponding to the t-component of the path and the nonstationary nature of the model itself, this approach also demonstrates poor out-of-sample performance (green trajectory in Figure 2).

3. EFM-signature of time-augmented Brownian motion: $X_t = \hat{W}_t^\lambda$. The third model relies on the EFM-signature of the Brownian motion observed on $(-\infty, T]$ introduced in this paper. As shown in the illustration, this model demonstrates the best out-of-sample performance and remains remarkably robust and close to the signal process, thanks to the time invariance embedded in the construction of the EFM-signature.

For the numerical illustration, we took $N = 6$ and $T = 2$ and simulated a trajectory with a sampling frequency of 3650 points per unit time. The parameters of $Y$ were chosen as $p = 5$ and $\mu = 10$. For each model, the interval $[0, 1]$ was used as the training sample, while the entire interval $[0, T]$ was used to calibrate the hyperparameters $\alpha$, $\omega$, and $\lambda$. The remaining part, $[2, 4]$, was not involved in the optimization problem and is provided to evaluate the out-of-sample model performance.

<figure><img src="image_2.png" /><figcaption>Figure 2: Trajectories predicted by the three regression models fitted to the stationary signal (black) given by (2.2): regression on $\hat{W}_t$ (brown), regression on $\hat{U}_t$ (green), and regression on $\hat{W}_t^\lambda$ (violet). Black vertical lines indicate the training interval (dotted) and the in-sample interval (solid).</figcaption></figure>

## 3 Exponentially fading memory signature and its first properties

**Notations.** Throughout this paper, $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in \mathbb{R}}, \mathbb{P})$ will denote a filtered probability space. We will use the notion of increment semimartingales, i.e., processes $X$ such that, for all $s \in \mathbb{R}$, the process $(X_{s+t} - X_s)_{t \ge 0}$ is a semimartingale in the usual sense with respect to $(\mathcal{F}_{s+t})_{t \ge 0}$. In this case, the notion of stochastic integral can be generalized, and integration over the whole real line becomes possible. For details, we refer to the works of Basse-O'Connor, Graversen, and Pedersen (2010, 2014). This theory allows us to manipulate stochastic integrals on the real line in exactly the same way as standard stochastic integrals on $\mathbb{R}_+$. With a slight abuse of terminology, we will henceforth use the terms semimartingales and increment semimartingales interchangeably.

An important example of an increment semimartingale we will often use is a two-sided Brownian motion: a centered Gaussian process $W = (W_t)_{t \in \mathbb{R}}$ such that $W_0 = 0$, and for any $s \in \mathbb{R}$ and $t > 0$, the increment $W_{s+t} - W_s$ is distributed as $\mathcal{N}(0, t)$ and is independent of $\mathcal{F}_s$.

For $t_0 \in \mathbb{R} \cup \{-\infty\}$, we will denote the Itô integral by $\int_{t_0} Y_t dX_t$ and the Stratonovich integral by $\int_{t_0} Y_t \circ dX_t$. If both $X$ and $Y$ are continuous semimartingales, then we have the relation

$$ \int_{t_0} Y_t \circ dX_t = \int_{t_0} Y_t dX_t + \frac{1}{2} \int_{t_0} d\langle X, Y \rangle_t. $$

We denote by $\odot$ the Hadamard product: for two vectors $\boldsymbol{x} = (x^1, \dots, x^d) \in \mathbb{R}^d$ and $\boldsymbol{y} = (y^1, \dots, y^d) \in \mathbb{R}^d$, we
define

$$ \boldsymbol{x} \odot \boldsymbol{y} := (x^i y^i)_{i=1,\dots,d} \in \mathbb{R}^d. $$

We write $\boldsymbol{x} \prec \boldsymbol{y}$ if $x^i < y^i$ for $i = 1, \dots, d$. We use $\lesssim$ to denote an inequality up to a positive constant, and $\lesssim_a$ to denote an inequality up to a positive constant depending on $a$.

## 3.1 Preliminaries on signatures

In this subsection, we recall the necessary notations related to the signatures. For a more detailed introduction of signatures and its properties, we refer to the work of Chevyrev and Kormilitzin (2016).

**Definition of signature.** Consider an $\mathbb{R}^d$-valued continuous semimartingale $X = (X_t)_{t \in [0,T]}$ defined on a finite interval $[0, T]$. Its signature over $[s, t] \subset [0, T]$ is nothing else but a collection of properly ordered iterated Stratonovich integrals of different orders $n \in \mathbb{N}_0$,

$$ \mathbb{X}_{s,t}^{i_1 \cdots i_n} := \int_{s<u_1<\cdots<u_n<t} \circ dX_{u_1}^{i_1} \cdots \circ dX_{u_n}^{i_n} = \int_s^t \mathbb{X}_{u_n}^{i_1 \cdots i_{n-1}} \circ dX_{u_n}^{i_n}, \quad (3.1) $$

where $i_1, \dots, i_n \in \{1, \dots, d\}$. The iterated integrals of order $n \in \mathbb{N}_0$ are then the elements of $(\mathbb{R}^d)^{\otimes n}$. Let $\{e_1, \dots, e_d\}$ denote the canonical basis of $\mathbb{R}^d$. Then, the basis of $(\mathbb{R}^d)^{\otimes n}$ is given by the vectors

$$ e_{i_1} \otimes \dots \otimes e_{i_n}, \quad i_1, \dots, i_n \in \{1, \dots, d\}, $$

identified with words $v = i_1 \dots i_n$ of length $|v| = n$ over the alphabet $A_d = \{1, \dots, d\}$. This explains the indexation in (3.1). For the particular case $n = 0$, we set $(\mathbb{R}^d)^{\otimes 0} \cong \mathbb{R}$, and denote the empty word corresponding to its basis vector by $\emptyset$. By convention, we set $\mathbb{X}_t^\emptyset = 1$.

This allows us to write the n-th level of signature $\mathbb{X}_{s,t}^n = (\mathbb{X}_{s,t}^v)|_{v=n} \in (\mathbb{R}^d)^{\otimes n}$ shortly as

$$ \mathbb{X}_{s,t}^n := \int_{s<u_1<\cdots<u_n<t} \circ dX_{u_1} \otimes \cdots \otimes \circ dX_{u_n}. $$

The path signature of $X$ is then given by a sequence of all signature levels, also called a tensor sequence:

$$ \mathbb{X}_{s,t} := (1, \mathbb{X}_{s,t}^1, \dots, \mathbb{X}_{s,t}^n), $$

which is an element of the extended tensor algebra:

$$ T((\mathbb{R}^d)) := \{\boldsymbol{\ell} = (\boldsymbol{\ell}^n)_{n \ge 0} : \boldsymbol{\ell}^n \in (\mathbb{R}^d)^{\otimes n}\}. $$

We will also denote $\mathbb{X}_t := \mathbb{X}_{0,t}$.

The signature plays a similar role to polynomials on path-space. Indeed, in dimension $d = 1$, the signature $\mathbb{X}_{s,t}$ of $X$ is the sequence of monomials $(\frac{1}{n!}(X_t - X_s)^n)_{n \in \mathbb{N}}$.

**Example 3.0.1.** In the case $d = 2$ and $X_t = \tilde{W}_t := (t, W_t)$ where $W$ is a 1-dimensional Brownian motion, the first orders of signature are given by

$$ \tilde{W}_t^0 = 1, \quad \tilde{W}_t^1 = \begin{pmatrix} t \\ W_t \end{pmatrix}, \quad \tilde{W}_t^2 = \begin{pmatrix} \frac{t^2}{2!} & \int_0^t s dW_s \\ \int_0^t W_s ds & \frac{W_t^2}{2!} \end{pmatrix} \quad (3.2) $$

**Tensor algebra.** The basis of $T((\mathbb{R}^d))$ is given by all the words $v$ over the alphabet $A_d$, so that each element $\boldsymbol{\ell} \in T((\mathbb{R}^d))$ can be represented as

$$ \boldsymbol{\ell} = \sum_{n \ge 0} \sum_{|v|=n} \boldsymbol{\ell}^v v. $$

We also define the truncated tensor algebra of order $N \in \mathbb{N}$ by

$$ T^N(\mathbb{R}^d) := \{\boldsymbol{\ell} \in T((\mathbb{R}^d)) : \boldsymbol{\ell}^n = 0, n > N\}, $$
and the tensor algebra $T(\mathbb{R}^d) := \bigcup_{N \ge 0} T^N(\mathbb{R}^d)$ as the algebra containing only finite tensor sequences. Similarly, the truncated signature of order $N \in \mathbb{N}$ is defined by

$$ \mathbb{X}_{s,t}^{\le N} := (1, \mathbb{X}_{s,t}^{1}, \dots, \mathbb{X}_{s,t}^{N}, 0, \dots, 0, \dots). $$

Along with the natural sum and multiplication by constants for the elements of $T((\mathbb{R}^d))$, we define the tensor product of $\mathbf{p}, \mathbf{q} \in T((\mathbb{R}^d))$ as

$$ \mathbf{p} \otimes \mathbf{q} = \left( \sum_{k=0}^{n} \mathbf{p}^{k} \otimes \mathbf{q}^{n-k} \right)_{n \ge 0}, $$

or, in coordinate form,

$$ (\mathbf{p} \otimes \mathbf{q})^{\mathbf{v}} = \sum_{\substack{\mathbf{u}, \mathbf{w} \\ \mathbf{uw}=\mathbf{v}}} p^{\mathbf{u}} q^{\mathbf{w}}, $$

where $\mathbf{uw}$ is the concatenation of the words $\mathbf{u}$ and $\mathbf{w}$, identified with the tensor product of vectors corresponding to $\mathbf{u}$ and $\mathbf{w}$.

We define the bracket $\langle \cdot, \cdot \rangle : T(\mathbb{R}^d) \times T((\mathbb{R}^d)) \to \mathbb{R}$ by

$$ \langle \mathbf{p}, \mathbf{q} \rangle = \sum_{n \ge 0} \sum_{|\mathbf{v}|=n} \mathbf{p}^{\mathbf{v}} \mathbf{q}^{\mathbf{v}}. $$

## 3.2 Exponentially fading memory signature

We now consider an $\mathbb{R}^d$-valued increment semimartingale $X = (X_t)_{t \in \mathbb{R}}$ and a vector $\mathbf{\lambda} = (\lambda^1, \dots, \lambda^d) \in \mathbb{R}^d$ with positive entries, i.e., $\mathbf{\lambda} > 0$. We will denote $\Delta := \{(s, t) \in \mathbb{R}^2: s \le t\}$.

To define the EFM-signature over the infinite time interval, we introduce exponential damping of the integrators $\circ dX_u^i$ by weights $e^{-\lambda^i(t-u)}$, for some positive constants $\lambda^i > 0$.

**Definition 3.1.** The $\mathbf{\lambda}$-exponential fading memory signature of $X$ is defined by

$$ \begin{gathered} \mathbb{X}_{s,t}^{\mathbf{\lambda}} : \Omega \times \Delta \to T((\mathbb{R}^d)) \\ (\omega, s, t) \mapsto \mathbb{X}_{s,t}^{\mathbf{\lambda}}(\omega) := (1, \mathbb{X}_{s,t}^{\mathbf{\lambda},1}(\omega), \dots, \mathbb{X}_{s,t}^{\mathbf{\lambda},n}(\omega), \dots), \end{gathered} $$

where

$$ \mathbb{X}_{s,t}^{\mathbf{\lambda},n} := \int_{s<u_1<\dots<u_n<t} e^{-\lambda(t-u_1)} \odot dX_{u_1} \odot \dots \odot e^{-\lambda(t-u_n)} \odot dX_{u_n} \quad (3.3) $$

takes values in $(\mathbb{R}^d)^{\otimes n}$, for $n \ge 0$. We also denote $\mathbb{X}_t^{\mathbf{\lambda}} := \mathbb{X}_{-\infty,t}^{\mathbf{\lambda}}$.

We postpone the question of the well-definedness of the iterated integrals in (3.3) as well as the fading memory property to Section 3.4.

**Remark 3.1.1.** Explicitly, the components of $\mathbb{X}_t^{\mathbf{\lambda},n}$ are given by

$$ \mathbb{X}_{s,t}^{\mathbf{\lambda},i_1 \cdots i_n} := \int_{s<u_1<\cdots<u_n<t} e^{-\lambda^{i_1}(t-u_1)} dX_{u_1}^{i_1} \cdots e^{-\lambda^{i_n}(t-u_n)} dX_{u_n}^{i_n}. $$

They can also be expressed in an iterated form:

$$ \mathbb{X}_{s,t}^{\mathbf{\lambda},i_1 \cdots i_n} = \int_s^t e^{-\lambda^{i_1}(t-u_1)} \mathbb{X}_{u_n}^{\mathbf{\lambda},i_1 \cdots i_{n-1}} \circ dX_{u_n}^{i_n}, \quad (3.4) $$

where we set

$$ \lambda^{i_1 \cdots i_n} := \sum_{k=1}^{n} \lambda^{i_k}. \quad (3.5) $$

In particular, we set $\lambda^\theta = 0$. It is clear that $\lambda^{\mathbf{u}\mathbf{v}} = \lambda^\mathbf{u} + \lambda^\mathbf{v}$ for all words $\mathbf{u}, \mathbf{v}$.
*Example 3.1.1.* For a one-dimensional process $X_t$ with $d=1$, its EFM-signature is given by

$$ \mathbb{X}_{s,t}^{\lambda} = \left( 1, (U_t - U_s), \frac{(U_t - U_s)^2}{2!}, \dots, \frac{(U_t - U_s)^n}{n!}, \dots \right), $$

where

$$ U_t := \int_{-\infty}^{t} e^{-\lambda^1(t-s)} dX_s. $$

In particular, if $X = W$ is a Brownian motion, its EFM-signature is nothing other than the powers of the increments of the corresponding stationary Ornstein-Uhlenbeck process with mean-reversion coefficient $\lambda^1$. In the general case, the process $U$ can be interpreted as a continuous-time version of the exponential moving average of the past increments of $X$. ■

*Example 3.1.2.* In the case $d=2$ and $X_t = \hat{W}_t := (t, W_t)$, the first levels of the signature are given by

$$ \hat{W}_t^{\lambda,0} = 1, \quad \hat{W}_t^{\lambda,1} = \begin{pmatrix} (\lambda^1)^{-1} \\ U_t \end{pmatrix}, \quad \hat{W}_t^{\lambda,2} = \left( \int_{-\infty}^{t} e^{-(\lambda^1+\lambda^2)(t-s)} U_s \, ds \quad \frac{(\lambda^1)^{-2}}{2!} \quad (\lambda^1)^{-1} \int_{-\infty}^{t} e^{-(\lambda^1+\lambda^2)(t-s)} \, dW_s \right), $$

where $U = (U_t)_{t \in \mathbb{R}}$ is a stationary Ornstein-Uhlenbeck process defined by

$$ U_t = \int_{-\infty}^{t} e^{-\lambda^2(t-s)} dW_s. $$

This is to be compared with the (standard) signature of $\hat{W}$ given in (3.2). ■

*Example 3.1.3.* The entries $\lambda^i$ of the vector $\lambda$ are hyperparameters that control how quickly the influence of the past of the $i$-th signal component $X^i$ fades, or equivalently, how rapidly the signature elements involving the integrators $dX^i$ revert to their mean. More precisely, the mean-reversion rate of $\mathbb{X}_t^{\lambda,\nu}$ is given by $\lambda^\nu$. Multiple time scales can be explored either by computing higher-order terms of the EFM-signature, as shown in the previous example, or by extending the path dimension duplicating path components and assigning them different values of $\lambda^i$. For instance, taking the path $\bar{X}_t = (X_t, X_t)$ and $\lambda = (\lambda^1, \lambda^2)$, we obtain

$$ \mathbb{X}_t^{\lambda,0} = 1, \quad \mathbb{X}_t^{\lambda,1} = \begin{pmatrix} U_t^{\lambda^1} \\ U_t^{\lambda^2} \end{pmatrix}, \quad \mathbb{X}_t^{\lambda,2} = \left( \int_{-\infty}^{t} e^{-(\lambda^1+\lambda^2)(t-s)} U_s^{\lambda^2} \circ dX_s \quad \frac{(U_t^{\lambda^1})^2}{2!} \quad \int_{-\infty}^{t} e^{-(\lambda^1+\lambda^2)(t-s)} U_s^{\lambda^1} \circ dX_s \quad \frac{(U_t^{\lambda^2})^2}{2!} \right), $$

where $U = (U_t^{\lambda^i})_{i \in \mathbb{R}}$ is given by

$$ U_t^{\lambda^i} = \int_{-\infty}^{t} e^{-\lambda^i(t-s)} dX_s. $$

This construction allows for explicit representation of Volterra processes $\int_{-\infty}^{t} K(t-s) dX_s$ with

$$ K(t) = \sum_{k=1}^{n} c_k e^{\lambda^k t}, \quad t \ge 0 \qquad (3.6) $$

and the polynomials of these processes. Due to the remarkable approximation power of kernels of the form (3.6), this also opens the door to a tractable approximation of nonlinear functionals of Volterra processes with more general kernels, see, e.g., Carmona and Coutin (1998) and Abi Jaber and El Euch (2019). ■

By definition, the EFM-signature is time-invariant in the sense of (1.1). This result is formalized in the following proposition.

**Proposition 3.2.** If $X = (X_t)_{t \in \mathbb{R}}$ is a continuous semimartingale and $Y = X_{\cdot+h}$ for some $h \in \mathbb{R}$, then

$$ \mathbb{Y}_t^{\lambda} = \mathbb{X}_{t+h}^{\lambda}. \qquad (3.7) $$

*Proof.* (3.7) follows immediately from the change of variables $v_i = u_i + h$ for $i = 1, \dots, n$ in (3.3). □
## 3.3 EFM-signature dynamics

We start by introducing two operators that are important for understanding the properties of the EFM-signature and for the analysis of its dynamics.

**Definition 3.3.** The operator $\Lambda: T((\mathbb{R}^d)) \to T((\mathbb{R}^d))$ is defined by

$$ \Lambda: \ell = \sum_{n \ge 0} \sum_{|v|=n} \ell^v \mapsto \Lambda \ell = \sum_{n \ge 0} \sum_{|v|=n} \lambda^v \ell^v, $$

where $\lambda^v$ is given by (3.5).

In other words, $\Lambda$ is a diagonal operator multiplying each entry of the tensor algebra element by the corresponding mean-reversion coefficient. We note that this operator is not continuous with respect to any norm on $T((\mathbb{R}^d))$.

**Definition 3.4.** For $\lambda \in \mathbb{R}^d$ and $h \in \mathbb{R}$, we define the operator $\mathbf{D}_h^\lambda: T((\mathbb{R}^d)) \to T((\mathbb{R}^d))$ by

$$ \mathbf{D}_h^\lambda: \ell = \sum_{n \ge 0} \sum_{|v|=n} \ell^v \mapsto \mathbf{D}_h^\lambda \ell = \sum_{n \ge 0} \sum_{|v|=n} e^{-\lambda^v h} \ell^v. \quad (3.8) $$

**Proposition 3.5.** The following properties hold

(i) $\mathbf{D}_h^\lambda$ is a dilation operator multiplying the i-th basis vector by $e^{-\lambda^i h}$.

(ii) For all $h_1, h_2 \in \mathbb{R}$, $\mathbf{D}_{h_1}^\lambda \mathbf{D}_{h_2}^\lambda = \mathbf{D}_{h_1+h_2}^\lambda$, so that $(\mathbf{D}_h^\lambda)_{h \ge 0}$ form a semigroup.

(iii) For all $\lambda_1, \lambda_2 \in \mathbb{R}^d$, $\mathbf{D}_h^{\lambda_1} \mathbf{D}_h^{\lambda_2} = \mathbf{D}_h^{\lambda_1+\lambda_2}$.

(iv) For all $c \in \mathbb{R}$, $\mathbf{D}_{ch}^\lambda = \mathbf{D}_h^{c\lambda}$.

(v) $\mathbf{D}_h^\lambda$ is an operator exponential of $-h\Lambda$, i.e., $\mathbf{D}_h^\lambda = e^{-h\Lambda} = \sum_{n \ge 0} \frac{(-h\Lambda)^n}{n!}$.

(vi) For $h \in \mathbb{R}$, $\frac{d}{dh}\mathbf{D}_h^\lambda = -\Lambda\mathbf{D}_h^\lambda = -\mathbf{D}_h^c\Lambda$.

(vii) For $\mathbf{p}, \mathbf{q} \in T((\mathbb{R}^d))$, we have $\mathbf{D}_h^\lambda(\mathbf{p} \otimes \mathbf{q}) = (\mathbf{D}_h^\lambda \mathbf{p}) \otimes (\mathbf{D}_h^\lambda \mathbf{q})$.

(viii) For $\mathbf{p}, \mathbf{q} \in T((\mathbb{R}^d))$, we have $\Lambda(\mathbf{p} \otimes \mathbf{q}) = (\Lambda\mathbf{p}) \otimes \mathbf{q} + \mathbf{p} \otimes (\Lambda\mathbf{q})$.

