Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models

Introduction

Methods based on the signature of a path represent a non-parametric way for extracting characteristic features from time series data which is essential in machine learning tasks. This explains why these techniques are more and more applied in econometrics and mathematical finance, see e.g. Buehler et al. 2020; Kalsi et al. 2020; Arribas et al. 2020; Lyons et al. 2020; Ni et al. 2020; Bayer et al. 2021; Cuchiero et al. 2021; Min and Hu 2021; Akyildirim et al. 2022 and the references therein. Indeed, signature-based methods allow for data-driven modeling approaches, while stylized facts or first principles from mathematical finance can still be easily guaranteed.

The notion of signature of a path goes back to Chen 1957; Chen 1977 (see also Magnus 1954; Fliess 1981; Sussmann 1988) and plays a prominent role in the context of rough path theory, pioneered by Lyons 1998. We also refer to the monographs by Friz and Victoir 2010 and Friz and Hairer 2014. It has initially been developed to deal with controlled differential equations driven by rough signals in a pathwise way. Indeed, signature can be understood as an enhancement of the input signal with iterated integrals, which in turn allows to express the solution of the rough differential equation as a continuous map of the input signal enhanced in this way.

More generally, the choice of the signature as feature map capturing the specific characteristics of the path, can be explained by a universal approximation theorem (UAT) according to which continuous (with respect to certain variation distances) functionals of continuous paths can be approximated on compact sets of paths by linear functionals of the time-extended signature. This result is however only proved for continuous paths and therefore leaves open the question of approximating continuous functionals of the more general set of càdlàg paths, which have particular relevance when it comes to financial modeling. Based on advances on the signature of càdlàg paths by Friz and Shekhar 2017 and using the Skorokhod-$J_{1}$-topology on (Lie group valued) càdlàg paths, we here present a UAT that solves this question. Firstly, we prove that continuous path functionals over the time interval $$can be uniformly approximated on compact sets of paths by linear functionals of the time-extended signature evaluated at the final time$1$. Secondly, we extend this result by considering functionals of the stopped paths, stopped at all deterministic times$t\in$, which can thus be associated with non-anticipative path-functionals (see Remark 3.14). We then prove that the coefficients of the linear functionals do not depend on$t\in$ but can be chosen uniformly in time. This is subject of Theorem 3.13, our main result in the first part of the paper.

Our principal motivation to treat this problem comes from signature-based models for finance that allow for an inclusion of jumps. Indeed, all signature models for asset prices that have been proposed so far, (see Arribas et al. 2020; Cuchiero et al. 2023b; Cuchiero et al. 2023a), build on stochastic processes with continuous trajectories. In order to extend them to processes with càdlàg trajectories and to show universality properties among classical jump-diffusion models in finance, the UAT proved in Section 3 is important, as will become apparent from our definition of Lévy-type signature models below.

To analyze tractability properties of models of the form (1.1), we first show that the truncated signature process of a generic multivariate Lévy process is a polynomial process on the truncated tensor algebra and derive its expected value via the so-called moment formula, thus solving a linear ODE (see Cuchiero et al. 2012; Filipović and Larsson 2020). Specifically, we prove that under moment assumptions on the Lévy measure, the expected signature of a Lévy process assumes the Lévy-Kintchine form described in Friz and Shekhar 2017.

Our results build on the theory of càdlàg rough paths as commenced in Friz and Shekhar 2017, but related concepts like semimartingale signature (see Teichmann 2021) could be pursued as well. Let us finally remark that another strand of research related to finance where (càdlàg) rough paths play a significant role is robust and model-free finance, see for example Perkowski and Prömel 2016; Allan et al. 2021a and in particular Allan et al. 2021b for càdlàg rough paths foundations for robust finance. Indeed, in this context the theory of rough integration allows one to go beyond classical Itô-integrals and to overcome issues associated with null sets that are inherent in stochastic models, when e.g. dealing with volatility uncertainty. This in turn opens the door to a worst case analysis and a robustification of financial models and strategies. Note that we here do not pursue this direction but interpret all integrals in our financial model setup as Itô-integrals.

The remainder of the paper is structured as follows. In Section 2, we introduce the algebraic setting proper of rough path theory and recall the notion of weakly geometric càdlàg rough paths and their Marcus signature. Particular attention is also given to the Marcus signature of càdlàg semimartingales. Section 3 is dedicated to the universal approximation theorem. In Section 4 we introduce Lévy-type signature models, discuss their universality properties and derive pricing and hedging formulas based on polynomial technology. All the technical proofs are given in the Appendix. Finally, in order to enhance the accessibility of the paper, in Section C we include some auxiliary results on the Marcus signature of weakly geometric càdlàg rough paths.

Preliminaries

The exponential and logarithm maps are defined as follows:

where the tensor multiplication is again always truncated beyond level $N$.

We equip it with the so-called Carnot-Caratheodory (CC) norm $\|\cdot\|_{CC}$ and the induced (left-invariant) metric, denoted by $d_{CC}$. We refer to Chapter 7 in Friz and Victoir 2010 for more details.

For two multi-indices $I\in\{1,\dots,d\}^{n}$, $J\in\{1,\dots,d\}^{m}$, and two indices of length $1$, $a,b\in\{1,\dots,d\}$, the shuffle product $\shuffle$ is defined recursively by

where $(I,a)$ denotes the concatenation of multi-indices.

The set of group-like elements is defined as follows

where $\epsilon_{I}\shuffle\epsilon_{J}:=\sum_{k=1}^{K}\epsilon_{I_{k}}$ where $K,I_{k}$ for $k=1,\ldots,K$ are determined via $I\shuffle J=\sum_{k=1}^{K}I_{k}$.