*Proof.* The properties (i)-(vi) follow immediately from the definition. For the last two properties, recall that

$$ (\mathbf{p} \otimes \mathbf{q})^v = \sum_{\substack{u,w \\ uw=v}} p^u q^w, $$

so that

$$ (\mathbf{D}_h^\lambda (\mathbf{p} \otimes \mathbf{q}))^v = e^{-\lambda^v h} \sum_{\substack{u,w \\ uw=v}} p^u q^w = \sum_{\substack{u,w \\ uw=v}} (e^{-\lambda^u h} p^u)(e^{-\lambda^w h} q^w) = ((\mathbf{D}_h^\lambda \mathbf{p}) \otimes (\mathbf{D}_h^\lambda \mathbf{q}))^v, $$

since $\lambda^v = \lambda^u + \lambda^w$ for all $u, w$ such that $uw = v$. Similarly,

$$ (\Lambda(\mathbf{p} \otimes \mathbf{q}))^v = \lambda^v \sum_{\substack{u,w \\ uw=v}} p^u q^w = \sum_{\substack{u,w \\ uw=v}} (\lambda^u p^u q^w + \lambda^w p^u q^w) = ((\Lambda\mathbf{p}) \otimes \mathbf{q})^v + (\mathbf{p} \otimes (\Lambda\mathbf{q}))^v. $$

**Proposition 3.5** allows us to rewrite the iterated integrals in the EFM-signature more concisely. Indeed, it follows from (3.4) that $X^\lambda$ can be written as

$$ X_{s,t}^{\lambda} = \int_{s}^{t} \mathbf{D}_{t-u}^{\lambda} (X_{u}^{\lambda} \otimes \circ dX_{u}), \quad -\infty \le s \le t. \quad (3.9) $$

The following proposition is a compact formulation of the variation of constants formula written in terms of the operators $\Lambda$ and $\mathbf{D}_h^\lambda$.
**Proposition 3.6.** Fix $s \in \mathbb{R} \cup \{-\infty\}$. Suppose that $\sigma = (\sigma_t)_{t \ge s}$ is a continuous $T((\mathbb{R}^d))$-valued semimartingale such that

$$ \int_s^t e^{-2\lambda^v(t-u)} d\langle \sigma^v \rangle_u < \infty. $$

for all words v and t ≥ s. Then, for all $\ell_s \in T((\mathbb{R}^d))$, the process

$$ \ell_t = \mathbf{D}_{t-s}^{\lambda} \ell_s + \int_s^t \mathbf{D}_{t-u}^{\lambda} d\sigma_u. $$

solves the equation

$$ d\ell_t = -\Lambda\ell_t dt + d\sigma_t, \quad t \ge s. $$

*Proof.* The proof follows immediately from Proposition 3.5 (vi). □

**Proposition 3.6** combined with (3.9) yield that the dynamics of the EFM-signature is given by

$$ d\mathbf{X}_{s,t}^{\lambda} = -\Lambda\mathbf{X}_{s,t}^{\lambda} dt + \mathbf{X}_{s,t}^{\lambda} \otimes \circ dX_t, \quad t \ge s. \qquad (3.10) $$

The second term in (3.10) corresponds to the standard signature dynamics equation, while the first one is the mean-reversion term appearing due to the exponential weights in (3.3).

*Remark 3.6.1.* This and all the following $T((\mathbb{R}^d))$-valued equations are understood element-wise. This means that for any truncation order $N$, the truncated equation reduces to a linear multidimensional Stratonovich SDE with coefficients depending only on the elements of $\mathbf{X}_{s,t}^{\lambda}$ up to level $N$. Hence, the SDE admits a unique strong solution. ■

The EFM-signature allows us to construct a fundamental solution of (3.10).

**Proposition 3.7.** Fix $\mathbf{x} \in T((\mathbb{R}^d))$. The solution to the equation

$$ \begin{aligned} d\mathbf{\Phi}_{t}^{s,x} &= -\Lambda\mathbf{\Phi}_{t}^{s,x} dt + \mathbf{\Phi}_{t}^{s,x} \otimes \circ dX_t, \quad t \ge s, \\ \mathbf{\Phi}_{s}^{s,x} &= \mathbf{x}. \end{aligned} \qquad (3.11) $$

is given by $\mathbf{\Phi}_{t}^{s,x} = (\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \mathbf{X}_{s,t}^{\lambda}$.

*Proof.* Using Proposition 3.5 (vi) and (3.10), we obtain

$$ \begin{aligned} d((\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \mathbf{X}_{s,t}^{\lambda}) &= d(\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \mathbf{X}_{s,t}^{\lambda} + (\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes d\mathbf{X}_{s,t}^{\lambda} \\ &= -(\Lambda\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \mathbf{X}_{s,t}^{\lambda} dt - (\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \Lambda\mathbf{X}_{s,t}^{\lambda} dt + (\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \mathbf{X}_{s,t}^{\lambda} \otimes \circ dX_t \\ &= -\Lambda((\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \mathbf{X}_{s,t}^{\lambda})dt + ((\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \mathbf{X}_{s,t}^{\lambda}) \otimes \circ dX_t, \end{aligned} $$

where we used Proposition 3.5 (viii) for the last equality. This shows that $(\mathbf{D}_{t-s}^{\lambda}\mathbf{x}) \otimes \mathbf{X}_{s,t}^{\lambda}$ satisfies (3.11). Observing that the system (3.11), truncated at any order $N \in \mathbb{N}$, admits a unique strong solution as a linear SDE concludes the proof. □

## 3.4 Well-definedness of the EFM-signature and fading-memory property

The goal of this subsection is to address two following questions. The first one is the conditions under which the EFM-signature (3.3) of a continuous semimartingale is well-defined. The second one is the notion of fading memory, in the spirit of Boyd and Chua (1985). We introduce all the definitions and notations from rough path theory necessary to state the main results. Their proofs will rely on the improper rough convolution, which will be introduced later in Section 7.

**Definition 3.8.** Following Boyd and Chua (1985), we say that the EFM-signature has the fading memory property with respect to a weighted metric $\rho$ on the path space $(\mathcal{X}, \rho)$ if, for all $s \le t$ and $n \in \mathbb{N}_0$, the mapping

$$ X \in \mathcal{X} \mapsto \mathbf{X}_{s,t}^{\lambda,n}, $$

is continuous with respect to $\rho$.
In words, this means that if two paths $X, Y \in \mathcal{X}$ are close in the recent past (proximity in the weighted metric $\rho$), then the EFM-signatures $\mathbb{X}_{s,t}^{\lambda,n}$ and $\mathbb{Y}_{s,t}^{\lambda,n}$ will also be close. Of course, to make this definition precise, one must explicitly specify the path space $\mathcal{X}$ and the distance $\rho$.

We first consider absolutely continuous paths $X = (X_t)_{t \le T}$ and define its weighted variation norm by

$$ \|X\|_{\mathrm{BV},\lambda;[s,t]} := \int_s^t |\mathbf{D}_{t-u}^{\lambda} \dot{X}_u| du, \quad \|X\|_{\mathrm{BV},\lambda} := \|X\|_{\mathrm{BV},\lambda;(-\infty,T]} $$

The proposition below establishes the fading memory property of the EFM-signature. Note that for continuously differentiable $X$, the fading memory property implies that each level of the EFM-signature is a fading memory operator for the signal $u_t = \dot{X}_t$ in the sense of Boyd and Chua (1985). Of course, the signature approach admits much rougher input signals, since we pass from $u$ to its integral $X$. This means that $u = \dot{X}$ may be not defined in terms of Boyd and Chua (1985). One can think about a Brownian motion $X = W$, which is almost surely nowhere differentiable; in this case, $u$ corresponds to the white noise, which is, of course, not a continuous process.

**Proposition 3.9.** For the absolutely continuous path $X = (X_t)_{t \le T}$ such that $\|X\|_{\mathrm{BV},\lambda} < \infty$, the EFM-signature is well-defined, and for all $s \le t \le T$, the following bound holds:

$$ |\mathbb{X}_{s,t}^{\lambda,n}| \le \frac{\|X\|_{\mathrm{BV},\lambda;[s,t]}^n}{n!}. \qquad (3.12) $$

Moreover, the EFM-signature has the fading memory property with respect to the weighted variation distance induced by $\| \cdot \|_{\mathrm{BV},\lambda}$.

*Proof.* Repeating the proof of (Lyons, 2014, Lemma 5.1), for $n \ge 1$, we obtain

$$ \begin{aligned} |\mathbb{X}_{s,t}^{\lambda,n}| &= \left| \int_{s<u_1<\dots<u_n<t} \mathbf{D}_{t-u_1}^{\lambda} \dot{X}_{u_1} \otimes \dots \otimes \mathbf{D}_{t-u_n}^{\lambda} \dot{X}_{u_n} du_1 \dots du_n \right| \\ &\le \int_{s<u_1<\dots<u_n<t} \left| \mathbf{D}_{t-u_1}^{\lambda} \dot{X}_{u_1} \otimes \dots \otimes \mathbf{D}_{t-u_n}^{\lambda} \dot{X}_{u_n} \right| du_1 \dots du_n \\ &= \int_{s<u_1<\dots<u_n<t} \prod_{k=1}^n \left| \mathbf{D}_{t-u_k}^{\lambda} \dot{X}_{u_k} \right| du_1 \dots du_n \\ &= \frac{1}{n!} \left( \int_s^t \left| \mathbf{D}_{t-u}^{\lambda} \dot{X}_u \right| du \right)^n = \frac{\|X\|_{\mathrm{BV},\lambda;[s,t]}^n}{n!}. \end{aligned} $$

To prove the continuity, we proceed by induction in $n$ and we claim that the continuity is uniform in $s, t$ once the absolutely continuous paths $X, Y$ satisfy

$$ \|X\|_{\mathrm{BV},\lambda} < M, \quad \|Y\|_{\mathrm{BV},\lambda} < M. \qquad (3.13) $$

For $n = 0$, $\mathbb{X}_{s,t}^{\lambda,n} = 1$, and the continuity follows trivially. Suppose that we proved the uniform continuity for all $\mathbb{X}_{s,t}^{\lambda,k} = 1$, $k \le n$, and recall that

$$ \mathbb{X}_{s,t}^{\lambda,n+1} = \int_s^t \mathbf{D}_{t-u}^{\lambda} (\mathbb{X}_{s,u}^{\lambda,n} \otimes dX_u). $$

for some constant $M > 0$. Then, the difference between two signatures reads

$$ \mathbb{X}_{s,t}^{\lambda,n+1} - \mathbb{Y}_{s,t}^{\lambda,n+1} = \int_s^t \mathbf{D}_{t-u}^{\lambda} ((\mathbb{X}_{s,u}^{\lambda,n} - \mathbb{Y}_{s,u}^{\lambda,n}) \otimes dX_u) + \int_s^t \mathbf{D}_{t-u}^{\lambda} (\mathbb{Y}_{s,u}^{\lambda,n} \otimes d(X_u - Y_u)). $$

Since $\|\mathbf{D}_{T-t}^{\lambda} X\|_{\mathrm{BV},\lambda;[s,t]} \le \|X\|_{\mathrm{BV},\lambda}$, we have $\|X\|_{\mathrm{BV},\lambda;[s,t]} \lesssim_{\lambda,t,T} \|X\|_{\mathrm{BV},\lambda}$ and

$$ |\mathbb{X}_{s,t}^{\lambda,n+1} - \mathbb{Y}_{s,t}^{\lambda,n+1}| \lesssim_{\lambda,t,T} \|X\|_{\mathrm{BV},\lambda} \sup_{s \le u \le T} |\mathbb{X}_{s,u}^{\lambda,n} - \mathbb{Y}_{s,u}^{\lambda,n}| + \|X - Y\|_{\mathrm{BV},\lambda} \sup_{s \le u \le T} |\mathbb{Y}_{s,u}^{\lambda,n}|. $$

We note that $\|X\|_{\mathrm{BV},\lambda}$ and $\sup_{s \le u \le T} |\mathbb{Y}_{s,u}^{\lambda,n}|$ are bounded thanks to (3.13) and (3.12), so that $|\mathbb{X}_{s,t}^{\lambda,n+1} - \mathbb{Y}_{s,t}^{\lambda,n+1}| \to 0$ uniformly in $s, t$ as $\|X - Y\|_{\mathrm{BV},\lambda} \to 0$ by the induction hypothesis. This finishes the proof. $\square$
We will show that the idea of fading memory can be generalized to encompass irregular drivers X thanks to rough path theory, namely, through the notion of the exponentially weighted rough path that we introduce below. To proceed, we need to introduce some notations related to Hölder spaces and rough path spaces.

**Hölder spaces.** In this subsection, as well as in Section 7, all Hölder functions $f$ take values in $(\mathbb{R}^d)^{\otimes n}$ for some $n \in \mathbb{N}_0$, where $d$ denotes the dimension of the driving path. This guarantees that $\mathbf{D}^\lambda f$ is well-defined as a restriction of the operator $\mathbf{D}^\lambda$ in Definition 3.4. For continuous functions $A = (A_{u,v})_{s \le u \le v \le t}$, we define the Hölder norm

$$ \|A\|_{\alpha;[s,t]} := \sup_{\substack{u,v \in [s,t] \\ 0 < |u-v| < 1}} \frac{|A_{u,v}|}{|u-v|^\alpha}. \qquad (3.14) $$

We adopt the standard notations

$$ \delta f_{s,t} := f_t - f_s, \quad \hat{\delta} f_{s,t} := f_t - \mathbf{D}_{t-s}^{\lambda} f_s, \quad s \le t, $$

where the operator $\mathbf{D}_{t-s}^{\lambda}$ is defined by (3.8).

The constraint $0 < |u - v| < 1$ in (3.14) is added to simplify the further analysis of Hölder functions on infinite time intervals. Notice that this additional constraint does not affect any rough path result since Hölder regularity is always considered as a local property.

Let $C^\alpha([s, t])$ denote the space of $\alpha$-Hölder paths on $[s, t]$ with finite seminorm

$$ \|f\|_{\alpha;[s,t]} := \|\hat{\delta} f\|_{\alpha;[s,t]}. $$

Similarly, $\hat{C}^\alpha([s, t])$ denotes the space of paths on $[s, t]$ with finite seminorm

$$ \|f\|_{\alpha;[s,t]}^{\wedge} := \|\hat{\delta} f\|_{\alpha;[s,t]}. $$

For functions $A = (A_{u,v})_{u \le v \le T}$ defined on infinite time horizons $(-\infty, T]$, we propose to use the exponentially weighted Hölder seminorms with coefficients $\rho \in \mathbb{R}^d$, $\rho > 0$:

$$ \|A\|_{\alpha, \rho} := \sup_{\substack{u,v \in (-\infty, T] \\ 0 < |u-v| < 1}} \frac{|\mathbf{D}_{T-u}^{\rho} A_{u,v}|}{|u-v|^{\alpha}}, $$

and we denote by $C^{\alpha, \rho}((-\infty, T])$ the space of functions $f$ such that $\|f\|_{\alpha, \rho} := \|\hat{\delta} f\|_{\alpha, \rho} < \infty$. Similarly, we define $\|f\|_{\alpha, \rho}^{\wedge} := \|\hat{\delta} f\|_{\alpha, \rho}$ and denote the corresponding Hölder space by $\hat{C}^{\alpha, \rho}((-\infty, T])$.

**Remark 3.9.1.** It is straightforward that the finiteness of $\|f\|_{\alpha, \rho}$ (respectively $\|f\|_{\alpha, \rho}^{\wedge}$) implies the finiteness of $\|f\|_{\alpha;[s,t]}$ (resp. $\|f\|_{\alpha;[s,t]}^{\wedge}$) for all $s < t \le T$. Moreover, if $0 < \rho < \rho'$, then $\|f\|_{\alpha, \rho'} \le \|f\|_{\alpha, \rho}$ (resp. $\|f\|_{\alpha, \rho'}^{\wedge} \le \|f\|_{\alpha, \rho}^{\wedge}$) and $C^{\alpha, \rho}((-\infty, T]) \subset C^{\alpha, \rho'}((-\infty, T])$ (resp. $\hat{C}^{\alpha, \rho}((-\infty, T]) \subset \hat{C}^{\alpha, \rho'}((-\infty, T]))$. ■

The following lemma clarifies the relationship between the weighted and unweighted norms.

**Lemma 3.10.** For $f \in C^{\alpha, \rho}((-\infty, T])$,

$$ \|f\|_{\alpha, \rho} = \sup_{R \le T} (\|\mathbf{D}_{T-R}^{\rho} f\|_{\alpha;[R,T]}). $$

*Proof.* Denote $C := \sup_{R \le T} (\|\mathbf{D}_{T-R}^{\rho}(\delta f)\|_{\alpha;[R,T]})$. For all $s, t \in [R, T]$ such that $0 < |t - s| < 1$, we have

$$ |\mathbf{D}_{T-R}^{\rho}(\delta f_{s,t})| \le |\mathbf{D}_{T-s}^{\rho}(\delta f_{s,t})| \le \|f\|_{\alpha, \rho}|t - s|^{\alpha}, $$

so that $\|\mathbf{D}_{T-R}^{\rho} f\|_{\alpha;[R,T]} \le \|f\|_{\alpha, \rho}$, and hence $C \le \|f\|_{\alpha, \rho}$.

Conversely, for all $R \le T$ and $t \in [R, T]$ such that $0 < |t - R| < 1$,

$$ |\mathbf{D}_{T-R}^{\rho}(\hat{\delta} f_{R,t})| \le \|\mathbf{D}_{T-R}^{\rho} f\|_{\alpha;[R,T]} |t - R|^{\alpha} \le C |t - R|^{\alpha} $$

Dividing by $|t - R|^\alpha$ and taking the supremum over $R$ and $t$ yields $\|f\|_{\alpha, \rho} \le C$. □

<figure><img src="image_3.png" /></figure>
It is clear that the same result holds for the “hatted” norms $\| \cdot \|_{\alpha, \rho}^{\wedge}$.

However, the finiteness of $\|f\|_{\alpha, \rho}^{\wedge}$ appears to be insufficient to properly define the EFM-signature. One should also require boundedness with exponential weight, i.e., the finiteness of the norm

$$ \|f\|_{\infty, \rho} := \sup_{t \in (-\infty, T]} |\mathbf{D}_{T-t}^{\rho} f_t|. $$

We denote the space of weighted bounded Hölder functions by

$$ \hat{C}_b^{\alpha, \rho} ((-\infty, T]) := \{ f \in \hat{C}^{\alpha, \rho} ((-\infty, T]) : \|f\|_{\alpha, \rho}^{\wedge} + \|f\|_{\infty, \rho} < \infty \}. \quad (3.15) $$

**Rough path spaces.** We recall the definition of the standard $\alpha$-Hölder rough path for $\alpha \in (\frac{1}{3}, \frac{1}{2}]$ over a finite interval. A more comprehensive introduction to rough path theory can be found in Friz and Hairer (2020).

**Definition 3.11.** For a finite time interval $[R, T]$ and $\alpha \in (\frac{1}{3}, \frac{1}{2}]$, the space $\mathcal{C}^{\alpha}([R, T], \mathbb{R}^d)$ of $\alpha$-Hölder rough paths over $\mathbb{R}^d$ consists of pairs $\mathbf{X} = (\mathbb{X}^1, \mathbb{X}^2)$ satisfying

$$ \|\mathbf{X}^1\|_{\alpha; [R, T]} < \infty, \quad \|\mathbf{X}^2\|_{2\alpha; [R, T]} < \infty, $$

as well as the Chen's identity

$$ \mathbf{X}_{s,t}^2 = \mathbf{X}_{s,u}^2 + \mathbf{X}_{u,t}^2 + \mathbf{X}_{s,u}^1 \otimes \mathbf{X}_{u,t}^1, \quad R \le s \le u \le t \le T \quad (3.16) $$

We denote

$$ \|\mathbf{X}\|_{\alpha; [R, T]} := \|\mathbf{X}^1\|_{\alpha; [R, T]} + \|\mathbf{X}^2\|_{2\alpha; [R, T]}. $$

The natural extension of this definition to paths defined on infinite time intervals $(-\infty, T]$ consists in replacing the Hölder space and norm conditions by their weighted counterparts.

**Definition 3.12** (Exponentially weighted rough path). For $\alpha \in (\frac{1}{3}, \frac{1}{2}]$ and $\rho \in \mathbb{R}^d$ with $\rho > 0$, the space $\mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d)$ of $(\alpha, \rho)$-Hölder rough paths over $\mathbb{R}^d$ consists of pairs $\mathbf{X} = (\mathbb{X}^1, \mathbb{X}^2)$ satisfying

$$ \|\mathbf{X}^1\|_{\alpha, \rho} < \infty, \quad \|\mathbf{X}^2\|_{2\alpha, \rho} < \infty, $$

and Chen's identity (3.16) for all $s \le u \le t \le T$. We also define the norm

$$ \|\mathbf{X}\|_{\alpha, \rho} := \|\mathbf{X}^1\|_{\alpha, \rho} + \|\mathbf{X}^2\|_{2\alpha, \rho}. $$

It follows from Remark 3.9.1 that the restriction of a rough path $\mathbf{X} \in \mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d)$ to $[R, T]$ lies in $\mathcal{C}^{\alpha}([R, T], \mathbb{R}^d)$ for all $R < T$. Moreover, for all $\rho'$ such that $0 < \rho < \rho'$, we have the inclusion

$$ \mathcal{C}^{\alpha, \rho} ((-\infty, T], \mathbb{R}^d) \subset \mathcal{C}^{\alpha, \rho'} ((-\infty, T], \mathbb{R}^d). $$

For $\mathbf{X}, \mathbf{Y} \in \mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d)$, we define the exponentially weighted rough path distance by

$$ \varrho_{\alpha, \rho}(\mathbf{X}, \mathbf{Y}) := \|\mathbf{X}^1 - \mathbf{Y}^1\|_{\alpha, \rho} + \|\mathbf{X}^2 - \mathbf{Y}^2\|_{2\alpha, \rho}, \quad (3.17) $$

which induces the exponentially weighted rough path topology on $\mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d)$.

Our main result concerning the existence of the EFM-signature states that the only condition for the EFM-signature with parameters $\lambda$ to be well-defined as an element of $\hat{C}_b^{\alpha, \rho}((-\infty, T])$ is that the rough path lift $\mathbf{X}$ belongs to $\mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d)$ for some $\rho$ such that $0 < \rho < \lambda$.

**Theorem 3.13.** Fix $T \in \mathbb{R}$. If $\mathbf{X} \in \mathcal{C}^{\alpha, \rho}((-\infty, T])$ for some $\alpha \in (\frac{1}{3}, \frac{1}{2}]$ and $0 < \rho < \lambda$, then the EFM-signature $\mathbf{X}^{\lambda}$ given by (3.3) is well-defined. Moreover, for all $s \in [-\infty, T]$ and all $n \ge 0$, we have

$$ \mathbf{X}_{s, \cdot}^{\lambda, n} \in \hat{C}_b^{\alpha, \rho}((s, T]). $$
*Proof.* For the proof, we refer to Section 7. The statement follows from Corollary 7.3 for finite $s$ and from Corollary 7.6 for $s = -\infty$. □

*Remark 3.13.1.* For standard rough path theory on finite intervals, the rough integral of a semimartingale $X$ coincides with either Itô integration or Stratonovich integration, depending on the choice of the lift:

$$X_{s,t}^{2, \text{Itô}} = \int_s^t X_u \otimes dX_u, \quad X_{s,t}^{2, \text{Strato}} = \int_s^t X_u \otimes dX_u.$$

This result can be readily extended to our setting on infinite intervals by taking the limit as in (7.3). In particular, we can define the EFM-signature of a semimartingale in terms of either the Itô or Stratonovich integral. Throughout this paper, we will adopt the Stratonovich lift for continuous semimartingales. ■

The fading memory property can now be understood as the continuity of the signature map with respect to the exponentially weighted rough path distance (3.17), providing a rough path generalization of the results of Boyd and Chua (1985) and Proposition 3.9.

**Theorem 3.14** (Fading memory property). Let $\mathbf{X}, \mathbf{Y} \in \mathcal{C}^{\alpha, \rho}((-\infty, T])$ for some $\alpha \in (\frac{1}{3}, \frac{1}{2}]$ and $0 < \rho < \lambda$. Then, the EFM-signature $\mathbf{X}^{\lambda}$ has the fading memory property w.r.t. the exponentially weighted rough path distance $\varrho_{\alpha, \rho}$ given by (3.17), that is, for all $-\infty \le s \le t \le T$ and $n \ge 0$,

$$|\mathbf{X}_{s,t}^{\lambda, n} - \mathbf{Y}_{s,t}^{\lambda, n}| \to 0 \quad \text{as} \quad \varrho_{\alpha, \rho}(\mathbf{X}, \mathbf{Y}) \to 0.$$

*Proof.* The proof is given in Subsection 7.2. □

## 3.5 EFM-signature as a stationary process

As a time-invariant transformation, EFM-signature is expected to produce a stationary output in response to the stationary input. Indeed, the EFM-signature $\mathbf{X}^{\lambda} = (\mathbf{X}_{t}^{\lambda})_{t \in \mathbb{R}}$ can be seen as a $T((\mathbb{R}^d))$-valued stochastic process, which exists by Kolmogorov's extension theorem. In this subsection, we will show that if $X$ is a process with stationary increments, then its EFM-signature is a stationary process, i.e., for all $t_1 < \dots < t_n$ and for all $h \in \mathbb{R}$, we have

$$\mathcal{L}(\mathbf{X}_{t_1}^{\lambda}, \dots, \mathbf{X}_{t_n}^{\lambda}) = \mathcal{L}(\mathbf{X}_{t_1+h}, \dots, \mathbf{X}_{t_n+h}).$$

The proof relies on the following technical lemma.

**Lemma 3.15.** Let $\eta_t$ and $\sigma_t$ be stationary càdlàg adapted processes (possibly multivariate), such that, for all $h \in \mathbb{R}$,

$$\mathcal{L}((\eta_t, \sigma_t, X_t)_{t \in \mathbb{R}}) = \mathcal{L}((\eta_{t+h}, \sigma_{t+h}, X_t^h)_{t \in \mathbb{R}}),$$

where $X_t^h := X_{t+h} - X_h$. Then, for all $\mu > 0$ and $i \in \{1, \dots, d\}$, $Y_t = \int_{-\infty}^t e^{-\mu(t-s)} \sigma_s \circ dX_s^i$ is stationary and

$$\mathcal{L}((\eta_t, Y_t, X_t)_{t \in \mathbb{R}}) = \mathcal{L}((\eta_{t+h}, Y_{t+h}, X_t^h)_{t \in \mathbb{R}}). \quad (3.18)$$

*Proof.* The proof of this statement follows the lines of the proof of (Sauri, 2017, Proposition 1), with the following straightforward modifications: considering a semimartingale with stationary increments instead of Brownian motion, using Stratonovich integration instead of Itô integration, and adding the equality (3.18), which also follows immediately from the referenced proof. □

**Theorem 3.16.** Suppose that $X = (X_t)$ is an $\mathbb{R}^d$-valued continuous semimartingale with stationary increments. Then, $\mathbf{X}^{\lambda} = (\mathbf{X}_t^{\lambda})_{t \in \mathbb{R}}$ is a stationary $T((\mathbb{R}^d))$-valued process. In particular, for all $t \in \mathbb{R}$,

$$\mathcal{L}(\mathbf{X}_t^{\lambda}) = \mathcal{L}(\mathbf{X}_0^{\lambda}).$$

*Proof.* We show the stationarity of the truncated EFM-signature $\mathbf{X}^{\lambda, \le N}$ for all $N$ by induction on words $v$, ordered first by length and then lexicographically (this order is denoted by $<$). For $v = \emptyset$, the EFM-signature is a constant process equal to 1.

For the induction step, consider a word $v = v'i$ and suppose that the process $(\mathbf{X}^{\lambda, w})_{w<\nu}$ is stationary, and that

$$\mathcal{L}((\mathbf{X}_t^{\lambda, w})_{w<\nu}, X_t)_{t \in \mathbb{R}} = \mathcal{L}((\mathbf{X}_{t+h}^{\lambda, w})_{w<\nu}, X_t^h)_{t \in \mathbb{R}}.$$
Recall that $\mathbb{X}^{\lambda, \nu}$ is given by (3.4). It follows from Lemma 3.15, with $\eta_t = (\mathbb{X}_t^{\lambda, w})_{w \le v}$, $\sigma_t = \mathbb{X}_t^{\lambda, v'}$, and $\mu = \lambda^\nu$, that $\mathbb{X}_t^{\lambda, \nu}$ is stationary, and for all $h \in \mathbb{R}$, we have

$$ \mathcal{L}((\mathbb{X}_t^{\lambda, w})_{w \le v}, X_t)_{t \in \mathbb{R}} = \mathcal{L}((\mathbb{X}_{t+h}^{\lambda, w})_{w \le v}, X_t^h)_{t \in \mathbb{R}}. $$

This proves the stationarity of the finite-dimensional processes $\mathbb{X}^{\lambda, \le N}$ for all $N \in \mathbb{N}$, and hence implies the stationarity of $\mathbb{X}^{\lambda}$. $\square$

Another advantage of increments stationarity is that it allows one to verify the conditions of Theorem 3.13, showing that $\mathbf{X} \in \mathcal{C}^{\alpha, \rho}((-\infty, T])$.

**Proposition 3.17.** Let $X = (X_t)_{t \le T}$ be a continuous $\mathbb{R}^d$-valued semimartingale with stationary increments, and suppose that for some interval $[t_0, t_0 + h]$, the Hölder constants satisfy

$$ \mathbb{E} [\|\mathbb{X}^1\|_{\alpha;[t_0, t_0+h]}] + \mathbb{E} [\|\mathbb{X}^2\|_{2\alpha;[t_0, t_0+h]}] < \infty. \quad (3.19) $$

Then, $\mathbf{X} \in \mathcal{C}^{\alpha, \rho}((-\infty, T])$ almost surely.

*Proof.* First, by the stationarity of increments, the law of $\|\mathbb{X}^1\|_{\alpha;[t_0, t_0+h]}$ and $\|\mathbb{X}^2\|_{2\alpha;[t_0, t_0+h]}$ does not depend on $t_0$. For $R = T - nh$, $n \in \mathbb{N}$, using the subadditivity of the Hölder seminorms, we have

$$ \begin{aligned} \|\mathbf{D}_{T-R}^{\rho}\mathbf{X}\|_{\alpha;[R,T]} &\le \sum_{k=1}^{n} \|\mathbf{D}_{T-R}^{\rho}\mathbf{X}\|_{\alpha;[T-kh,T-(k-1)h]} \\ &\le \sum_{k=1}^{+\infty} \|\mathbf{D}_{kh}^{\rho}\mathbf{X}\|_{\alpha;[T-kh,T-(k-1)h]} \end{aligned} $$

The case of arbitrary $R < T$ follows similarly. Taking the supremum over $R$ and applying Lemma 3.10 yields

$$ \|\mathbf{X}\|_{\alpha, \rho} \le \sum_{k=1}^{+\infty} \|\mathbf{D}_{kh}^{\rho} \mathbf{X}\|_{\alpha;[T-kh, T-(k-1)h]} $$

Taking expectations and using the stationarity to observe that $\mathbb{E}[\|\mathbf{X}\|_{\alpha;[T-kh, T-(k-1)h]}]$ is finite and independent of $k$, we conclude that

$$ \mathbb{E}[\|\mathbf{X}\|_{\alpha, \rho}] < \infty, $$

hence $\|\mathbf{X}\|_{\alpha, \rho}$ is almost surely finite. $\square$

*Remark 3.17.1.* The condition (3.19) can be verified using the Kolmogorov continuity theorem for rough paths (see, e.g., (Friz and Hairer, 2020, Theorem 3.1)). Proposition 3.17, in particular, establishes the well-definedness of the EFM-signature of $d$-dimensional (possibly time-augmented) Brownian motion $\mathbb{W}^{\lambda}$ ($\tilde{\mathbb{W}}^{\lambda}$ respectively). $\blacksquare$

# 4 Properties of the EFM-signature

In this section, we establish results on the algebraic and analytic properties of EFM-signatures, which closely resemble those of standard signatures.

## 4.1 Chen's identity

Chen's identity for signatures (see Chen (1958)) states that for any $0 \le s \le u \le t \le T$,

$$ \mathbb{X}_{s,t} = \mathbb{X}_{s,u} \otimes \mathbb{X}_{u,t}. $$

In other words, to compute the signature of a concatenated path $X * Y$, one can compute independently the signatures of $X$ and $Y$, and then take their tensor product.

When dealing with EFM-signatures, the algebraic structure of iterated integrals remains unchanged, except that each component $\mathbb{X}_{s,u}^{\lambda, \nu}$ of the EFM-signature over $[s, u]$ must be multiplied by $e^{-\lambda^\nu(t-u)}$ to compensate for
the difference in exponential weights in (3.3). This means that the EFM-signature $X_{s,u}^{\lambda}$ must be appropriately discounted using the operator $D_{t-u}^{\lambda}$ before applying the tensor product. In other words, the contribution of past segments to the EFM-signature decays exponentially at rate $\lambda^i$ for the $i$-th component of the path. More precisely, the following result holds.

**Proposition 4.1.** Fix $\lambda \in \mathbb{R}^d$, $\lambda > 0$ and let $X = (X_t)_{t \in \mathbb{R}}$ be continuous $\mathbb{R}^d$-valued semimartingale such that $X^{\lambda}$ is well-defined. Then,

$$X_{s,t}^{\lambda} = (D_{t-u}^{\lambda} X_{s,u}) \otimes X_{u,t}^{\lambda} \quad (4.1)$$

for all $s \le u \le t$.

*Proof.* The proof follows from Proposition 3.7 together with the EFM-signature dynamics (3.10). $\square$

## 4.2 EFM-signature of linear path

Chen's identity is a crucial property for the computation of signatures. Indeed, a standard approach to computing the signature of a discrete path is to approximate it by a piecewise linear path, compute the signature of each linear segment—which is known explicitly—and then recover the signature of the entire path by performing the tensor multiplication of the segment signatures.

We begin by noticing that when all the components of $\lambda$ are equal, the EFM-signature of a linear path can be computed in closed form as an exponential, similarly to the standard signature. For $\ell \in T((\mathbb{R}^d))$, we define the tensor product exponential as the series

$$e^{\otimes \ell} := \sum_{k \ge 0} \frac{\ell^{\otimes k}}{k!}.$$

**Lemma 4.2.** Suppose that $\lambda = (\lambda, \dots, \lambda)$ for some $\lambda > 0$. For a linear path $X_t = tx$ with $x \in \mathbb{R}^d$, its EFM-signature $X_{s,t}^{\lambda} = (X_{s,t}^{\lambda,n})_{n \ge 0}$ is given explicitly by

$$X_{s,t}^{\lambda} = e^{\otimes \left( \frac{1-e^{-\lambda(t-s)}}{\lambda(t-s)} (X_t - X_s) \right)}. \quad (4.2)$$

In particular, the EFM-signature of a linear path over $(-\infty, t]$ is constant and equals

$$X_t^{\lambda} = \exp \left( \otimes \frac{x}{\lambda} \right). \quad (4.3)$$

Moreover, the EFM-signature of a piecewise-linear path with breakpoints $t_0 < t_1 < \dots < t_n$ is given recursively by

$$X_{t_0, t_n}^{\lambda} = (D_{t_n - t_{n-1}}^{\lambda} X_{t_0, t_{n-1}}^{\lambda}) \otimes e^{\otimes \left( \frac{1 - e^{-\lambda(t_n - t_{n-1})}}{\lambda(t_n - t_{n-1})} (X_{t_n} - X_{t_{n-1}}) \right)}. \quad (4.4)$$

*Proof.* A straightforward computation shows that

$$X_{s,t}^{\lambda,n} = \frac{1}{n!} \left( \frac{1 - e^{-\lambda(t-s)}}{\lambda(t-s)} (X_t - X_s) \right)^{\otimes n}, \quad n \ge 0,$$

so that (4.2) holds. (4.3) follows as a limit as $s \to -\infty$. The formula (4.4) follows from Chen's identity (4.1). $\square$

The case where $\lambda$ has different components is more delicate, as the computation involves integrals of the form

$$\int_{s<u_1<\dots<u_n<t} e^{-\lambda^{i_1}(t-u_1)} \dots e^{-\lambda^{i_n}(t-u_n)} du_1 \dots du_n, \quad (4.5)$$

which are no longer symmetric, making it impossible to use formulas like (4.2). Fortunately, these integrals can still be computed explicitly via the following lemma.
**Lemma 4.3.** Fix $\lambda = (\lambda^1, \dots, \lambda^d) \in \mathbb{R}^d$, $\lambda > 0$. For each word $v = i_1 \dots i_n$, the components of the EFM-signature $X^\lambda$ of the linear path $X_t = tx$ are given by

$$X_{s,t}^{\lambda, v} = (x^{i_1} \cdots x^{i_n}) \sum_{k=0}^{n} c_k e^{-\mu_k (t-s)}, \qquad X_t^{\lambda, v} = (x^{i_1} \cdots x^{i_n}) \prod_{k=1}^{n} \frac{1}{\mu_k} \qquad (4.6)$$

where $c_k = \prod_{l \neq k} \frac{1}{\mu_l - \mu_k}$ and $\mu_k = \sum_{j=1}^k \lambda^{i_j}$, $k = 0, \dots, n$, with $\mu_0 = 0$.

*Proof.* The proof is given in Appendix A.

<figure><img src="image_4.png" /></figure>

We can now state the result for general $\lambda$. The EFM-signature of a linear path can be also written in exponential form, as in (4.2), using the notion of Magnus exponential.

We rely on the classical result due to Magnus (1954), adapted to the right multiplication case. It states that the solution to the linear system on $T^N(\mathbb{R}^d)$

$$\begin{aligned} \dot{x}_t^N &= x_t^N \otimes A(t), \quad t \ge s, \\ x_s^N &= \emptyset, \end{aligned}$$

is given by

$$x_t^N = \exp \left( \otimes \sum_{k \ge 1} \Omega_k(s,t) \right),$$

where

$$\begin{aligned} \Omega_1(s,t) &= \int_s^t A(u_1) du_1, \\ \Omega_2(s,t) &= \frac{1}{2} \int_s^t du_1 \int_s^{u_1} du_2 [A(u_1), A(u_2)], \\ \Omega_3(s,t) &= \frac{1}{6} \int_s^t du_1 \int_s^{u_1} du_2 \int_s^{u_2} du_3 ( [A(u_1), [A(u_2), A(u_3)]] + [[A(u_1), A(u_2)], A(u_3)]), \quad (4.7) \\ \Omega_n(s,t) &= \sum_{j=1}^{n-1} \frac{B_j}{j!} \sum_{\substack{k_1+\dots+k_j=n-1 \\ k_1 \ge 1, \dots, k_j \ge 1}} \int_s^t \mathrm{ad}_{\Omega_{k_1}(s,u)} \dots \mathrm{ad}_{\Omega_{k_j}(s,u)} A(u) du, \end{aligned}$$

the bracket $[\cdot, \cdot]$ is the inverse commutator $[p, q] := q \otimes p - p \otimes q$, $\mathrm{ad}_B A := [B, A]$, and $B_j$ stands for the Bernoulli numbers defined through the generating function

$$\frac{x}{e^x - 1} = \sum_{n \ge 0} \frac{B_n}{n!} x^n.$$

The following result requires the inversion of the operator $\Lambda$ introduced in Definition 3.3. Although $\Lambda$ is diagonal, it is not injective, since $\Lambda\emptyset = 0$. For this reason, we will use its generalized inverse $\Lambda^\dagger$, defined by

$$\Lambda^\dagger: \ell = \sum_{n \ge 0} \sum_{|v|=n} \ell^v \mapsto \Lambda^\dagger \ell = \sum_{n \ge 1} \sum_{|v|=n} \frac{1}{\lambda^v} \ell^v.$$

**Theorem 4.4.** Fix $\lambda = (\lambda^1, \dots, \lambda^d) \in \mathbb{R}^d$, $\lambda > 0$. For a linear path $X_t = tx$ with $x \in \mathbb{R}^d$, its EFM-signature $X_{s,t}^\lambda$ is given by

$$X_{s,t}^{\lambda} = \exp \left( \otimes \sum_{k \ge 1} \Omega_k^{\lambda, x}(s,t) \right), \quad X_t^{\lambda} = \exp \left( \otimes \sum_{k \ge 1} \Omega_k^{\lambda, x}(-\infty,t) \right) = \sum_{k \ge 0} \mathbf{H}_x^k \emptyset, \quad (4.8)$$
where $\Omega_k^{\lambda, x}(s,t)$ is given by (4.7) with $\mathbf{A}(u) = \mathbf{D}_{t-u}^{\lambda}\mathbf{x}$ and $\mathbf{H}_{\mathbf{x}}: \ell \mapsto \Lambda^{\dagger}(\ell \otimes \mathbf{x})$. The coefficients of the EFM-signature are given explicitly by (4.6). Moreover, if $\lambda^1 = \dots = \lambda^d =: \lambda$, then

$$ \mathbf{X}_{s,t}^{\lambda} = e^{\otimes \left( \frac{1-e^{-\lambda(t-s)}}{\lambda} \right)} \mathbf{x}, \quad \mathbf{X}_{t}^{\lambda} = \exp \left( \otimes \frac{\mathbf{x}}{\lambda} \right). \qquad (4.9) $$

*Proof.* In the case of the linear path, the EFM-signature equation (3.9) reads

$$ \mathbf{X}_{s,t}^{\lambda} = \int_{s}^{t} \mathbf{D}_{t-u}^{\lambda} (\mathbf{X}_{s,u}^{\lambda} \otimes \mathbf{x}) du, \quad t \geq s. \qquad (4.10) $$

This implies that $y_t := \mathbf{D}_{s-t}^{\lambda} \mathbf{X}_{s,t}^{\lambda}$ satisfies the linear equation

$$ \begin{gathered} dy_t = y_t \otimes (\mathbf{D}_{s-t}^{\lambda} \mathbf{x}) dt, \quad t \geq s, \\ y_0 = \emptyset, \end{gathered} $$

so that

$$ \mathbf{X}_{s,t}^{\lambda} = \mathbf{D}_{t-s}^{\lambda} y_t = \mathbf{D}_{t-s}^{\lambda} \exp \left( \otimes \sum_{k \geq 1} \tilde{\Omega}_k^{\lambda, x}(s,t) \right), $$

where $\tilde{\Omega}_k^{\lambda, x}(s,t)$ are given by (4.7) with $\mathbf{A}(u) = \mathbf{D}_{s-u}^{\lambda}\mathbf{x}$. Thanks to Proposition 3.5 (vii), one has

$$ \mathbf{D}_h^{\lambda} e^{\otimes \ell} = e^{\otimes \mathbf{D}_h^{\lambda} \ell}, \quad h \in \mathbb{R}, \quad \ell \in T((\mathbb{R}^d)). $$

Applying again Proposition 3.5 (vii), we obtain that

$$ \mathbf{D}_{t-s}^{\lambda} \tilde{\Omega}_k^{\lambda, x}(s,t) = \Omega_k^{\lambda, x}(s,t), \quad k \geq 1, $$

so that the exponential representation in (4.8) holds.

The second part of (4.8) follows from the application of the Picard iterations to (4.10) and noticing that for all $\ell$ such that $\ell^0 = 0$, we have

$$ \int_{s}^{t} \mathbf{D}_{t-u}^{\lambda} \ell \, du = \mathbf{\Lambda}^{\dagger}(\mathrm{Id} - \mathbf{D}_{t-s}^{\lambda}) \ell, \quad \int_{-\infty}^{t} \mathbf{D}_{t-u}^{\lambda} \ell \, du = \mathbf{\Lambda}^{\dagger} \ell. \qquad (4.11) $$

In the case $\lambda^1 = \dots = \lambda^d = \lambda$, the generator $\mathbf{A}(u)$ becomes commutative, so that $[\mathbf{A}(u_1), \mathbf{A}(u_2)] = 0$ and $\Omega_k^{\lambda, x}(s,t) \equiv 0$ for $k \geq 2$. The formula (4.9) follows immediately from (4.11) with $\ell = \mathbf{x}$. $\square$

**Corollary 4.5.** The EFM-signature of a piecewise-linear path with breakpoints $t_0 < t_1 < \dots < t_n$ is given recursively by

$$ \mathbf{X}_{t_0, t_n}^{\lambda} = (\mathbf{D}_{t_n - t_{n-1}}^{\lambda} \mathbf{X}_{t_0, t_{n-1}}^{\lambda}) \otimes \exp \left( \otimes \sum_{k \ge 1} \Omega_k^{\lambda, \frac{x_{t_n} - x_{t_{n-1}}}{t_n - t_{n-1}}} (t_{n-1}, t_n) \right). \qquad (4.12) $$

**Corollary 4.6.** The EFM-signature $\mathbf{X}_{s,t}^{\lambda, \le N}$, truncated at order $N$, belongs to the standard Lie group $\mathcal{G}^N$ corresponding to the Lie algebra $\mathfrak{g}^N$ generated by $\mathbf{1}, \dots, \mathbf{d}$.

*Proof.* Since $\mathbf{A}(u) = \mathbf{D}_{t-u}^{\lambda}\mathbf{x} = \sum_{i=1}^{d} e^{-\lambda^i(t-u)}x^i \in \mathfrak{g}^N$, the formula (4.8) implies that $\mathbf{X}_{s,t}^{\lambda, \le N} \in \mathcal{G}^N$. It remains to notice that $\mathbf{D}_h^{\lambda}$ preserves the group as a dilation operator multiplying the $i$-th component of the path by $e^{-\lambda^i h}$, so that the EFM-signature of piecewise linear paths also belongs to $\mathcal{G}^N$ thanks to (4.12). Finally, an arbitrary path can be approximated by a piecewise linear one and the statement follows from the closedness of $\mathcal{G}^N$. $\square$

Thus, the EFM-signature process $\mathbf{X}_t^{\lambda, \le N}$, being a Lie group-valued process, can be seen as a mean-reverting analogue of the signature. Moreover, since the signature of $d$-dimensional Brownian motion $\mathbb{W}$ is the Brownian motion on $\mathcal{G}^N$ in the sense of (Baudoin, 2004, Definition 1.2), the EFM-signature $\mathbb{W}^{\lambda}$ plays the role of the Ornstein-Uhlenbeck process on $\mathcal{G}^N$.
## 4.3 Time reparametrization

It is well-known that the signature of path is invariant under time reparametrizations. Namely, for any smooth reparametrization $\psi: [0, T] \to [0, T]$, denote $\tilde{X}_t := X_{\psi_t}$. It follows that

$$ \int_{0<u_1<\dots<u_n<T} \circ dX_{u_1} \otimes \dots \otimes \circ dX_{u_n} = \int_{0<u_1<\dots<u_n<T} \circ d\tilde{X}_{u_1} \otimes \dots \otimes \circ d\tilde{X}_{u_n}. $$

Similarly, for the EFM-signatures,

$$ \begin{aligned} & \int_{s<u_1<\dots<u_n<t} e^{-\lambda(t-u_1)} \odot dX_{u_1} \otimes \dots \otimes \odot e^{-\lambda(t-u_n)} \odot dX_{u_n} \\ & = \int_{s<u_1<\dots<u_n<t} e^{-\lambda(\psi_t-\psi_{u_1})} \odot d\tilde{X}_{u_1} \otimes \dots \otimes \odot e^{-\lambda(\psi_t-\psi_{u_n})} \odot d\tilde{X}_{u_n}. \end{aligned} $$

At first glance, it may seem that the EFM-signature is not invariant under time reparametrization. However, the notion of time-invariance does not exist without a time structure. Therefore, any path $(X_t)_{t \in \mathbb{R}}$ parametrized by $t \in \mathbb{R}$ should be interpreted as a time-evolving process, equipped with a corresponding time process $(\tau_t)_{t \in \mathbb{R}}$. In this setting, it is natural to define the EFM-signature as $\tilde{X}_{s,t} = (\tilde{X}_{s,t}^{\lambda,n})_{n \ge 0}$, where

$$ \tilde{X}_{s,t}^{\lambda,n} := \int_{s<u_1<\dots<u_n<t} e^{-\lambda(\tau_t-\tau_{u_1})} \odot dX_{u_1} \otimes \dots \otimes \odot e^{-\lambda(\tau_t-\tau_{u_n})} \odot dX_{u_n}, $$

which is indeed invariant under the simultaneous reparametrization of both the path $\tilde{X}_t = X_{\psi_t}$ and the time process $\tilde{\tau}_t = \tau_{\psi_t}$, for any smooth reparametrization $\psi: \mathbb{R} \to \mathbb{R}$. To simplify notation, we will always assume that the parametrization of the path is chosen so that $\tau_t = t$ for all $t \in \mathbb{R}$.

## 4.4 Uniqueness of the EFM-signature

The uniqueness of the signature means that the signature determines the path up to tree-like equivalence, as shown by Hambly and Lyons (2010). In particular, the signature of the time-augmented path $\tilde{X}_t = (t, X_t)$, $t \in [0, T]$ determines the path uniquely; see, for instance, (Cuchiero et al., 2023a, Lemma 2.6) for a simple proof in the case of continuous semimartingales. However, this proof is not applicable in the EFM-signature framework, since polynomials of the form $(X_t^i)^n$ cannot be represented as elements of the EFM-signature. Nevertheless, the uniqueness result remains true.

For notational simplicity, when working with time-augmented paths $\tilde{X} = (t, X_t)$, we denote the time component by the letter 0, and the spatial components of X by the letters 1, ..., d.

**Theorem 4.7.** Let $X = (X_t)_{-\infty<t\le T}$ and $Y = (Y_t)_{-\infty<t\le T}$ be two continuous $\mathbb{R}^d$-valued semimartingales. Consider the time-augmented paths $\tilde{X}_t := (t, X_t)$ and $\tilde{Y}_t := (t, Y_t)$ with EFM-signatures $\tilde{X}_t^{\lambda}$ and $\tilde{Y}_t^{\lambda}$. Then $\tilde{X}_T^{\lambda} = \tilde{Y}_T^{\lambda}$ if and only if $X_t = Y_t + c$ for all $t \in (-\infty, T]$ and some real constant $c \in \mathbb{R}$.

*Proof.* It is clear that $X_t = Y_t + c$ for $t \le T$ implies equality of the EFM-signatures. We will prove the converse in two steps.

**Step 1.** We will show that $\tilde{X}_T^{\lambda} = \tilde{Y}_T^{\lambda}$ implies

$$ \int_{-\infty}^{T} e^{-\mu(T-t)} \langle \mathbf{i}, \tilde{X}_t^{\lambda} \rangle dt = \int_{-\infty}^{T} e^{-\mu(T-t)} \langle \mathbf{i}, \tilde{Y}_t^{\lambda} \rangle dt, \quad \mu > \lambda^0 + \lambda^i, \quad i \in \{1, \dots, d\}. \quad (4.13) $$

First, it follows from Lemma C.1 that for $k \ge 1$,

$$ \int_{-\infty}^{T} e^{-(k\lambda^0+\lambda^i)(T-t)} \langle \mathbf{i}, \tilde{X}_t^{\lambda} \rangle dt = \int_{-\infty}^{T} e^{-(k\lambda^0+\lambda^i)(T-t)} \langle \mathbf{i}, \tilde{Y}_t^{\lambda} \rangle dt, \quad i \in \{1, \dots, d\}. $$

Denoting $Z_t^i := \langle \mathbf{i}, \tilde{X}_t^{\lambda} \rangle - \langle \mathbf{i}, \tilde{Y}_t^{\lambda} \rangle$, this can be written as

$$ J_k^i := \int_{-\infty}^{T} e^{-k\lambda^0(T-t)} (Z_t^i e^{-(\lambda^0+\lambda^i)(T-t)}) dt = 0, \quad k \ge 0, $$
and we want to show that

$$J_{\mu}^{i} := \int_{-\infty}^{T} e^{-(\mu-\lambda^{0}-\lambda^{i})(T-t)} (Z_{t}^{i} e^{-(\lambda^{0}+\lambda^{i})(T-t)}) dt = 0, \quad \mu > \lambda^{0} + \lambda^{i}.$$

By Theorem 3.13, the EFM-signature elements belong to $\hat{C}_b^{\alpha, \rho}((-\infty, T])$, so that $(Z_t^i e^{-\lambda^i(T-t)})$ is almost surely bounded and the integral

$$\int_{-\infty}^{T} |Z_{t}^{i}| e^{-(\lambda^{0}+\lambda^{i})(T-t)} dt$$

is almost surely finite.

Note that $e^{-(\mu-\lambda^0-\lambda^i)(T-t)}$ can be approximated arbitrarily well by

$$e^{-(\mu-\lambda^0-\lambda^i)(T-t)} \approx \sum_{k=0}^{N} c_k e^{-k\lambda^0(T-t)}, \quad t \ge 0,$$

uniformly on $\mathbb{R}^+$. Indeed, taking $x = e^{-\lambda^0(T-t)} \in (0,1]$ makes this approximation equivalent to the uniform approximation by a polynomial

$$x^{\beta} \approx \sum_{k=0}^{N} c_k x^k, \quad x \in (0,1],$$

with $\beta = \frac{\mu-\lambda^0-\lambda^i}{\lambda^0} > 0$, which holds since the function $x \mapsto x^\beta$ is continuous on $[0,1]$. This means that for any $\epsilon > 0$, there exist $N$ and coefficients $c_0, c_1, \dots, c_N$ such that

$$|J_{\mu}^{i}| = \left| J_{\mu}^{i} - \sum_{k=0}^{N} c_k J_{k}^{i} \right| \le \epsilon,$$

so that $J_{\mu}^{i} = 0$.

**Step 2.** Differentiating (4.13) with respect to $\mu$, we obtain

$$(-1)^n \int_{-\infty}^{T} e^{-\mu(T-t)} (T-t)^n Z_t^i dt = 0,$$

for all $n \in \mathbb{N}$ and $\mu > \lambda^0 + \lambda^i$. Since the Laguerre polynomials form a basis of $L^2(\mathbb{R}^+, e^{-\mu x} dx)$, we conclude that $Z_t^i \equiv 0$ for $i = 1, \dots, d$, by continuity. As $\langle i, \hat{X}_t^\lambda \rangle$ and $\langle i, \hat{Y}_t^\lambda \rangle$ satisfy the equations

$$d \langle i, \hat{X}_t^\lambda \rangle = -\lambda^i \langle i, \hat{X}_t^\lambda \rangle dt + dX_t^i,$$

$$d \langle i, \hat{Y}_t^\lambda \rangle = -\lambda^i \langle i, \hat{Y}_t^\lambda \rangle dt + dY_t^i,$$

it follows that $d(X_t^i - Y_t^i) \equiv 0$ for $i = 1, \dots, d$, which implies the statement of the theorem. $\square$

The uniqueness result allows us to conclude that any functional of the path $X$ can be expressed as a function of the EFM-signature of this path:

$$F((X_s)_{s \le t}) = \tilde{F}(\hat{X}_t^\lambda),$$

for some function $\tilde{F}$, provided that the EFM-signature $\hat{X}_t^\lambda$ captures all the information contained in the path $(X_s)_{s \le t}$.

## 4.5 Linearization property

Another important operation on $T((\mathbb{R}^d))$ is the shuffle product, which allows for the linearization of products of two linear functionals of the signature; see (Lyons, Caruana, and Lévy, 2007, Theorem 2.15). For basis vectors $v, w \in T((\mathbb{R}^d))$ and letters $i, j$, the shuffle product is defined recursively by

$$(\text{vi}) \sqcup (\text{wj}) = (v \sqcup (\text{wj}))i + ((\text{vi}) \sqcup w)j,$$

$$w \sqcup \emptyset = \emptyset \sqcup w = w,$$
and extended to $T((\mathbb{R}^d))$ by linearity. The following proposition establishes the shuffle property for EFM-signatures.

**Proposition 4.8.** If $\ell_1, \ell_2 \in T(\mathbb{R}^d)$, then $\ell_1 \sqcup \ell_2 \in T(\mathbb{R}^d)$ and

$$ \langle \ell_1, \mathbb{X}_t^\lambda \rangle \langle \ell_2, \mathbb{X}_t^\lambda \rangle = \langle \ell_1 \sqcup \ell_2, \mathbb{X}_t^\lambda \rangle. $$

*Proof.* By linearity, it is enough to prove the proposition for $\ell_1 = \mathbf{u}$ and $\ell_2 = \mathbf{v}$ with $\mathbf{u}, \mathbf{v} \in V$. We proceed by induction on $|\mathbf{u}| + |\mathbf{v}|$. If $\mathbf{u} = \emptyset$ or $\mathbf{v} = \emptyset$, the equality is trivial.

Therefore, we consider $\mathbf{u} = \tilde{\mathbf{u}}\mathbf{i}$ and $\mathbf{v} = \tilde{\mathbf{v}}\mathbf{j}$ for some $i, j \in A_d$. Since

$$ \begin{aligned} d \langle \mathbf{u}, \mathbb{X}_t^\lambda \rangle &= -\lambda^u \langle \mathbf{u}, \mathbb{X}_t^\lambda \rangle dt + \langle \tilde{\mathbf{u}}, \mathbb{X}_t^\lambda \rangle \circ dX_t^i, \\ d \langle \mathbf{v}, \mathbb{X}_t^\lambda \rangle &= -\lambda^v \langle \mathbf{v}, \mathbb{X}_t^\lambda \rangle dt + \langle \tilde{\mathbf{v}}, \mathbb{X}_t^\lambda \rangle \circ dX_t^j, \end{aligned} $$

we have

$$ \begin{aligned} d \langle (\mathbf{u}, \mathbb{X}_t^\lambda) \langle \mathbf{v}, \mathbb{X}_t^\lambda \rangle \rangle &= d \langle \mathbf{u}, \mathbb{X}_t^\lambda \rangle \langle \mathbf{v}, \mathbb{X}_t^\lambda \rangle + \langle \mathbf{u}, \mathbb{X}_t^\lambda \rangle d \langle \mathbf{v}, \mathbb{X}_t^\lambda \rangle \\ &= -(\lambda^u + \lambda^v) \langle \mathbf{u}, \mathbb{X}_t^\lambda \rangle \langle \mathbf{v}, \mathbb{X}_t^\lambda \rangle dt + \langle \mathbf{v}, \mathbb{X}_t^\lambda \rangle \langle \tilde{\mathbf{u}}, \mathbb{X}_t^\lambda \rangle \circ dX_t^i + \langle \mathbf{u}, \mathbb{X}_t^\lambda \rangle \langle \tilde{\mathbf{v}}, \mathbb{X}_t^\lambda \rangle \circ dX_t^j, \end{aligned} $$

so that

$$ \begin{aligned} \langle \mathbf{u}, \mathbb{X}_t^\lambda \rangle \langle \mathbf{v}, \mathbb{X}_t^\lambda \rangle &= \int_{-\infty}^{t} e^{-(\lambda^u + \lambda^v)(t-s)} [\langle \mathbf{v}, \mathbb{X}_s^\lambda \rangle \langle \tilde{\mathbf{u}}, \mathbb{X}_s^\lambda \rangle \circ dX_s^i + \langle \mathbf{u}, \mathbb{X}_s^\lambda \rangle \langle \tilde{\mathbf{v}}, \mathbb{X}_s^\lambda \rangle \circ dX_s^j] \\ &= \int_{-\infty}^{t} e^{-(\lambda^u + \lambda^v)(t-s)} [\langle \mathbf{v} \sqcup \tilde{\mathbf{u}}, \mathbb{X}_s^\lambda \rangle \circ dX_s^i + \langle \mathbf{u} \sqcup \tilde{\mathbf{v}}, \mathbb{X}_s^\lambda \rangle \circ dX_s^j] \\ &= \langle (\mathbf{v} \sqcup \tilde{\mathbf{u}})\mathbf{i} + (\mathbf{u} \sqcup \tilde{\mathbf{v}})\mathbf{j}, \mathbb{X}_s^\lambda \rangle, \end{aligned} $$

where we have used the induction hypothesis and (3.4). $\square$

*Remark 4.8.1.* An alternative proof of the shuffle property consists in noticing that the EFM-signature $\mathbb{X}_T$ can be seen as the standard signature of the weighted process $X^T = (X_t^T)_{t \le T}$ defined by

$$ dX_t^T = \mathbf{D}_{T-t}^\lambda(dX_t), \quad t \le T. $$

We denote this signature by $\mathbb{X}_t^T$ and observe that the shuffle property holds for it. Finally, since

$$ \mathbb{X}_T^\lambda = \mathbb{X}_T^T, $$

the shuffle property holds for the EFM-signature as well. This perspective also provides an alternative proof of Chen's identity (4.1), which follows from the classical Chen's identity for $\mathbb{X}_t^T$ combined with the multiplicative property of $\mathbf{D}_h^\lambda$ given by Proposition 3.5 (ii). $\blacksquare$

## 4.6 Universal approximation theorem

The goal of this section is to show that a sufficiently large class of functions of path $F((X_t)_{t \le T})$ can be approximated by a linear functional of the EFM-signature $\langle \ell, \hat{\mathbb{X}}_T^\lambda \rangle$ of the time-augmented path $\tilde{X}_t = (t, X_t)$. In particular, if the approximation holds for $F$, the time-invariant output signal (1.1) is approximated uniformly in $t \in \mathbb{R}$, i.e.,

$$ Y_t = F((X_s)_{s \le t}) \approx \langle \ell, \hat{\mathbb{X}}_t^\lambda \rangle, \quad t \in \mathbb{R}. $$

As for signatures, this result is based on the Stone-Weierstrass approximation theorem (Rudin, 1976, Theorem 7.32).

**Theorem 4.9 (Stone-Weierstrass).** Let $K \subset \mathcal{X}$ be a compact set and let $\mathcal{A}$ denote the class of functions $f: \mathcal{X} \to \mathbb{R}$ satisfying:

1. Every $f \in \mathcal{A}$ is continuous;
2. the constant function $1 \in \mathcal{A}$;

3. if $f, g \in \mathcal{A}$, then $fg \in \mathcal{A}$;

4. for $x, y \in \mathcal{X}$, such that $x \neq y$, there exists $f \in \mathcal{A}$, such that $f(x) \neq f(y)$.

Then, for any continuous function $F: \mathcal{X} \to \mathbb{R}$ and for any $\epsilon > 0$, there exists $f \in \mathcal{A}$, such that $|F(x)-f(x)| < \epsilon$ for all $x \in K$.

Since for standard signatures on a finite interval $[0, T]$, the natural path space is the space of rough paths $\mathcal{C}^{\alpha}([0, T], \mathbb{R}^d)$, see, for example, Cuchiero et al. (2023a), we will use $\mathcal{X} = \mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d)$ with $0 < \rho < \lambda$, and consider functionals

$$F: \mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d) \to \mathbb{R}, \quad \mathbf{X} \mapsto F(\mathbf{X}), \qquad (4.14)$$

continuous in the exponentially weighted rough path topology. Since $\mathbf{X}$ contains all the increments of $X$, the set of functionals of the form (4.14) contains all dependencies of the form $Y_t = F((X_t - X_s)_{s \le t})$.

**Theorem 4.10.** Consider a compact set $K \subset \mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d)$ and a continuous function $F: \mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d) \to \mathbb{R}$. Then, for any $\epsilon > 0$, there exists $\ell \in T(\mathbb{R}^d)$ such that, for all $\mathbf{X} \in K$,

$$|F(\mathbf{X}) - \langle \ell, \hat{\mathbf{X}}_T^\lambda \rangle| < \epsilon.$$

*Proof.* The proof consists in the application of the Stone–Weierstrass theorem. As the class $\mathcal{A}$, we choose the class of finite linear functionals of the EFM-signature:

$$\mathcal{A} := \{ \mathbf{X} \mapsto \langle \ell, \hat{\mathbf{X}}_T^\lambda \rangle : \ell \in T(\mathbb{R}^d) \}.$$

Each functional in this class is continuous by Theorem 3.14. The second condition of the Stone–Weierstrass theorem is trivially satisfied since $\langle \emptyset, \hat{\mathbf{X}}_T^\lambda \rangle = 1$. The third condition holds due to the shuffle property given by Proposition 4.8. The point-separation condition 4 follows from the uniqueness result, Theorem 4.7. $\square$

This result can be seen as a combination of the classical universal approximation theorem for signatures and Theorem 1 in the paper by Boyd and Chua (1985) considering the approximation of operators with fading memory, which is essentially the continuity in the weighted distance.

*Remark 4.10.1.* One can establish a similar result for regular processes by choosing the space of absolutely continuous paths with finite norm $\|X\|_{BV, \lambda} < \infty$. The continuity of the signature map in this case follows from Proposition 3.9. $\blacksquare$

## 4.7 Moment problem

The universal approximation theorem allows us to establish that a measure defined on a compact set of paths is determined by the expected EFM-signature. Namely, the following proposition holds.

**Proposition 4.11.** Let $\mathbb{P}$ and $\mathbb{Q}$ be probability measures on a compact set $K \subset \mathcal{C}^{\alpha, \rho}((-\infty, T], \mathbb{R}^d)$. If their expected (time-augmented) EFM-signatures are equal, i.e.

$$\mathbb{E}^{\mathbb{P}}[\hat{\mathbf{X}}_T^\lambda] = \mathbb{E}^{\mathbb{Q}}[\hat{\mathbf{X}}_T^\lambda],$$

then $\mathbb{P} = \mathbb{Q}$.

*Proof.* It is sufficient to prove that for any continuous function $F: K \to \mathbb{R}$,

$$\mathbb{E}^{\mathbb{P}}[F(\mathbf{X})] = \mathbb{E}^{\mathbb{Q}}[F(\mathbf{X})]. \qquad (4.15)$$

By Theorem 4.10, for any $\epsilon > 0$, there exists $\ell \in T(\mathbb{R}^d)$ such that

$$|F((X_s)_{s \le T}) - \langle \ell, \hat{\mathbf{X}}_T^\lambda \rangle| < \epsilon.$$
Since $\mathbb{E}^P[\langle \ell, \hat{\mathbf{X}}_T^\lambda \rangle] = \mathbb{E}^Q[\langle \ell, \hat{\mathbf{X}}_T^\lambda \rangle]$, it follows that

$$|\mathbb{E}^P[F(\mathbf{X})] - \mathbb{E}^Q[F(\mathbf{X})]| < 2\epsilon.$$

As $\epsilon > 0$ can be chosen arbitrarily small, the equality (4.15) holds, which completes the proof. $\square$

*Remark 4.11.1.* The compactness assumption above can be weakened in the context of the signature moments problem, as shown by Chevyrev and Oberhauser (2022). We believe that the same ideas and techniques can be extended to the setting of EFM-signatures to obtain stronger results. For instance, we conjecture that the notions of tensor normalization and robust signature moments, in the spirit of Chevyrev and Oberhauser (2022), can be suitably defined for the EFM-signature as well. ■

# 5 EFM-signature of the time-augmented Brownian motion

In this section, we focus on a particularly interesting case where the path is the time-augmented standard Brownian motion $\hat{W}_t = (t, W_t^1, \dots, W_t^d)$. Recall that the EFM-signature of $\hat{W}$ is well-defined, see Remark 3.17.1. While its standard signature $\hat{W}$ inherits several key properties of Brownian motion—namely, $\hat{W}_0 = \emptyset$, and $\hat{W}_{t,t+h}$ is independent of $\mathcal{F}_t$ with a law depending only on $h$—the EFM-signature $\hat{W}^\lambda$ can be viewed as an Ornstein–Uhlenbeck process on the Lie group. Indeed, beyond the mean-reverting dynamics

$$d\hat{W}_t^\lambda = -\Lambda\hat{W}_t^\lambda dt + \hat{W}_t^\lambda \otimes d\hat{W}_t, \quad t \in \mathbb{R},$$

and the stationarity established in Theorem 3.16, we will show that it is an ergodic Markov process. Moreover, we prove that the non-stationary version $\hat{W}_{0,t}^\lambda$ converges exponentially fast to the stationary distribution in the Wasserstein distance. Finally, we compute the expected EFM-signature in both the stationary and non-stationary settings, in analogy with the celebrated result of Fawcett (2003).

## 5.1 Markov property for $\hat{W}^\lambda$

The EFM-signature $\hat{W}^\lambda$ can be considered as an adapted $T((\mathbb{R}^{d+1}))$-valued stochastic process. We equip the state space $T((\mathbb{R}^{d+1}))$ with the norm

$$\|\ell\|_2 := \sqrt{\sum_{n \ge 0} \sum_{|v|=n} |\ell^v|^2},$$

and denote

$$L^2T((\mathbb{R}^{d+1})) = \{\ell \in T((\mathbb{R}^{d+1})) : \|\ell\|_2 < +\infty\}.$$

When equipped with a scalar product

$$\langle \ell, \sigma \rangle_2 := \sum_{n \ge 0} \sum_{|v|=n} \ell^v \sigma^v,$$

it becomes a separable Hilbert space isomorphic to the standard $\ell^2$ space. We denote by $\mathcal{B}(L^2T((\mathbb{R}^{d+1})))$ the Borel sigma-algebra generated by the open sets of $L^2T((\mathbb{R}^{d+1}))$.

The following lemma ensures that $\hat{W}_t^\lambda$ and $\hat{W}_{0,t}^\lambda$ are indeed elements of $L^2T((\mathbb{R}^{d+1}))$ for any $t \in \mathbb{R}$.

**Lemma 5.1.** There exists a constant $M_\lambda > 0$, such that

$$\mathbb{E}[\|\hat{W}_t^\lambda\|_2^2] \le M_\lambda, \quad t \in \mathbb{R} \quad \text{and} \quad \sup_{t \ge 0} \mathbb{E}[\|\hat{W}_{0,t}^\lambda\|_2^2] \le M_\lambda.$$

*Proof.* The proof is given in Appendix B $\square$

Lemma 5.1 states that $\hat{W}^\lambda = (\hat{W}_t^\lambda)_{t \in \mathbb{R}}$ can be considered as a general Markov process with state space $(L^2T((\mathbb{R}^{d+1})), \mathcal{B}(L^2T((\mathbb{R}^{d+1})))$, as shown in the following proposition.
**Theorem 5.2.** The process $\widehat{W}^{\lambda} = (\widehat{W}_{t}^{\lambda})_{t \in \mathbb{R}}$ is a Markov process in the sense that for any measurable bounded function $f: L^{2}T((\mathbb{R}^{d+1})) \to \mathbb{R}$, the Markov property is verified:

$$ \mathbb{E}[f(\widehat{W}_{t}^{\lambda}) | \mathcal{F}_{s}] = \mathbb{E}[f(\widehat{W}_{t}^{\lambda}) | \widehat{W}_{s}^{\lambda}], \quad t \geq s. $$

*Proof.* For a measurable bounded $f$ and $t \geq 0$, we have

$$ \mathbb{E}[f(\widehat{W}_t^\lambda) | \mathcal{F}_s] = \mathbb{E}[f(D_{t-s}^\lambda \widehat{W}_s^\lambda \otimes \widehat{W}_{s,t}^\lambda) | \mathcal{F}_s] = \mathbb{E}[f(D_{t-s}^\lambda \ell \otimes \widehat{W}_{s,t}^\lambda) | \mathcal{F}_s] \Big|_{\ell=\widehat{W}_s^\lambda} = \mathbb{E}[f(D_{t-s}^\lambda \ell \otimes \widehat{W}_{s,t}^\lambda)] \Big|_{\ell=\widehat{W}_s^\lambda}. $$

Here we have used the Chen's identity, the measurability of $\widehat{W}_s^\lambda$ with respect to $\mathcal{F}_s$ and finally, independence of $\widehat{W}_{s,t}$ and $\mathcal{F}_s$. $\square$

## 5.2 Ergodicity of $\widehat{W}^{\lambda}$

As we have already shown in Theorem 3.16 and Theorem 5.2, $\widehat{W}_t^\lambda$ is a stationary Markov process. The process $(\widehat{W}_{0,t}^\lambda)_{t \geq 0}$ corresponds to its nonstationary version, starting from $\widehat{W}_{0,0}^\lambda = \emptyset$ (i.e., we assume that the underlying path is constant on the negative half-line: $\widehat{W}_t \equiv 0$ for $t < 0$). As a one-dimensional analogy, one can think of the stationary Ornstein-Uhlenbeck process $\int_{-\infty}^{t} e^{-\lambda(t-s)} dW_s$ and its nonstationary counterpart $\int_{0}^{t} e^{-\lambda(t-s)} dW_s$.

A natural question arises: How quickly does the law of $\widehat{W}_{0,t}^\lambda$ converge to the stationary distribution $\mathcal{L}(\widehat{W}_t^\lambda)$?

In this section, we will show that, exactly as in the Ornstein-Uhlenbeck case, this convergence is geometric in the Wasserstein distance

$$ W_2(\mu, \nu) = \sqrt{\inf_{\pi \in \Pi(\mu, \nu)} \int \|x-y\|_2^2 \pi(dx, dy)}, \quad \mu, \nu \in \mathcal{P}(L^2T((\mathbb{R}^d))), \quad (5.1) $$

where $\Pi(\mu, \nu)$ stands for a set of all coupling of $\mu$ and $\nu$.

More precisely, we will prove that the dependence on the initial condition decays exponentially with rate $\min_i \lambda^i$. In particular, this will imply the uniqueness of the invariant distribution and the ergodicity of the EFM-signature process.

**Theorem 5.3.** Let $W = (W_t)_{t \in \mathbb{R}}$ denote a Brownian motion and let $x = (x_t)_{t \leq 0}$ be a continuous semimartingale. Consider the path obtained by concatenation of $(x_t)_{t \leq 0}$ with $(W_t)_{t \geq 0}$ at $t = 0$:

$$ X_t = \begin{cases} x_t, & t \leq 0, \\ x_0 + W_t, & t > 0, \end{cases} $$

Assume that $x$ and $(W_t - W_0)_{t \geq 0}$ are independent and that the EFM-signature $\hat{x}_t^\lambda$ of the time-augmented path $\hat{x}_t = (t, x_t)$ is well-defined as a process in $L^2T((\mathbb{R}^{d+1}))$ for $t \leq 0$ with $\mathbb{E}[\|\hat{x}_0^\lambda\|_2^2] < \infty$. Let $\hat{X}_t^\lambda$ denote the EFM-signature of $\hat{X}_t = (t, X_t)$, $t \in \mathbb{R}$ and set $\mu_t = \mathcal{L}(\hat{X}_t^\lambda)$, $t \geq 0$. Then, there exists a unique ergodic measure $\mu_* = \mathcal{L}(\widehat{W}_t^\lambda)$ and a constant $M_\lambda > 0$ such that

$$ W_2^2(\mu_t, \mu_*) \leq 2M_\lambda (\mathbb{E}[\|\hat{x}_0^\lambda\|_2^2] + M_\lambda) e^{-2\lambda^{\min}t}, $$

where $\lambda^{\min} = \min_i \lambda^i$. In particular, taking $x \equiv 0$, one obtains

$$ W_2^2(\mathcal{L}(\widehat{W}_{0,t}^\lambda), \mathcal{L}(\widehat{W}_t^\lambda)) \leq 2M_\lambda^2 e^{-2\lambda^{\min}t}, $$

*Proof.* First, note that $\hat{X}_t^\lambda = \hat{x}_t^\lambda$ for $t \leq 0$. By Chen's identity (4.1), we have, for $t \geq 0$,

$$ \hat{X}_t^\lambda = D_t^\lambda \hat{X}_0^\lambda \otimes \widehat{W}_{0,t}^\lambda, \quad \widehat{W}_t^\lambda = D_t^\lambda \widehat{W}_0^\lambda \otimes \widehat{W}_{0,t}^\lambda, $$
and

$$ \hat{\mathbf{X}}_t^{\lambda} - \hat{\mathbf{W}}_t^{\lambda} = \mathbf{D}_t^{\lambda} (\hat{\mathbf{X}}_0^{\lambda} - \hat{\mathbf{W}}_0^{\lambda}) \otimes \hat{\mathbf{W}}_{0,t}^{\lambda}. $$

We note that the norm $\| \cdot \|_2$ is compatible with the tensor product, i.e., for any $\ell, \mathbf{p} \in T((\mathbb{R}^{d+1}))$ such that $\|\ell\|_2 < +\infty$ and $\|\mathbf{p}\|_2 < +\infty$, we have

$$ \|\ell \otimes \mathbf{p}\|_2 = \|\ell\|_2 \|\mathbf{p}\|_2 < +\infty. $$

Then,

$$ \|\hat{\mathbf{X}}_t^{\lambda} - \hat{\mathbf{W}}_t^{\lambda}\|_2^2 = \|\mathbf{D}_t^{\lambda} (\hat{\mathbf{X}}_0^{\lambda} - \hat{\mathbf{W}}_0^{\lambda})\|_2^2 \|\hat{\mathbf{W}}_{0,t}^{\lambda}\|_2^2. $$

Conditioning with respect to $x$ and then taking the expectation, we obtain

$$ \begin{aligned} \mathbb{E} \left[ \| \hat{\mathbf{X}}_t^{\lambda} - \hat{\mathbf{W}}_t^{\lambda} \|_2^2 \right] &= \mathbb{E} \left[ \left\| \mathbf{D}_t^{\lambda} (\hat{\mathbf{X}}_0^{\lambda} - \hat{\mathbf{W}}_0^{\lambda}) \right\|_2^2 \right] \mathbb{E} \left[ \| \hat{\mathbf{W}}_{0,t}^{\lambda} \|_2^2 \right] \\ &\le e^{-2\lambda^{\min h}} \mathbb{E} \left[ \left\| \hat{\mathbf{X}}_0^{\lambda} - \hat{\mathbf{W}}_0^{\lambda} \right\|_2^2 \right] \mathbb{E} \left[ \| \hat{\mathbf{W}}_{0,t}^{\lambda} \|_2^2 \right] \tag{5.2} \\ &\le 2M_{\lambda} \left( \mathbb{E} \left[ \left\| \hat{\mathbf{X}}_0^{\lambda} \right\|_2^2 \right] + M_{\lambda} \right) e^{-2\lambda^{\min h}}, \end{aligned} $$

where we applied Lemma 5.1 and the trivial observation that $\|\mathbf{D}_h^{\lambda}\ell\|_2 \le e^{-\lambda^{\min h}} \|\ell\|_2$ for all $\ell \in T((\mathbb{R}^{d+1}))$ such that $\ell^\circ = 0$.

By definition (5.1), the same bound holds for $W_2^2(\mu_t, \mu_*)$. In particular, (5.2) implies the uniqueness of the invariant measure $\mu_*$. Hence, this invariant measure is ergodic (Hairer, 2006, Corollary 5.12). $\square$

**Corollary 5.4.** The continuous-time version of the Birkhoff-Khinchin ergodic theorem (Kallenberg, 1997, Corollary 10.9) then implies that, for any measurable integrable function $f: T((\mathbb{R}^{d+1})) \to \mathbb{R}$,

$$ \lim_{T \to \infty} \frac{1}{T} \int_0^T f(\hat{\mathbf{W}}_t^{\lambda}) dt = \mathbb{E} [f(\hat{\mathbf{W}}_0^{\lambda})]. \qquad (5.3) $$

The convergence in (5.3) is important for applications, as it allows for the estimation of relevant statistics (e.g., signature moments) and enables learning the law of the stationary process from a single trajectory observed over a sufficiently long time horizon.

## 5.3 Expected EFM-signature of the Brownian motion.

The expected signature of Brownian motion is given by Fawcett's formula (see Fawcett (2003)).

In this section, we aim to compute the expected EFM-signatures:

1. $\hat{\mathcal{E}}^{\lambda} := \mathbb{E}[\hat{\mathbf{W}}_t^{\lambda}]$, corresponding to the stationary distribution;

2. $\hat{\mathcal{E}}_{t-s}^{\lambda} := \mathbb{E}[\hat{\mathbf{W}}_{s,t}^{\lambda}]$, the expected EFM-signature on $[s, t]$.

Since $\mathcal{L}(\hat{\mathbf{W}}_{s,t}^{\lambda})$ converges weakly to the stationary law $\mathcal{L}(\hat{\mathbf{W}}_t^{\lambda})$ as $s \to -\infty$, and since $\sup_{t \ge s} \mathbb{E}[\|\hat{\mathbf{W}}_{s,t}^{\lambda}\|_2^2] \le M_{\lambda}$ by Lemma 5.1, the expected EFM-signatures converge as well:

$$ \hat{\mathcal{E}}_h^{\lambda} \xrightarrow{h \to \infty} \hat{\mathcal{E}}^{\lambda}. $$

The importance of the Magnus expansion for the computation of expected signatures in a general semimartingale context was highlighted by Friz, Hager, and Tapia (2022, 2024). We will show that for the time-augmented Brownian motion path, the expected EFM-signature computation problem reduces to the computation of an EFM-signature of a linear path, a problem we have already treated in Subsection 4.2 using the Magnus exponential. Indeed, let us denote $\tilde{0} := 0$ and $\tilde{\mathbf{i}} := \mathbf{ii}$ for $\mathbf{i} \in \{1, \dots, d\}$ and the alphabet $\tilde{A}_{d+1} = \{\tilde{0}, \tilde{\mathbf{i}}, \dots, \tilde{d}\}$. Note that the words formed by the letters of $\tilde{A}_{d+1}$ form a subset of words in the initial alphabet $A_{d+1} = \{0, 1, \dots, d\}$ that correspond to the non-zero elements of the expected EFM-signature.
Let $\mathbb{T}^{\tilde{\lambda}}$ denote the EFM-signature of the linear path $T_t = t \cdot (1, \frac{1}{2}, \dots, \frac{1}{2})$ corresponding to the new alphabet $\tilde{A}_{d+1}$ and $\tilde{\lambda} = (\lambda^{\tilde{0}}, \lambda^{\tilde{1}}, \dots, \lambda^{\tilde{d}})$. We recall that this signature is given in closed form by Theorem 4.4.

**Theorem 5.5.** Let $\hat{\mathcal{E}}^{\lambda}$ and $\hat{\mathcal{E}}_{t-s}^{\lambda}$ be defined as above. Then,

$$ \hat{\mathcal{E}}_{t-s}^{\lambda} = \exp \left( \otimes \sum_{k \ge 1} \hat{\Omega}_k^{\hat{\mathcal{E}}^{\lambda}} (s,t) \right), \quad \hat{\mathcal{E}}^{\lambda} = \exp \left( \otimes \sum_{k \ge 1} \hat{\Omega}_k^{\hat{\mathcal{E}}^{\lambda}} (-\infty, t) \right) = \sum_{k=0}^{\infty} \mathbf{H}^k \binom{1+\frac{1}{2}\sum_{i=1}^d \mathrm{ii}}{\mathrm{ii}} \phi, $$

where $\hat{\Omega}_k^{\hat{\mathcal{E}}^{\lambda}}(s,t)$ is given by (4.7) with $\mathbf{A}(u) = \mathbf{D}_{t-u}^{\lambda} \binom{1+\frac{1}{2}\sum_{i=1}^d \mathrm{ii}}{\mathrm{ii}}$, and the operator $\mathbf{H}$ is defined in Theorem 4.4. The non-zero coefficients are given explicitly by

$$ (\hat{\mathcal{E}}_{t-s}^{\lambda})^{\tilde{1}_1 \dots \tilde{1}_n} = \mathbb{T}_{0,t-s}^{\tilde{\lambda}, \tilde{1}_1 \dots \tilde{1}_n}, \quad (\hat{\mathcal{E}}^{\lambda})^{\tilde{1}_1 \dots \tilde{1}_n} = \mathbb{T}_{0}^{\tilde{\lambda}, \tilde{1}_1 \dots \tilde{1}_n}. $$

Moreover, if $\lambda^0 = 2\lambda^1 = \dots = 2\lambda^d =: \lambda$, then

$$ \hat{\mathcal{E}}_{t-s}^{\lambda} = \exp \left( \otimes \left( \frac{1 - e^{-\lambda(t-s)}}{\lambda} \right) \binom{1 + \frac{1}{2} \sum_{i=1}^{d} \mathrm{ii}}{\mathrm{ii}} \right), \quad \hat{\mathcal{E}}^{\lambda} = \exp \left( \otimes \left( \frac{1}{\lambda} \binom{1 + \frac{1}{2} \sum_{i=1}^{d} \mathrm{ii}}{\mathrm{ii}} \right) \right). $$

*Proof.* Recall that the signature dynamics reads:

$$ d\hat{\mathbb{W}}_{s,t}^{\lambda} = -\Lambda\hat{\mathbb{W}}_{s,t}^{\lambda}dt + \hat{\mathbb{W}}_{s,t}^{\lambda} \otimes \mathbf{1}dt + \sum_{i=1}^{d} \hat{\mathbb{W}}_{s,t}^{\lambda} \otimes \mathbf{i} \circ dW_t^i, \quad t \ge s. $$

Converting the Stratonovich integrals to Itô integrals, we deduce

$$ d\hat{\mathbb{W}}_{s,t}^{\lambda} = -\Lambda\hat{\mathbb{W}}_{s,t}^{\lambda}dt + \hat{\mathbb{W}}_{s,t}^{\lambda} \otimes \binom{1+\frac{1}{2}\sum_{i=1}^{d}\mathrm{ii}}{\mathrm{ii}} dt + \sum_{i=1}^{d} \hat{\mathbb{W}}_{s,t}^{\lambda} \otimes \mathbf{i} dW_t^i, \quad t \ge s. $$

Using the variation of constant and taking the expectation, one obtains

$$ \mathbb{E}[\hat{\mathbb{W}}_{s,t}^{\lambda}] = \int_{s}^{t} \mathbf{D}_{t-u}^{\lambda} \left( \mathbb{E}[\hat{\mathbb{W}}_{s,u}^{\lambda}] \otimes \binom{1+\frac{1}{2}\sum_{i=1}^{d}\mathrm{ii}}{\mathrm{ii}} \right) du, \quad t \ge s. \quad (5.4) $$

Since $\binom{1+\frac{1}{2}\sum_{i=1}^{d}\mathrm{ii}}{\mathrm{ii}} = \binom{\tilde{1}+\frac{1}{2}\sum_{i=1}^{d}\tilde{\mathrm{i}}}{\tilde{\mathrm{i}}}$, the solution of (5.4) corresponds exactly to the stationary signature of a linear path $T_t = (t, \frac{t}{2}, \dots, \frac{t}{2})$ with $\tilde{\lambda} = (\lambda^{\tilde{0}}, \lambda^{\tilde{1}}, \dots, \lambda^{\tilde{d}})$ indexed by the words in the alphabet $\{\tilde{1}, \dots, \tilde{d}\}$. We conclude by applying Theorem 4.4. $\square$

*Remark 5.5.1.* One can equivalently use the recursive formulas obtained by a straightforward computation: for all words v and i, j ∈ {1, ..., d} such that i ≠ j, we have

$$ \hat{\mathcal{E}}^{\lambda, v0} = \frac{1}{\lambda^{v0}} \hat{\mathcal{E}}^{\lambda, v}, \quad \hat{\mathcal{E}}^{\lambda, vii} = \frac{1}{2} \frac{1}{\lambda^{vii}} \hat{\mathcal{E}}^{\lambda, v}, \quad \hat{\mathcal{E}}^{\lambda, v0i} = 0, \quad \hat{\mathcal{E}}^{\lambda, vij} = 0, $$

and

$$ \hat{\mathcal{E}}_{t}^{\lambda, v0} = \int_{0}^{t} e^{-\lambda^{v0}(t-s)} \hat{\mathcal{E}}_{s}^{\lambda, v} ds, \quad \hat{\mathcal{E}}_{t}^{\lambda, vii} = \frac{1}{2} \int_{0}^{t} e^{-\lambda^{vii}(t-s)} \hat{\mathcal{E}}_{s}^{\lambda, v} ds, \quad \hat{\mathcal{E}}_{t}^{\lambda, v0i} = 0, \quad \hat{\mathcal{E}}_{t}^{\lambda, vij} = 0. $$

The coefficients of the expected EFM-signature are no longer symmetric in the sense that

$$ \hat{\mathcal{E}}^{\lambda, ii0} = \frac{1}{\lambda^{ii0} \lambda^{ii}} \neq \frac{1}{\lambda^{0ii} \lambda^{0}} = \hat{\mathcal{E}}^{\lambda, 0ii}, \quad i \in \{1, \dots, d\}, $$
as they are in Fawcett's formula until one has $\lambda^0 = 2\lambda^1 = \dots = 2\lambda^d$. The choice $\lambda^0 = 2\lambda^i$ comes naturally from the fact that the quadratic variation of the Ornstein-Uhlenbeck processes

$$ \widehat{W}_t^{\lambda,i} = \int_{-\infty}^{t} e^{-\lambda(t-u)} dW_u^i, \quad i = 1, \dots, d, $$

appearing at the first level of the EFM-signature, equals in this case

$$ \widehat{W}_t^{\lambda,0} = \int_{-\infty}^{t} e^{-2\lambda(t-u)} du. $$

*Example 5.5.1.* As an illustration of the expected EFM-signature, we consider a prediction problem. Suppose that the output signal $Y = (Y_t)_{t \in \mathbb{R}}$ is given as a linear combination of EFM-signature elements:

$$ Y_t = \langle \ell, \widehat{W}_t^{\lambda} \rangle, \quad t \in \mathbb{R}. $$

As follows immediately from Chen's identity (4.1), the conditional expectation and variance are given by

$$ \mathbb{E}[Y_{t+h} | \mathcal{F}_t] = \langle \ell, \mathbf{D}_h^{\lambda} \widehat{W}_t^{\lambda} \otimes \hat{\mathcal{E}}_h^{\lambda} \rangle, \quad \mathrm{Var}[Y_{t+h} | \mathcal{F}_t] = \langle \ell^{\perp 2}, \mathbf{D}_h^{\lambda} \widehat{W}_t^{\lambda} \otimes \hat{\mathcal{E}}_h^{\lambda} \rangle - \langle \ell, \mathbf{D}_h^{\lambda} \widehat{W}_t^{\lambda} \otimes \hat{\mathcal{E}}_h^{\lambda} \rangle^2. \quad (5.5) $$

Figure 3 illustrates the use of this prediction model for a stationary signal

$$ Y_t = \frac{1}{1.01 + \sin(U_t)}, \quad dU_t = -\kappa U_t dt + \nu dW_t, $$

with $\kappa = 15$ and $\nu = 1.5$. First, we learn the coefficients $\ell$ via standard linear regression against the EFM-signature $\widehat{W}_t^{\lambda}$, truncated at order 5, over the interval $[0, 1]$. The signature parameters are chosen as $\lambda = (10, 10)$. We then apply the prediction formula (5.5) to estimate the conditional expectation of the signal given $\mathcal{F}_1$. The shaded region represents one standard deviation.

<figure><img src="image_5.png" /><figcaption>Figure 3: Trajectory of the signal $Y_t$ (brown), the regression process $\langle \ell, \widehat{W}_t^{\lambda} \rangle$ (dashed green), and the prediction process (dotted green) defined by (5.5).</figcaption></figure>

# 6 Linear functionals of EFM-signatures

As we saw in previous sections, the linear functional of the EFM-signature $\langle \ell, \widehat{W}_t^{\lambda} \rangle$ of the time-augmented Brownian motion, with $\ell \in T(\mathbb{R}^d)$, is a natural candidate for time-invariant path-dependent modeling, since:
• $\widehat{W}_t^\lambda$ is a stationary Markov process by Theorem 5.2;

• $\widehat{W}_t^\lambda$ uniquely determines the path $\widehat{W}$ up to time $t$ by Theorem 4.7;

• functionals of the form $\langle \ell, \widehat{W}_t^\lambda \rangle$ are universal approximators of path functionals by Theorem 4.10;

• the expected EFM-signature $\mathbb{E}[\widehat{W}_t^\lambda]$ can be computed explicitly by Proposition 5.5.

This section is dedicated to studying the properties of linear functionals of the EFM-signature $\langle \ell, \widehat{W}_t^\lambda \rangle$. For simplicity and clarity of the proposed proofs, we establish Itô's decomposition only in the case where the coefficient $\ell \in T(\mathbb{R}^d)$ is finite. Analogous results for possibly infinite linear functionals of the signature of extended Brownian motion can be found in the works by Abi Jaber and Gérard (2025) and Abi Jaber, Gérard, and Huang (2024a). We expect that the approach proposed in these works can be extended to the case of EFM-signatures.

## 6.1 Itô's decomposition

We start this subsection by considering a general continuous semimartingale path $X$ in $\mathbb{R}^d$. The following definition is useful for obtaining the semimartingale decomposition of linear functionals of the signature.

**Definition 6.1.** For a tensor sequence $\ell \in T((\mathbb{R}^d))$ and a word $u$, the projection $\ell|_u$ is defined as

$$ \ell|_u = \sum_{n \ge 0} \sum_{v \in V_n} \ell^{vu} v. $$

In other words, the projection retains only the coefficients of $\ell$ corresponding to words that end with $u$, and replaces each such word $vu$ by its prefix $v$. The connection between this projection and differentiation becomes evident in the following decomposition:

$$ \langle \ell, X_t \rangle = \langle \ell, X_0 \rangle + \sum_{i=1}^{d} \int_{0}^{t} \langle \ell|_i, X_s \rangle \circ dX_s^i, $$

or, expressed in terms of tensor sequences,

$$ \ell = \ell^{\Theta} + \sum_{i=1}^{d} \ell|_i i. $$

This decomposition leads to the Itô formula for (possibly infinite) linear functionals of the signature of semi-martingales, as shown by (Abi Jaber, Gérard, and Huang, 2024a, Theorem 3.3). In the case of EFM-signatures, a similar result holds, with the only difference being the presence of exponential kernels in their definition, which introduces an additional mean-reverting term.

**Proposition 6.2.** Let $\ell_t \in T(\mathbb{R}^d)$ be continuously differentiable (element-wise). Then the linear functional $\langle \ell_t, X_t^\lambda \rangle$ satisfies the stochastic differential equation:

$$ d \langle \ell_t, X_t^\lambda \rangle = \langle \dot{\ell}_t - \Lambda \ell_t, X_t^\lambda \rangle dt + \sum_{i=1}^{d} \langle \ell_t|_i, X_t^\lambda \rangle \circ dX_t^i. $$

*Proof.* Recall that by (3.10), for any non-empty word $v = i_1 \dots i_n$, the corresponding EFM-signature component satisfies

$$ dX_t^{\lambda,v} = -\lambda^v X_t^{\lambda,v} dt + X_t^{\lambda,v'} \circ dX_t^{i_n}, $$

where $v' := i_1 \dots i_{n-1}$. The result follows by plugging this expression into the definition

$$ \langle \ell_t, X_t^\lambda \rangle = \sum_{n \ge 0} \sum_{v \in V_n} \ell_t^v X_t^{\lambda,v}, $$
and applying the product rule:

$$
\begin{align*}
d \langle \ell_t, \mathbb{X}_t^\lambda \rangle &= \sum_{n \ge 0} \sum_{v \in V_n} \dot{\ell}_t^v \mathbb{X}_t^{\lambda, v} dt + \sum_{n \ge 0} \sum_{v \in V_n} \ell_t^v d\mathbb{X}_t^{\lambda, v} \\
&= \sum_{n \ge 0} \sum_{v \in V_n} \left( \dot{\ell}_t^v \mathbb{X}_t^{\lambda, v} - \lambda^v \ell_t^v \mathbb{X}_t^{\lambda, v} \right) dt + \sum_{n \ge 0} \sum_{v \in V_n} \ell_t^v \mathbb{X}_t^{\lambda, v'} \circ dX_t^{i_n} \\
&= \langle \dot{\ell}_t - \Lambda \ell_t, \mathbb{X}_t^\lambda \rangle dt + \sum_{i=1}^d \langle \ell_t|_i, \mathbb{X}_t^\lambda \rangle \circ dX_t^i.
\end{align*}
$$

□

This proposition allows us to derive the Itô formula for the EFM-signature $\widehat{W}_t^\lambda$ of the extended standard Brownian motion $\widehat{W}_t = (t, W_t^1, \dots, W_t^d)$.

**Proposition 6.3.** Let $\ell_t \in T(\mathbb{R}^{d+1})$ be continuously differentiable (element-wise). Then the process $\langle \ell_t, \widehat{W}_t^\lambda \rangle$ satisfies:

$$
d \langle \ell_t, \widehat{W}_t^\lambda \rangle = \left\langle \dot{\ell}_t - \Lambda \ell_t + \ell_t|_0 + \frac{1}{2} \sum_{i=1}^d \ell_{t|ii} \cdot \widehat{W}_t^\lambda \right\rangle dt + \sum_{i=1}^d \langle \ell_t|_i \cdot \widehat{W}_t^\lambda \rangle dW_t^i. \quad (6.1)
$$

*Proof.* Applying Proposition 6.2 to the EFM-signature $\widehat{W}_t^\lambda$ yields:

$$
d \langle \ell_t, \widehat{W}_t^\lambda \rangle = \langle \dot{\ell}_t - \Lambda \ell_t + \ell_t|_0, \widehat{W}_t^\lambda \rangle dt + \sum_{i=1}^d \langle \ell_t|_i, \widehat{W}_t^\lambda \rangle \circ dW_t^i. \quad (6.2)
$$

To convert the Stratonovich integral to an Itô integral, we compute the quadratic covariation between $W^{i-1}$ and $\langle \ell_t|_i, \widehat{W}_t^\lambda \rangle$. Applying again Proposition 6.2 to $\langle \ell_t|_i, \widehat{W}_t^\lambda \rangle$, we find:

$$
d \langle W^i, \langle \ell|_i, \widehat{W}^\lambda \rangle \rangle_t = \langle \ell|_{ii}, \widehat{W}^\lambda \rangle dt.
$$

Therefore, rewriting (6.2) in Itô form yields the desired result (6.1).

□

*Example 6.3.1 (OU representation revisited).* The Itô decomposition can be used to find an approximation of an Ornstein–Uhlenbeck process $Y_t = \int_{-\infty}^t e^{-\mu(t-s)} dW_s$ satisfying

$$
dY_t = -\mu Y_t dt + dW_t,
$$

in the form $\langle \ell^\mu, \widehat{W}_t^\lambda \rangle$ for fixed $\lambda$ and for arbitrary $\mu > 0$. Indeed, applying Itô's formula and equating the drift and the volatility of $Y_t$ and $\langle \ell^\mu, \widehat{W}_t^\lambda \rangle$, we obtain

$$
\begin{cases}
-\Lambda \ell^\mu + \ell^\mu|_0 = -\mu \ell^\mu, \\
\ell^\mu|_1 = \emptyset.
\end{cases}
$$

It is clear that $\ell^\mu$ satisfies this condition if it is a solution to the algebraic equation

$$
\ell^\mu = (\Lambda(\ell^\mu) - \mu\ell^\mu)0 + 1 = \mathcal{B}\ell^\mu + 1, \quad (6.3)
$$

where the operator $\mathcal{B}: T((\mathbb{R}^d)) \to T((\mathbb{R}^d))$ is defined by $\mathcal{B}: \ell \mapsto (\Lambda(\ell) - \mu\ell)0$. This is a linear equation which can be solved with the resolvent:

$$
\ell^\mu = \sum_{k \ge 0} \mathcal{B}^k 1,
$$

which corresponds to the Picard iterations for the equation (6.3). Note that this can be rewritten explicitly as

$$
\ell^\mu = \sum_{k \ge 0} c_k 10^{\otimes k},
$$
with

$$ \begin{cases} c_0 = 1, \\ c_{k+1} = (k\lambda^0 + \lambda^1 - \mu)c_k, \quad k \ge 0. \end{cases} $$

This is the formula we used in the introductory Subsection 2.1. If we denote

$$ \ell^{\mu,N} = \sum_{k=0}^{N} c_k \mathbf{1}_{10^{\otimes k}}, \quad Y_t^N = \langle \ell^{\mu,N}, \widehat{W}_t^{\lambda} \rangle, \qquad (6.4) $$

one can verify that

$$ d(Y_t - Y_t^N) = -\mu(Y_t - Y_t^N)dt + \langle c_{N+1} \mathbf{1}_{10^{\otimes N}}, \widehat{W}_t^{\lambda} \rangle, $$

so that

$$ \mathbb{E}[(Y_t - Y_t^N)^2] = \mathbb{E} \left[ \left( \int_{-\infty}^{t} e^{-\mu(t-s)} \langle c_{N+1} \mathbf{1}_{10^{\otimes N}}, \widehat{W}_s^{\lambda} \rangle ds \right)^2 \right] \le \frac{c_{N+1}^2}{2\mu} \langle (10^{\otimes N})^{\boxplus 2}, \widehat{\mathcal{E}}^{\lambda} \rangle. $$

A cumbersome but straightforward computation shows that the right-hand side expression converges to 0 as $N \to +\infty$, so that $Y_t^N \to Y_t$ in $L^2$. ■

## 6.2 A remark on the characteristic function of linear functional of $\widehat{W}_t^{\lambda}$

In this section, we show how the approach introduced in Abi Jaber and Gérard (2025) can be adapted to compute the characteristic functions of the EFM-signature:

$$ \phi(\ell) := \mathbb{E} \left[ e^{i \langle \ell, \widehat{W}_t^{\lambda} \rangle} \right] = \mathbb{E} \left[ e^{i \langle \ell, \widehat{W}_0^{\lambda} \rangle} \right], \quad t \in \mathbb{R}, $$

and

$$ \phi_T(\ell) := \mathbb{E} \left[ e^{i \langle \ell, \widehat{W}_{0,T}^{\lambda} \rangle} \right], \quad T > 0, $$

for a given $\ell \in T(\mathbb{R}^{d+1})$.

The following theorem provides a representation of these characteristic functions in terms of the solution to an infinite-dimensional system of mean-reverting Riccati equations. In this characterization, we allow for infinite linear combinations of EFM-signature elements. To that end, and only in this subsection, we assume the necessary convergence of the infinite series involved.

**Theorem 6.4.** Let $T > 0$. Suppose that $\psi_t \in T((\mathbb{R}^{d+1}))$, $t \ge 0$, is a smooth solution to the mean-reverting Riccati equation

$$ \begin{cases} \dot{\psi}_t = -\Delta\psi_t + F(\psi_t), \quad t \ge 0, \\ \psi_0 = i\ell, \end{cases} \qquad (6.5) $$

with $F(\psi) = \psi|_0 + \frac{1}{2} \sum_{i=1}^d \psi|_{ii} + \frac{1}{2} \sum_{i=1}^d (\psi|_i)^{\boxplus 2}$, such that the processes $M = (M_t)_{t \le 0}$ and $N^T = (N_t^T)_{t \in [0,T]}$,

$$ M_t := e^{\langle \psi_{-t}, \widehat{W}_t^{\lambda} \rangle}, \quad N_t^T = e^{\langle \psi_{T-t}, \widehat{W}_{0,t}^{\lambda} \rangle}. $$

are well-defined and the Itô's formula can be applied. Then $M$ and $N^T$ are local martingales. If, in addition, they are true martingales, then the following expression for the characteristic functions holds:

$$ M_t = \mathbb{E} \left[ e^{i \langle \ell, \widehat{W}_0^{\lambda} \rangle} \mid \mathcal{F}_t \right], \quad t \le 0, \qquad (6.6) $$

$$ N_t^T = \mathbb{E} \left[ e^{i \langle \ell, \widehat{W}_{0,T}^{\lambda} \rangle} \mid \mathcal{F}_t \right], \quad t \in [0, T]. $$

In particular, the characteristic functions are given by

$$ \phi(\ell) = M_{-\infty} = \lim_{T \to \infty} e^{\psi_T^o} \quad \text{and} \quad \phi_T(\ell) = N_0^T = e^{\psi_T^o}. $$
Proof. Applying the Itô's formula to $\langle \psi_{-t}, \widehat{W}_t^\lambda \rangle$, we obtain

$$d \langle \psi_{-t}, \widehat{W}_t^\lambda \rangle = \left\langle -\dot{\psi}_{-t} - \mathbf{\Lambda} \psi_{-t} + \psi_{-t}|_0 + \sum_{i=1}^d \psi_{-t}|_{ii}, \widehat{W}_t^\lambda \right\rangle dt + \sum_{i=1}^d \langle \psi_{-t}|_i, \widehat{W}_t^\lambda \rangle dW_t^i.$$

The application of Itô's formula to $M$ for $t \le 0$ yields

$$dM_t = M_t d \langle \psi_{-t}, \widehat{W}_t^\lambda \rangle + \frac{1}{2} M_t \sum_{i=1}^d \langle \psi_{-t}|_i, \widehat{W}_t^\lambda \rangle^2 dt \\ = M_t \left\langle -\dot{\psi}_{-t} - \mathbf{\Lambda} \psi_{-t} + \psi_{-t}|_0 + \sum_{i=1}^d \psi_{-t}|_{ii} + \sum_{i=1}^d (\psi_{-t}|_i)^{\otimes 2}, \widehat{W}_t^\lambda \right\rangle dt + M_t \sum_{i=1}^d \langle \psi_{-t}|_i, \widehat{W}_t^\lambda \rangle dW_t^i.$$

Since $\psi_t$ satisfies (6.5), the $dt$-term vanishes, so that $M$ is a local martingale. The same argument shows that $N^T$ is a local martingale as well.

Now, note that $M_0 = e^{\langle \psi_0, \widehat{W}_0^\lambda \rangle} = e^{i \langle \ell, \widehat{W}_{0,T}^\lambda \rangle}$ and $N_T^T = e^{\langle \psi_0, \widehat{W}_{0,T}^\lambda \rangle} = e^{i \langle \ell, \widehat{W}_{0,T}^\lambda \rangle}$, so that (6.6) holds for $t = 0$ and $t = T$, respectively. If $M$ and $N^T$ are true martingales, the equality (6.6) will hold for all $t \le 0$ and $t \in [0, T]$, correspondingly.

By (6.6), $M$ is uniformly integrable, so that it is closable and we set $M_{-\infty} = \lim_{t \to -\infty} M_t$. We note that

$$\phi(\ell) = M_{-\infty}, \quad \phi_T(\ell) = N_0^T = e^{\langle \psi_T, \widehat{W}_{0,0}^\lambda \rangle} = e^{\psi_T^\circ}.$$

Since the law of $\widehat{W}_{0,T}^\lambda$ converges weakly to the law of $\widehat{W}_0^\lambda$ as $T \to \infty$, so does the characteristic function:

$$e^{\psi_T^\circ} = \mathbb{E} \left[ e^{i \langle \ell, \widehat{W}_{0,T}^\lambda \rangle} \right] \underset{T \to \infty}{\longrightarrow} \mathbb{E} \left[ e^{i \langle \ell, \widehat{W}_0^\lambda \rangle} \right].$$

This finishes the proof.

We note that the existence of the solution $\psi$ satisfying Itô's decomposition (6.1) and the true martingality of $M$ and $N^T$ are the assumptions that are difficult to verify even in the one-dimensional case $d = 1$. Some partial results concerning $T((\mathbb{R}^d))$-valued Riccati ODEs were obtained by Cuchiero, Svaluto-Ferro, and Teichmann (2023b) and Abi Jaber, Li, and Lin (2024b).

**Example 6.4.1.** The equation (6.5) can be solved numerically using Euler's scheme or the predictor-corrector semi-integrated scheme described in Section D. Note that the only difference from the Riccati equation corresponding to the standard signature is the linear term $-\mathbf{\Lambda}\psi_t$. Appearing thanks to the mean reversion of the EFM-signature, it stabilizes the equation and forces all the coefficients except for $\psi_t^\circ$ to vanish as $t \to \infty$. A numerical illustration in Figure 4 shows that the characteristic function $\phi_T(\ell)$, obtained by numerically solving the mean-reverting Riccati equation using the predictor-corrector scheme, converges to the stationary value $\phi(\ell)$ computed by the Monte Carlo method.
<figure><img src="image_6.png" /><figcaption>Figure 4: Real (on the left) and imaginary (on the right) parts of $\phi_T(\ell) = e^{\psi_T^\ell}$ for $\ell = 1^{\otimes 4}$ and $\lambda = (1, 1)$ for different truncation orders $N$. The horizontal black line corresponds to the stationary value $\phi(\ell)$ estimated by the Monte Carlo method.</figcaption></figure>

# 7 Rough path view on the EFM-signature

This section is devoted to the construction of the EFM-signature with rough path techniques. Namely, our goal is to make sense of the improper rough convolution on $(-\infty, T]$. Indeed, we would like to define the $(n+1)$-th level of the EFM-signature recursively as

$$ \mathbb{X}_{s,t}^{\lambda,n+1} = \int_s^t \mathbf{D}_{t-u}^{\lambda}(\mathbb{X}_{s,u}^{\lambda,n} d\mathbf{X}_u). $$

The case of finite $s > -\infty$ is completely covered by the rough convolution relying on the mild sewing lemma as presented by Gerasimovics and Hairer (2018) and Gubinelli and Tindel (2010). However, this framework does not cover the case of an infinite integration domain $s = -\infty$. For this reason, we extend the rough convolution result of Gerasimovics and Hairer (2018) by exploiting the ideas of improper integration used by Bellingeri, Friz, and Gerencsér (2022).

Notice that the operator $\mathbf{D}^\lambda$ satisfies the condition

$$ \|\mathbf{D}_t^\lambda u - u\| \lesssim \|u\|, $$

needed for the applicability of the rough convolution results of Gerasimovics and Hairer (2018), since $\|\mathbf{D}_h^\lambda\| \le 1$ for all $h \ge 0$. As in the standard rough integration case, the rough convolution relies on the notion of controlled paths.

**Definition 7.1.** Let $\mathbf{X} = (\mathbb{X}^1, \mathbb{X}^2) \in \mathcal{C}^\alpha([s, t])$ for some $\alpha \in (\frac{1}{3}, \frac{1}{2}]$. We say that $(Y, Y') \in \mathcal{C}^\alpha([s, t]) \times \mathcal{C}^\alpha([s, t])$ is controlled by $\mathbf{X}$ according to the semigroup $\mathbf{D}^\lambda$, if the remainder term $R^Y$ defined by

$$ R_{u,v}^Y = \delta Y_{u,v} - \mathbf{D}_{v-u}^\lambda (Y_u' X_{u,v}), $$

belongs to $\mathcal{C}^{2\alpha}([s, t])$. We write $(Y, Y') \in \mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha}([s, t])$ and define the seminorm on $\mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha}([s, t])$ by

$$ \|Y, Y'\|_{X,2\alpha} = \|Y'\|_{\alpha}^{\wedge} + \|R^Y\|_{2\alpha}. $$

The convolution integral is then defined by

$$ Z_t = \int_s^t \mathbf{D}_{t-u}^{\lambda}(Y_u d\mathbf{X}_u) = \lim_{|P| \to 0} \sum_{[u,v] \in P} \mathbf{D}_{t-u}^{\lambda}(Y_u X_{u,v} + Y_u' \mathbb{X}_{u,v}^2), \quad (7.1) $$

where the limit is taken over all partitions $P$ of the interval $[s, t]$. We cite the result on the rough convolution (Gerasimovics and Hairer, 2018, Theorem 3.5), slightly simplified for our purposes.
**Theorem 7.2.** Fix $s \le t$ and $\mathbf{X} = (\mathbb{X}^1, \mathbb{X}^2) \in \mathcal{C}^\alpha([s, t])$ for some $\alpha \in (\frac{1}{3}, \frac{1}{2}]$. Let $(Y, Y') \in \mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha}([s, t])$. Then the integral $Z$ given by (7.1) exists as an element of $\mathcal{C}^\alpha([s, t])$. Moreover, $(Z, Y) \in \mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha}([s, t])$, and one has the bounds

$$ \left| \int_{u}^{v} \mathbf{D}_{v-u}^{\lambda}(Y_r d\mathbf{X}_r) - \mathbf{D}_{v-u}^{\lambda}(Y_u \mathbf{X}_{u,v} + Y_u' \mathbf{X}_{u,v}^2) \right| \le (\|R^Y\|_{2\alpha} \|\mathbb{X}^1\|_{\alpha} + \|Y'\|_{\alpha}^{\wedge} \|\mathbb{X}^2\|_{2\alpha}) |v-u|^{3\alpha}, \quad (7.2) \\ \|Z, Y\|_{X,2\alpha} \lesssim \|Y\|_{\alpha}^{\wedge} + \|\mathbf{X}\|_{\alpha}(\|Y_0'\| + \|Y, Y'\|_{X,2\alpha}). $$

This result allows us to define recursively the EFM-signature over finite intervals.

**Corollary 7.3.** Fix $s \le t$ and let $\mathbf{X} = (\mathbb{X}^1, \mathbb{X}^2) \in \mathcal{C}^\alpha([s, t])$ for some $\alpha \in (\frac{1}{3}, \frac{1}{2}]$. Then the EFM-signature

$$ \mathbf{X}_{s,t}^{\lambda,n+1} = \int_{s}^{t} \mathbf{D}_{t-u}^{\lambda}(\mathbf{X}_{s,u}^{\lambda,n} d\mathbf{X}_{u}), $$

is well-defined as a rough convolution of $(\mathbf{X}_{s,t}^{\lambda,n}, \mathbf{X}_{s,t}^{\lambda,n-1}) \in \mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha}([s, t])$ with respect to $\mathbf{X}$.

## 7.1 Improper rough convolution

The goal of this subsection is to define the rough convolution over $(-\infty, T]$ as an improper integral

$$ Z_t = \int_{-\infty}^{t} \mathbf{D}_{t-u}^{\lambda}(Y_u d\mathbf{X}_u) := \lim_{R \to -\infty} \int_{R}^{t} \mathbf{D}_{t-u}^{\lambda}(\mathbf{X}_{u}^{\lambda,n} d\mathbf{X}_u). $$

The space $\hat{C}_b^{\alpha, \rho}((-\infty, T])$ defined in (3.15) appears to fit well with our purposes, so that we will consider $Y \in \hat{C}_b^{\alpha, \rho}((-\infty, T])$. This naturally leads to the following definition of the corresponding controlled path space and allows us to formulate the integration result.

**Definition 7.4.** Let $\mathbf{X} = (\mathbb{X}^1, \mathbb{X}^2) \in \mathcal{C}^{\alpha, \rho}((-\infty, T])$ for some $\alpha \in (\frac{1}{3}, \frac{1}{2}]$ and $\rho > 0$. We say that $(Y, Y') \in \mathcal{C}_b^{\alpha, \rho}((-\infty, T]) \times \mathcal{C}_b^{\alpha, \rho}((-\infty, T])$ is controlled by $\mathbf{X}$ according to the semigroup $\mathbf{D}^\lambda$, if the remainder term $R^Y$ defined by

$$ R_{s,t}^{Y} = \hat{\delta} Y_{s,t} - \mathbf{D}_{t-s}^{\lambda}(Y'_{s} X_{s,t}), $$

belongs to $\mathcal{C}^{2\alpha, \rho}((-\infty, T])$. We write $(Y, Y') \in \mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha, \rho}((-\infty, T])$ and define the seminorm on $\mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha, \rho}((-\infty, T])$ by

$$ \|Y, Y'\|_{2\alpha, \rho}^{\wedge} = \|Y'\|_{\alpha, \rho}^{\wedge} + \|R^Y\|_{2\alpha, \rho}. $$

**Theorem 7.5 (Improper rough convolution).** Let $Y \in \mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha, \rho}((-\infty, T])$ and $\mathbf{X} \in \mathcal{C}^{\alpha, \rho}((-\infty, T])$ for some $\alpha \in (\frac{1}{3}, \frac{1}{2}]$ and $0 < \rho < \lambda$. Then, for $t \le T$, the improper rough convolution

$$ Z_t = \lim_{R \to -\infty} \int_{R}^{t} \mathbf{D}_{t-u}^{\lambda}(Y_u d\mathbf{X}_u), \quad (7.3) $$

is well-defined as an element of $\hat{C}_b^{\alpha, \rho}((-\infty, T])$ and

$$ \left| Z_t - \int_{R}^{t} \mathbf{D}_{t-u}^{\lambda}(Y_u d\mathbf{X}_u) \right| \lesssim \|\mathbf{D}_{T-R}^{\lambda-\rho} \mathbf{X}\|_{\alpha, \rho}(\|Y\|_{\infty, \rho} + \|Y'\|_{\infty, \rho} + \|Y, Y'\|_{2\alpha, \rho}^{\wedge}). \quad (7.4) $$

Moreover, the couple $(Z, Y)$ belongs to $\mathcal{D}_{\mathbf{D}^\lambda, X}^{2\alpha, \rho}((-\infty, T])$ and the following bounds hold

$$ \|Z\|_{\alpha, \rho}^{\wedge} + \|Z\|_{\infty, \rho} \lesssim \|\mathbf{X}\|_{\alpha, \rho}(\|Y\|_{\infty, \rho} + \|Y'\|_{\infty, \rho} + \|Y, Y'\|_{2\alpha, \rho}^{\wedge}), \quad (7.5) $$

$$ \|Z, Y\|_{2\alpha, \rho}^{\wedge} \lesssim \|\mathbf{X}\|_{\alpha, \rho}(\|Y'\|_{\infty, \rho} + \|Y, Y'\|_{2\alpha, \rho}^{\wedge}) + \|Y, Y'\|_{2\alpha, \rho}^{\wedge}. $$

*Proof.* The proof consists of three steps.
**Step 1. Existence of the limit (7.3).** Without loss of generality suppose that $t = T$. Let us denote

$$Z^R = \int_R^T \mathbf{D}_{T-u}^{\lambda}(Y_u d\mathbf{X}_u), \quad R \le T.$$

Take $S < R \le T$ such that $|R - S| \le 1$. We have

$$Z^R - Z^S = \int_S^R \mathbf{D}_{T-u}^{\lambda}(Y_u d\mathbf{X}_u) = \int_S^R \mathbf{D}_{R-u}^{\lambda} \mathbf{D}_{T-R}^{\lambda}(Y_u d\mathbf{X}_u) = \int_S^R \mathbf{D}_{R-u}^{\lambda}(\mathbf{D}_{T-R}^{\lambda} Y_u) d(\mathbf{D}_{T-R}^{\lambda} \mathbf{X}_u),$$

where we used Proposition 3.5 (vii) for the last equality. By (7.2), we have

$$|Z^R - Z^S| \le |\mathbf{D}_{T-S}^{\lambda}(Y_S X_{S,R} + Y_S' \mathbf{X}_{S,R}^2)| + (\|\mathbf{D}_{T-R}^{\lambda} R^Y\|_{2\alpha;[S,R]} \|\mathbf{D}_{T-R}^{\lambda} X\|_{\alpha;[S,R]} + \|\mathbf{D}_{T-R}^{\lambda} Y'\|_{\alpha;[S,R]}^{\wedge} \|\mathbf{D}_{T-R}^{\lambda} \mathbf{X}^2\|_{2\alpha;[S,R]}) \quad (7.6)$$

Since for all $f$, by Lemma 3.10,

$$\|\mathbf{D}_{T-R}^{\lambda} f\|_{\alpha;[S,R]} = \|\mathbf{D}_{T-S}^{\rho} \mathbf{D}_{T-R}^{\lambda-\rho} \mathbf{D}_{R-S}^{\lambda-\rho} f\|_{\alpha;[S,R]} \le \|\mathbf{D}_{T-R}^{\lambda-\rho} \mathbf{D}_{R-S}^{\lambda-\rho} f\|_{\alpha,\rho} \lesssim_{\rho} \|\mathbf{D}_{T-R}^{\lambda-\rho} f\|_{\alpha,\rho},$$

the second and third terms of the right-hand side of (7.6) are bounded by

$$\|\mathbf{D}_{T-R}^{\lambda-\rho} R^Y\|_{2\alpha,\rho} \|\mathbf{D}_{T-R}^{\lambda-\rho} X\|_{\alpha,\rho} + \|\mathbf{D}_{T-R}^{\lambda-\rho} Y'\|_{\alpha,\rho}^{\wedge} \|\mathbf{D}_{T-R}^{\lambda-\rho} \mathbf{X}^2\|_{2\alpha,\rho}. \quad (7.7)$$

The first term of (7.6) reads

$$|\mathbf{D}_{T-S}^{\lambda}(Y_S X_{S,T} + Y_S' \mathbf{X}_{S,T}^2)| = |\mathbf{D}_{T-S}^{\lambda-\rho}(\mathbf{D}_{T-S}^{\rho} Y_S \mathbf{D}_{T-S}^{\rho} X_{S,R} + \mathbf{D}_{T-S}^{\rho} Y_S' \mathbf{D}_{T-S}^{\rho} \mathbf{X}_{S,R}^2)| \quad (7.8) \\ \le \|Y\|_{\infty,\rho} \|\mathbf{D}_{T-S}^{\lambda-\rho} X\|_{\alpha,\rho} + \|Y'\|_{\infty,\rho} \|\mathbf{D}_{T-S}^{\lambda-\rho} \mathbf{X}^2\|_{2\alpha,\rho}.$$

Combining (7.7) and (7.8), we obtain

$$|Z^R - Z^S| \lesssim \|\mathbf{D}_{T-R}^{\lambda-\rho} \mathbf{X}\|_{\alpha,\rho}(\|Y\|_{\infty,\rho} + \|Y'\|_{\infty,\rho} + \|Y, Y'\|_{2\alpha,\rho}^{\wedge}), \quad 0 \le R - S \le 1.$$

For the general $S < R \le T$, there exist $\Delta \in (0, 1]$ and $n \in \mathbb{N}$ such that $R - S = n\Delta$. Hence, we can bound

$$|Z^R - Z^S| \le \sum_{k=0}^{n-1} |Z^{R-k\Delta} - Z^{R-(k+1)\Delta}| \lesssim \sum_{k=0}^{\infty} \|\mathbf{D}_{T-R+k\Delta}^{\lambda-\rho} \mathbf{X}\|_{\alpha,\rho}(\|Y\|_{\infty,\rho} + \|Y'\|_{\infty,\rho} + \|Y, Y'\|_{2\alpha,\rho}^{\wedge}) \\ \lesssim_{\lambda,\rho} \|\mathbf{D}_{T-R}^{\lambda-\rho} \mathbf{X}\|_{\alpha,\rho}(\|Y\|_{\infty,\rho} + \|Y'\|_{\infty,\rho} + \|Y, Y'\|_{2\alpha,\rho}^{\wedge}).$$

This implies the Cauchy criterion for $(Z^R)_{R \le T}$ and the existence of the limit $Z = \lim_{R \to -\infty} Z^R$ satisfying (7.4).

**Step 2.** Showing that $Z \in \hat{C}_b^{\alpha, \rho}((-\infty, T])$. For all $s, t \in (-\infty, T]$ such that $|t - s| \le 1$, we apply (7.2):

$$|\mathbf{D}_{T-s}^{\rho} \hat{\delta} Z_{s,t}| = \left| \int_s^t \mathbf{D}_{t-u}^{\lambda} \mathbf{D}_{T-s}^{\rho}(Y_u d\mathbf{X}_u) \right| \\ \le |\mathbf{D}_{t-s}^{\lambda} \mathbf{D}_{T-s}^{\rho}(Y_s X_{s,t})| + |\mathbf{D}_{t-s}^{\lambda} \mathbf{D}_{T-s}^{\rho}(Y_s' \mathbf{X}_{s,t}^2)| + (\|R^Y\|_{2\alpha,\rho} \|\mathbf{X}^1\|_{\alpha,\rho} + \|Y'\|_{\alpha,\rho}^{\wedge} \|\mathbf{X}^2\|_{2\alpha,\rho}) |t-s|^{3\alpha} \\ \le \|Y\|_{\infty,\rho} \|\mathbf{X}^1\|_{\alpha,\rho} |t-s|^{\alpha} + \|Y'\|_{\infty,\rho} \|\mathbf{X}^2\|_{2\alpha,\rho} |t-s|^{2\alpha} + (\|R^Y\|_{2\alpha,\rho} \|\mathbf{X}^1\|_{\alpha,\rho} + \|Y'\|_{\alpha,\rho}^{\wedge} \|\mathbf{X}^2\|_{2\alpha,\rho}) |t-s|^{3\alpha},$$

so that

$$\|Z\|_{\alpha,\rho}^{\wedge} \le \|Y\|_{\infty,\rho} \|\mathbf{X}^1\|_{\alpha,\rho} + \|Y'\|_{\infty,\rho} \|\mathbf{X}^2\|_{2\alpha,\rho} + (\|R^Y\|_{2\alpha,\rho} \|\mathbf{X}^1\|_{\alpha,\rho} + \|Y'\|_{\alpha,\rho}^{\wedge} \|\mathbf{X}^2\|_{2\alpha,\rho}) \\ \le \|\mathbf{X}\|_{\alpha,\rho}(\|Y\|_{\infty,\rho} + \|Y'\|_{\infty,\rho} + \|Y, Y'\|_{2\alpha,\rho}^{\wedge}).$$
To bound $||Z||_{\infty, \rho}$, we take $s \le T$ and write

$$ \begin{aligned} |\mathbf{D}_{T-s}^{\rho} Z_s| &= |\mathbf{D}_{T-s}^{\rho} \int_{-\infty}^{s} \mathbf{D}_{s-u}^{\lambda}(Y_u d\mathbf{X}_u)| \\ &\le \sum_{k \ge 0} |\mathbf{D}_{T-s}^{\rho} \mathbf{D}_{k}^{\lambda} \int_{s-k-1}^{s-k} \mathbf{D}_{(s-k)-u}^{\lambda}(Y_u d\mathbf{X}_u)| \\ &= \sum_{k \ge 0} |\mathbf{D}_{k}^{\lambda-\rho} \int_{s-k-1}^{s-k} \mathbf{D}_{(s-k)-u}^{\lambda} \mathbf{D}_{T-(s-k)}^{\rho}(Y_u d\mathbf{X}_u)|. \end{aligned} $$

Applying the bound (7.2) to each integral and proceeding as in Step 1 of the proof, we can show that

$$ \begin{aligned} |\mathbf{D}_{T-s}^{\rho} Z_s| &\lesssim_{\lambda, \rho} \sum_{k \ge 0} \|\mathbf{D}_{k}^{\lambda-\rho} \mathbf{X}\|_{\alpha, \rho}(\|Y\|_{\infty, \rho} + \|Y'\|_{\infty, \rho} + \|Y, Y'\|_{\hat{2\alpha, \rho}}) \\ &\lesssim_{\lambda, \rho} \|\mathbf{X}\|_{\alpha, \rho}(\|Y\|_{\infty, \rho} + \|Y'\|_{\infty, \rho} + \|Y, Y'\|_{\hat{2\alpha, \rho}}). \end{aligned} $$

Taking the supremum in $s$ finishes the proof of (7.5).

**Step 3. Showing that** $(Z, Y) \in \mathcal{D}_{\mathbf{D}^{\lambda}, X}^{2\alpha, \rho}((-\infty, T])$. For all $s < t \le T$ such that $|t-s| \le 1$, we have

$$ \begin{aligned} |\mathbf{D}_{T-s}^{\rho}(\delta Y_{s,t})| &\le |\mathbf{D}_{t-s}^{\lambda}(\mathbf{D}_{T-s}^{\rho} Y_s' \mathbf{D}_{T-s}^{\rho} X_{s,t})| + |\mathbf{D}_{T-s}^{\rho} R_{s,t}^{Y}| \\ &\le \|Y'\|_{\infty, \rho} \|\mathbf{X}^{1}\|_{\alpha, \rho} |t-s|^{\alpha} + \|R^{Y}\|_{2\alpha, \rho} |t-s|^{2\alpha}, \end{aligned} $$

so that

$$ \|Y\|_{\alpha, \rho} \le \|Y'\|_{\infty, \rho} \|\mathbf{X}^1\|_{\alpha, \rho} + \|R^Y\|_{2\alpha, \rho} \le \|\mathbf{X}\|_{\alpha, \rho}(\|Y'\|_{\infty, \rho} + \|Y, Y'\|_{\hat{2\alpha, \rho}}) + \|Y, Y'\|_{\hat{2\alpha, \rho}}. $$

To bound $\|R^Z\|_{2\alpha, \rho}$, we again use (7.2) and proceed as in Step 2:

$$ \begin{aligned} |\mathbf{D}_{T-s}^{\rho} R_{s,t}^{Z}| &= |\mathbf{D}_{T-s}^{\rho} \delta Z_{s,t} - \mathbf{D}_{T-s}^{\rho} \mathbf{D}_{t-s}^{\lambda}(Y_s X_{s,t})| \\ &= \left| \int_{s}^{t} \mathbf{D}_{t-u}^{\lambda} \mathbf{D}_{T-s}^{\rho}(Y_u d\mathbf{X}_u) - \mathbf{D}_{t-s}^{\lambda} \mathbf{D}_{T-s}^{\rho}(Y_s X_{s,t}) \right| \\ &\le |\mathbf{D}_{t-s}^{\lambda} \mathbf{D}_{T-s}^{\rho}(Y_s' \mathbf{X}_{s,t}^2)| + (\|R^Y\|_{2\alpha, \rho} \|\mathbf{X}^1\|_{\alpha, \rho} + \|Y'\|_{\hat{\alpha}, \rho} \|\mathbf{X}^2\|_{2\alpha, \rho}) |t-s|^{3\alpha} \\ &\le \|Y'\|_{\infty, \rho} \|\mathbf{X}^2\|_{2\alpha, \rho} |t-s|^{2\alpha} + (\|R^Y\|_{2\alpha, \rho} \|\mathbf{X}^1\|_{\alpha, \rho} + \|Y'\|_{\hat{\alpha}, \rho} \|\mathbf{X}^2\|_{2\alpha, \rho}) |t-s|^{3\alpha}, \end{aligned} $$

and we obtain

$$ \begin{aligned} \|R^Z\|_{2\alpha, \rho} &\le \|Y'\|_{\infty, \rho} \|\mathbf{X}^2\|_{2\alpha, \rho} + (\|R^Y\|_{2\alpha, \rho} \|\mathbf{X}^1\|_{\alpha, \rho} + \|Y'\|_{\hat{\alpha}, \rho} \|\mathbf{X}^2\|_{2\alpha, \rho}) \\ &\le \|\mathbf{X}\|_{\alpha, \rho}(\|Y'\|_{\infty, \rho} + \|Y, Y'\|_{\hat{2\alpha, \rho}}). \end{aligned} $$

This finishes the proof. □

**Corollary 7.6.** If $\mathbf{X} \in \mathcal{C}^{\alpha, \rho}((-\infty, T])$ for some $\alpha \in (\frac{1}{3}, \frac{1}{2}]$ and $0 < \rho < \lambda$, then the EFM-signature $\mathbf{X}^{\lambda}$ is well-defined. More precisely, for all $n \ge 0$, we have $\mathbf{X}^{\lambda, n} \in \hat{\mathcal{C}}_{b}^{\alpha, \rho}((-\infty, T])$ and $(\mathbf{X}^{\lambda, n}, \mathbf{X}^{\lambda, n-1}) \in \mathcal{D}_{\mathbf{D}^{\lambda}, X}^{2\alpha, \rho}((-\infty, T])$, and the following bounds hold (with $\mathbf{X}^{\lambda, k} \equiv 0$ for $k < 0$):

$$ \|\mathbf{X}^{\lambda, n}\|_{\alpha, \rho}^{\wedge} + \|\mathbf{X}^{\lambda, n}\|_{\infty, \rho} \lesssim \|\mathbf{X}\|_{\alpha, \rho}(\|\mathbf{X}^{\lambda, n-1}\|_{\infty, \rho} + \|\mathbf{X}^{\lambda, n-2}\|_{\infty, \rho} + \|\mathbf{X}^{\lambda, n-1}, \mathbf{X}^{\lambda, n-2}\|_{\hat{2\alpha, \rho}}), \quad (7.9) $$

$$ \|\mathbf{X}^{\lambda, n}, \mathbf{X}^{\lambda, n-1}\|_{\hat{2\alpha, \rho}} \lesssim \|\mathbf{X}\|_{\alpha, \rho}(\|\mathbf{X}^{\lambda, n-2}\|_{\infty, \rho} + \|\mathbf{X}^{\lambda, n-1}, \mathbf{X}^{\lambda, n-2}\|_{\hat{2\alpha, \rho}}) + \|\mathbf{X}^{\lambda, n-1}, \mathbf{X}^{\lambda, n-2}\|_{\hat{2\alpha, \rho}}. \quad (7.10) $$

**Proof.** We have $\mathbf{X}_{t}^{\lambda, 0} \equiv 1$ and we define

$$ \mathbf{X}_{t}^{\lambda, 1} = \int_{-\infty}^{t} \mathbf{D}_{t-u}^{\lambda}(\mathbf{X}_{u}^{\lambda, 0} d\mathbf{X}_{u}) $$

as an improper convolution of $(\mathbf{X}^{\lambda, 0}, 0)$ using Theorem 7.5. Indeed, $(\mathbf{X}^{\lambda, 0}, 0) \in \mathcal{D}_{\mathbf{D}^{\lambda}, X}^{2\alpha, \rho}((-\infty, T])$, since

$$ R_{s,t}^{\mathbf{X}^{\lambda, 0}} = \mathbf{X}_{t}^{\lambda, 0} - \mathbf{D}_{t-s}^{\lambda} \mathbf{X}_{s}^{\lambda, 0} \equiv 0. $$

Theorem 7.5 implies that $\mathbf{X}^{\lambda, 1} \in \mathcal{C}_{b}^{\alpha, \rho}((-\infty, T])$ and $(\mathbf{X}^{\lambda, 1}, \mathbf{X}^{\lambda, 0}) \in \mathcal{D}_{\mathbf{D}^{\lambda}, X}^{2\alpha, \rho}((-\infty, T])$.

For $n \ge 2$, we proceed by induction, defining $\mathbf{X}_{t}^{\lambda, n}$ as an improper convolution with $Y_t = \mathbf{X}_{t}^{\lambda, n-1}$ and $Y'_t = \mathbf{X}_{t}^{\lambda, n-2}$ and applying the theorem. □
## 7.2 Fading memory property: proof of Theorem 3.14

To prove the continuity at a fixed $\mathbf{X}$, we can suppose that there exists a constant $M$ such that

$$ \|\mathbf{X}\|_{\alpha, \rho} \leq M, \quad \|\mathbf{Y}\|_{\alpha, \rho} \leq M, \qquad (7.11) $$

since $\|\mathbf{X}\|_{\alpha, \rho} = \varrho_{\alpha, \rho}(\mathbf{0}, \mathbf{X})$.

Note that for $s > -\infty$, the result follows from the stability result for rough convolutions by (Gerasimovics and Hairer, 2018, Lemma 3.13). Thus, it is sufficient to consider only the case $s = -\infty$.

We proceed by induction in $n$. For $n = 0$, $X_t^{\lambda,n} \equiv Y_t^{\lambda,n} \equiv 1$, and the statement holds trivially. Suppose that $\mathbf{X} \mapsto X_t^{\lambda,k}$ is continuous for all $k \le n$ and $t \le T$.

Fix $t \le T$ and $\varepsilon > 0$. We will show that one can choose $r > 0$ such that $|X_t^{\lambda,n+1} - Y_t^{\lambda,n+1}| < \varepsilon$ for all $\mathbf{X}, \mathbf{Y}$ satisfying $\varrho_{\alpha, \rho}(\mathbf{X}, \mathbf{Y}) < r$. Recall that

$$ X_t^{\lambda,n+1} = \int_{-\infty}^{t} \mathbf{D}_{t-u}(X_u^{\lambda,n} d\mathbf{X}_u) = \lim_{R \to -\infty} \int_{R}^{t} \mathbf{D}_{t-u}(X_u^{\lambda,n} d\mathbf{X}_u). \qquad (7.12) $$

The bound (7.4) yields

$$ \left| X_t^{\lambda,n+1} - \int_R^t \mathbf{D}_{t-u}(X_u^{\lambda,n} d\mathbf{X}_u) \right| \lesssim \|\mathbf{D}_{t-R}^{-\rho} \mathbf{X}\|_{\alpha, \rho} (\|X_t^{\lambda,n}\|_{\infty, \rho} + \|X_t^{\lambda,n-1}\|_{\infty, \rho} + \|X_t^{\lambda,n}, X_t^{\lambda,n-1}\|_{2\alpha, \rho}). $$

Applying recursively (7.9), (7.10), and using (7.11), we can show that

$$ \left| X_t^{\lambda,n+1} - \int_R^t \mathbf{D}_{t-u}(X_u^{\lambda,n} d\mathbf{X}_u) \right| \lesssim_M \|\mathbf{D}_{T-R}^{-\rho} \mathbf{X}\|_{\alpha, \rho} \lesssim_M e^{-\mu(T-R)}, $$

where $\mu = \min_i(\lambda^i - \rho^i)$. This shows that the convergence (7.12) is uniform in $\mathbf{X}$ satisfying (7.11), so that we can choose $R < T$ such that

$$ \left| X_t^{\lambda,n+1} - \int_R^t \mathbf{D}_{t-u}(X_u^{\lambda,n} d\mathbf{X}_u) \right| \le \frac{\varepsilon}{4}, $$

for all $\mathbf{X}$ satisfying (7.11). It remains to control the difference

$$ \left| \int_R^t \mathbf{D}_{t-u}(X_u^{\lambda,n} d\mathbf{X}_u) - \int_R^t \mathbf{D}_{t-u}(Y_u^{\lambda,n} d\mathbf{Y}_u) \right|. \qquad (7.13) $$

We note that $X_u^{\lambda,n}$ satisfies the Chen's identity

$$ X_u^{\lambda,n} = \sum_{k=0}^{n} (\mathbf{D}_{u-R}^{\lambda} X_R^{\lambda,k}) \otimes X_{R,u}^{\lambda,n-k}, $$

so that

$$ \begin{aligned} \int_R^t \mathbf{D}_{t-u}(X_u^{\lambda,n} d\mathbf{X}_u) &= \sum_{k=0}^n \int_R^t (\mathbf{D}_{t-R}^{\lambda} X_R^{\lambda,k}) \otimes \mathbf{D}_{t-u}^{\lambda}(X_{R,u}^{\lambda,n-k} d\mathbf{X}_u) \\ &= \sum_{k=0}^n (\mathbf{D}_{t-R}^{\lambda} X_R^{\lambda,k}) \otimes \int_R^t \mathbf{D}_{t-u}^{\lambda}(X_{R,u}^{\lambda,n-k} d\mathbf{X}_u) \\ &= \sum_{k=0}^n (\mathbf{D}_{t-R}^{\lambda} X_R^{\lambda,k}) \otimes X_{R,t}^{\lambda,n-k+1}. \end{aligned} $$

Hence, it is sufficient to control

$$ \begin{aligned} & (\mathbf{D}_{t-R}^{\lambda} X_R^{\lambda,k}) \otimes X_{R,t}^{\lambda,n-k+1} - (\mathbf{D}_{t-R}^{\lambda} Y_R^{\lambda,k}) \otimes Y_{R,t}^{\lambda,n-k+1} \\ &= (\mathbf{D}_{t-R}^{\lambda} X_R^{\lambda,k}) \otimes (X_{R,t}^{\lambda,n-k+1} - Y_{R,t}^{\lambda,n-k+1}) + (\mathbf{D}_{t-R}^{\lambda} X_R^{\lambda,k} - \mathbf{D}_{t-R}^{\lambda} Y_R^{\lambda,k}) \otimes Y_{R,t}^{\lambda,n-k+1}, \end{aligned} $$
for all $k = 0, \dots, n$. In the first term, we have

$$|\mathbf{D}_{t-R}^{\lambda} \mathbf{X}_R^{\lambda,k}| = |\mathbf{D}_{t-R}^{\lambda-\rho} \mathbf{D}_{t-R}^{\rho} \mathbf{X}_R^{\lambda,k}| \lesssim_{R,t,T,\rho} \|\mathbf{X}_R^{\lambda,k}\|_{\infty,\rho} \lesssim_M 1.$$

where we recursively applied (7.9) and (7.10). The second part of the first term

$$|\mathbf{X}_{R,t}^{\lambda,n-k+1} - \mathbf{Y}_{R,t}^{\lambda,n-k+1}| \lesssim_{R,t} \|\mathbf{X}_{R,t}^{\lambda,n-k+1} - \mathbf{Y}_{R,t}^{\lambda,n-k+1}\|_{\alpha;[R,T]}.$$

By the stability result for rough convolutions (Gerasimovics and Hairer, 2018, Lemma 3.13), this norm tends to 0 as $\varrho_{\alpha,\rho}(\mathbf{X}, \mathbf{Y}) \to 0$.

In the second term, the second multiplier is bounded by the standard scaling argument

$$|\mathbf{Y}_{R,t}^{\lambda,n-k+1}| \lesssim_{R,t} \left( \|\mathbf{Y}\|_{\alpha;[R,t]} + \sqrt{\|\mathbf{Y}^2\|_{2\alpha;[R,t]}} \right)^{n-k+1} \lesssim_{R,T,\rho,M} 1,$$

while the first one

$$|\mathbf{D}_{t-R}^{\lambda}(\mathbf{X}_R^{\lambda,k} - \mathbf{Y}_R^{\lambda,k})| \to 0,$$

as $\varrho_{\alpha,\rho}(\mathbf{X}, \mathbf{Y}) \to 0$ by the induction hypothesis. Thus, one can choose $r > 0$ such that

$$|(\mathbf{D}_{t-R}^{\lambda} \mathbf{X}_R^{\lambda,k}) \otimes \mathbf{X}_{R,t}^{\lambda,n-k+1} - (\mathbf{D}_{t-R}^{\lambda} \mathbf{Y}_R^{\lambda,k}) \otimes \mathbf{Y}_{R,t}^{\lambda,n-k+1}| \le \frac{\varepsilon}{2(n+1)},$$

for $\mathbf{X}, \mathbf{Y}$ such that $\varrho_{\alpha,\rho}(\mathbf{X}, \mathbf{Y}) < r$. This implies that (7.13) is less than $\frac{\varepsilon}{2}$ and concludes the proof.

## A Proof of Lemma 4.3

The computation reduces to evaluating the iterated integral (4.5). After a change of variables

$$u_j = t - \sum_{k=1}^{n-j-1} \tau_k, \quad j = 1, \dots, n,$$

this integral becomes

$$\int_{\tau_j \ge 0, \sum_{j=1}^n \tau_j \le t-s} e^{-\sum_{j=1}^n \lambda^{ij} \sum_{k=1}^{n-j-1} \tau_k} d\tau_1 \dots d\tau_n = \int_{\tau_j \ge 0, \sum_{j=1}^n \tau_j \le t-s} e^{-\sum_{k=1}^n \left(\sum_{j=1}^{n-k-1} \lambda^{ij}\right) \tau_k} d\tau_1 \dots d\tau_n.$$

Setting $\mu_k = \sum_{j=1}^{n-k-1} \lambda^{ij}$ for $k = 1, \dots, n$ and introducing $\mu_{n+1} = 0$, as well as $\tau_{n+1} = (t-s) - \sum_{k=1}^n \tau_k$, the integral becomes one over the simplex $C := \{ \tau = (\tau_1, \dots, \tau_{n+1}) \in \mathbb{R}^{n+1} : \sum_{k=1}^{n+1} \tau_k = t-s \}$:

$$\int_C e^{-\sum_{k=1}^{n+1} \mu_k \tau_k} d\tau.$$

By induction, this integral can be written as the convolution of exponential functions $f_k(t) = e^{-\mu_k t}$:

$$\int_C e^{-\sum_{k=1}^{n+1} \mu_k \tau_k} d\tau = (f_1 \star \dots \star f_{n+1})(t-s).$$

The Laplace transform of each $f_k$ is $L[f_k](w) = \frac{1}{w + \mu_k}$, so

$$L\left[\int_C e^{-\sum_{k=1}^{n+1} \mu_k \tau_k} d\tau\right] (w) = \prod_{k=1}^{n+1} \frac{1}{w + \mu_k} = \sum_{k=1}^{n+1} \frac{c_k}{w + \mu_k},$$
where the coefficients $c_k = \prod_{l \neq k} \frac{1}{\mu_l - \mu_k}$ come from the partial fraction decomposition, valid when all $\mu_k$ are distinct. Applying the inverse Laplace transform yields

$$ \int_C e^{-\sum_{k=1}^{n+1} \mu_k \tau_k} d\tau = \sum_{k=1}^{n+1} c_k e^{-\mu_k (t-s)}. $$

A straightforward reindexing $k \mapsto n+1-k$ concludes the proof of the first formula. The second one is obtained by taking the limit $(t-s) \to +\infty$.

## B Proof of Lemma 5.1

*Proof.* We prove the lemma only in the case of one-dimensional Brownian motion $d = 1$. The multidimensional generalization is straightforward. To prove the convergence of the series

$$ \mathbb{E}[\|\hat{W}_{0,t}^{\lambda}\|_2^2] = \sum_{n \ge 0} \sum_{|v|=n} \mathbb{E}[\|\hat{W}_{0,t}^{\lambda,v}\|_2^2], \quad (B.1) $$

we will recursively bound $\mathbb{E}[\|\hat{W}_{0,t}^{\lambda,v}\|_2^2]$ for each $v$. For a word $v$ such that $|v| = n > 0$, we consider three cases. In the first case, where $v = v'0$, apply Hölder's inequality:

$$ \mathbb{E}\left[\left(\int_0^t e^{-\lambda^v(t-s)} \hat{W}_{0,s}^{\lambda,v'} ds\right)^2\right] \le \int_0^t e^{-\lambda^v(t-s)} \mathbb{E}\left[\|\hat{W}_{0,s}^{\lambda,v'}\|^2\right] ds \int_0^t e^{-\lambda^v(t-s)} ds \le \frac{1}{(\lambda^v)^2} \sup_{s \ge 0} \mathbb{E}\left[\|\hat{W}_{0,s}^{\lambda,v'}\|^2\right]. $$

For $v = v'1$, with $v'$ not ending in $1$, the Stratonovich integral coincides with the Itô integral, and we have

$$ \mathbb{E}\left[\left(\int_0^t e^{-\lambda^v(t-s)} \hat{W}_{0,s}^{\lambda,v'} \circ dW_s\right)^2\right] = \int_0^t e^{-2\lambda^v(t-s)} \mathbb{E}\left[\|\hat{W}_{0,s}^{\lambda,v'}\|^2\right] ds \le \frac{1}{2\lambda^v} \sup_{s \ge 0} \mathbb{E}\left[\|\hat{W}_{0,s}^{\lambda,v'}\|^2\right]. $$

Finally, if $v = v''11$, then

$$ \begin{aligned} \mathbb{E}\left[\left(\int_0^t e^{-\lambda^v(t-s)} \hat{W}_{0,s}^{\lambda,v'} \circ dW_s\right)^2\right] &\le \mathbb{E}\left[\left(\int_0^t e^{-\lambda^v(t-s)} \hat{W}_{0,s}^{\lambda,v'} dW_s\right)^2\right] + \mathbb{E}\left[\left(\frac{1}{2}\int_0^t e^{-\lambda^v(t-s)} \hat{W}_{0,s}^{\lambda,v''} ds\right)^2\right] \\ &\le \frac{1}{2\lambda^v} \sup_{s \ge 0} \mathbb{E}\left[\|\hat{W}_{0,s}^{\lambda,v'}\|^2\right] + \frac{1}{(2\lambda^v)^2} \sup_{s \ge 0} \mathbb{E}\left[\|\hat{W}_{0,s}^{\lambda,v''}\|^2\right], \end{aligned} $$

where the integrals are bounded as in the first two cases.

Combining these three bounds and noting that the bound does not depend on $t$, we obtain

$$ \begin{aligned} \sup_{t \ge 0} \mathbb{E} \left[ \| \hat{W}_{0,t}^{\lambda,v} \|^2 \right] &\le \left( \frac{1}{(\lambda^v)^2} \vee \frac{1}{2\lambda^v} \right) \sup_{t \ge 0} \mathbb{E} \left[ \| \hat{W}_{0,t}^{\lambda,v'} \|^2 \right] + \frac{1}{(2\lambda^v)^2} \sup_{s \ge 0} \mathbb{E} \left[ \| \hat{W}_{0,s}^{\lambda,v''} \|^2 \right] \quad (B.2) \\ &\le \frac{C_\lambda}{n} \sup_{t \ge 0} \mathbb{E} \left[ \| \hat{W}_{0,t}^{\lambda,v'} \|^2 \right] + \frac{C_\lambda}{n^2} \sup_{t \ge 0} \mathbb{E} \left[ \| \hat{W}_{0,t}^{\lambda,v''} \|^2 \right], \end{aligned} $$

since $\lambda^v \ge n \min_{i \in \{0,1\}} \lambda^i$, with a constant $C_\lambda \ge 1$ depending only on $\lambda$.

Applying (B.2) recursively, we obtain

$$ \sup_{t \ge 0} \mathbb{E} \left[ \| \hat{W}_{0,t}^{\lambda,v} \|^2 \right] \le \frac{(2C_\lambda)^n}{n!}. $$

Finally, noting that there are $2^n$ terms in (B.1) corresponding to words of length $n$, and taking the supremum in $t$, we obtain

$$ \sup_{t \ge 0} \mathbb{E}[\|\hat{W}_{0,t}^{\lambda}\|_2^2] \le \sum_{n \ge 0} 2^n \frac{(2C_{\lambda})^n}{n!} < +\infty, $$

which completes the proof of the second bound. The first one can be shown analogously. $\square$
C Useful lemma

**Lemma C.1.** *The following relationship holds:*

$$
\int_{-\infty}^{T} e^{-(k\lambda^{0}+\lambda^{i})(T-t)} \langle \mathbf{i}, \hat{\mathbf{X}}_{t}^{\lambda} \rangle dt = \sum_{m=1}^{k} b_{k,m} \langle \mathbf{i}0^{\otimes m}, \hat{\mathbf{X}}_{T}^{\lambda} \rangle, \quad k \ge 1, \qquad (C.1)
$$

$$
\text{with } b_{k,m} = (-\lambda^0)^{m-1} \frac{(k-1)!}{(k-m)!}, \quad m = 1, \dots, k.
$$

*Proof.* Applying Itô’s decomposition to the both parts of (C.1), and equating the dt-terms, we obtain

$$
-(k\lambda^0 + \lambda^i) \int_{-\infty}^{T} e^{-(k\lambda^0 + \lambda^i)(T-t)} \langle \mathbf{i}, \hat{\mathbf{X}}_t^\lambda \rangle dt + \langle \mathbf{i}, \hat{\mathbf{X}}_T^\lambda \rangle = -(m\lambda^0 + \lambda^i) \sum_{m=1}^{k} b_{k,m} \langle \mathbf{i}0^{\otimes m}, \hat{\mathbf{X}}_T^\lambda \rangle + \sum_{m=1}^{k} b_{k,m} \langle \mathbf{i}0^{\otimes (m-1)}, \hat{\mathbf{X}}_T^\lambda \rangle,
$$

or, equivalently,

$$
-(k\lambda^0 + \lambda^i) \sum_{m=1}^{k} b_{k,m} \langle \mathbf{i}0^{\otimes m}, \hat{\mathbf{X}}_T^{\lambda} \rangle + \langle \mathbf{i}, \hat{\mathbf{X}}_T^{\lambda} \rangle = \sum_{m=1}^{k} \left( -(m\lambda^0 + \lambda^i)b_{k,m} + b_{k,m+1} \right) \langle \mathbf{i}0^{\otimes (m-1)}, \hat{\mathbf{X}}_T^{\lambda} \rangle + b_{k,1} \langle \mathbf{i}, \hat{\mathbf{X}}_T^{\lambda} \rangle.
$$

with $b_{k,k+1} := 0$. This leads to

$$
b_{k,1} = 1, \quad b_{k,m+1} = \lambda^0(m-k)b_{k,m} = (\lambda^0)^{m-1}(m-k)(m-1-k)\dots(1-k) = (-\lambda^0)^m \frac{(k-1)!}{(k-m-1)!},
$$

and completes the proof.

<figure><img src="image_7.png" /></figure>

D Numerical scheme for infinite-dimensional ODEs

In this section, we tackle the problem of numerical solution of an infinite-dimensional ODE on $\psi_t \in T((\mathbb{R}^d))$:

$$
\begin{cases}
\dot{\psi}_t = -\mathbf{\Lambda}\psi_t + F(\psi_t), & t \ge 0, \\
\psi_0 = \ell,
\end{cases}
\tag{D.1}
$$

By Proposition 3.6, the equation (D.1) can be rewritten in the integral form

$$
\psi_t = \mathbf{D}_t^{\lambda}(\ell) + \int_0^t \mathbf{D}_{t-s}^{\lambda} F(\psi_s) \, ds.
$$

We denote by $\hat{\psi}$ an approximated solution truncated at order $N$ and corresponding to the discretization step $h$. We also denote by $\hat{F}(\psi)$ the value of $F(\psi)$ truncated at the same order $N$. This allows us to construct a semi-integrated scheme for numerical resolution of (D.1):

$$
\psi_{t+h} = \mathbf{D}_h^\lambda(\psi_t) + \int_t^{t+h} \mathbf{D}_{t+h-s}^\lambda F(\psi_s) ds \approx \mathbf{D}_h^\lambda(\psi_t) + \int_t^{t+h} \mathbf{D}_{t+h-s}^\lambda ds F(\psi_t),
$$

so that

$$
\hat{\psi}_{t+h} = \mathbf{D}_h^{\lambda} \hat{\psi}_t + C_h^{\lambda} \hat{F}(\hat{\psi}_t),
$$

where the operator $C_h: T((\mathbb{R}^d)) \rightarrow T((\mathbb{R}^d))$ is defined by

$$
C_h : \ell = \sum_{n \ge 0} \sum_{|v|=n} \ell^v \mapsto C_h \ell = \sum_{n \ge 0} \sum_{|v|=n} \left( \frac{1 - e^{-\lambda^v h}}{\lambda^v} \right) \ell^v
$$

$$
\text{with convention } \left( \frac{1 - e^{-\lambda^{\nu} h}}{\lambda^{\nu}} \right) \Big|_{\lambda^{\nu}=0} = h.
$$
To improve the numerical stability, we use the predictor-corrector scheme:

$$ \hat{\psi}_{t+h}^{P} = D_{h}^{\lambda} \hat{\psi}_{t} + C_{h}^{\lambda} F(\hat{\psi}_{t}), $$

$$ \hat{\psi}_{t+h} = D_{h}^{\lambda} \hat{\psi}_{t} + C_{h}^{\lambda} \left( \frac{F(\hat{\psi}_{t}) + F(\hat{\psi}_{t+h}^{P})}{2} \right). $$

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