2 Càdlàg rough paths and Marcus signature

In this section, we recall the definition of weakly geometric (w.g.) càdlàg rough paths and Marcus signature, as originally formulated in Friz and Shekhar 2017, to which we refer for a deeper discussion. We adopt here the Lie group-valued point of view and refer to Appendix C.1 for an equivalent definition.

The notion of w.g. rough paths naturally extends to arbitrary low regularity. We restricted the presentation only to paths of finite $p$-variation for $p\in[1,3)$, as this is what matters the most in our stochastic analysis setting.

Finally, the Marcus signature of a w.g. càdlàg rough path is defined as the unique path extension provided by Theorem 2.4.

Roughly speaking, a solution of a Marcus-type RDE Solutions of Marcus-type RDEs are also called geometric solutions. In the presence of jumps, this notion has to be distinguished with the concept of forward solution, introduced in Friz and Zhang 2017. driven by $\mathbf{X}$ is defined as the $\tau$-time changed solution of the corresponding continuous RDE driven by $\mathbf{X}^{\phi}$, that is the continuous path built via $\phi$. For some given suitable vector field $V$, such a RDE is then denoted by

Observe that the same type of equation has also been considered in a semimartingale context by Kurtz et al. 1995.

whose explicit form can be written using (2.3) and the notation of Section 2.2 as

The integral in (2.11) is understood as a rough integral and the summation term is well defined as an absolutely summable series.

3 Signature of càdlàg semimartingales

Càdlàg semimartingales fit well into the theory of càdlàg rough paths. Indeed, every semimartingale admits a canonical lift which is a.s. a Marcus-like càdlàg $p$-rough path for any $p>2$. For the proof of the subsequent proposition, we refer to Friz and Shekhar 2017 and Chevyrev and Friz 2019.

The signature of semimartingales can be computed by adopting stochastic integration methods instead of resorting to rough integrals. Proposition 4.16 in Chevyrev and Friz 2019 accounts for this. For completeness, we report an adaptation of this statement to our specific context.

coincides a.s. with the Marcus SDE (see Kurtz et al. 1995)

The explicit form of equation (2.12) Existence and uniqueness of the solution of the Marcus SDE (2.12) are provided by Theorem 3.2 in Kurtz et al. 1995. is given by

where $\exp(\Delta X_{s})=\sum_{k=0}^{\infty}\frac{1}{k!}(\Delta X_{s})^{\otimes k}$. This means that for each multi-index $I$ with entries in $\{1,\dots,d\}$, we get

Observe that here we consider Itô type integrals.

Universal approximation theorem

In this section, we present a universal approximation theorems (UAT) for continuous functionals of w.g. càdlàg rough paths. We start by specifying the topology and the subspace of paths that will be considered.

Let $(E,d)$ be a metric space and $D(,E)$ the space of càdlàg paths on it. Denote by $\Lambda$ the set of all strictly increasing bijections of $$to itself. The Skorokhod$J_{1}$-metric on$D(,E)$ is defined via

where, $|\lambda|:=\sup_{s\in}|\lambda(s)-s|$.

Next, we introduce a particular subspace of w.g. càdlàg rough paths.

Let $p\in[1,3)$. The subspace of time-extended w.g. càdlàg $p$-rough paths is defined as follows: For the explanation of the index $-1$ used here we refer to Notation 3.4 below.

In Definition 3.3 we use the index $-1$ to denote the time component of a time-extended w.g. càdlàg rough paths. This notation will be used throughout the paper.

This specific choice is needed in order to recover consistency with the classical Young integral (see Lemma C.4), which is key for showing condition (iii) in the proof of Proposition 3.6 (see also Corollary 3.10).

2 UAT for continuous path functionals

We start by proving that continuous path functionals can be uniformly approximated on compact sets by linear functionals of the time-extended signature evaluated at the final time.

The proof of Proposition 3.6 can be found in Appendix A.1.

On the subset of continuous paths the $J_{1}$-metric corresponds to the sup one defined via

Therefore, Theorem 3.6 states in particular that $d_{\infty}$-continuous functionals of continuous paths can be approximated, uniformly on compact sets, by linear functionals of the time-extended signature evaluated at the final point. For different versions of approximations theorems for continuous functionals of continuous paths we refer to Levin et al. 2013; Kiraly and Oberhauser 2019; Lyons et al. 2020.

Requiring the set $K$ to be bounded with respect to the $p$-variation norm is not redundant. For instance, on the subset of continuous paths the sup distance $d_{\infty}$ is dominated by the $p$-variation one. This implies that not every $J_{1}$-compact set is bounded with respect to the $p$-variation norm.

Note that the continuous functional $F$ in Corollary 3.8 can of course be a functional that only involves $X$ and not its Marcus lift.

As a direct result of the proof of Proposition 3.6, the following statement concerning uniqueness of the signature of time-extended w.g. càdlàg rough paths holds true.

This result follows directly from the proof of condition (iii) in Proposition 3.6. The reasoning therein does not depend on the set $K$, nor on the metric $\sigma_{\infty}$, showing that a time-extended w.g. càdlàg $p$-rough path is completely characterized by its signature evaluated at the final time $1$. ∎

We now present the main result of the first part of the paper: an extended version of Proposition 3.6 which leads to an approximation of paths functionals not only on compact sets of paths but also uniformly in time. For its formulation, we first need to introduce the notion of a stopped w.g. càdlàg rough path.

The proof of the following theorem can be found in Appendix A.2.

A similar result in the setting of continuous rough paths has been formulated in Kalsi et al. 2020, where, however, different techniques are used. Indeed, in contrast to our approach, Kalsi et al. 2020 consider non-anticipative functionals on the space of continuous rough paths equipped with an appropriate topology (fully clarified in Bayer et al. 2021).

Similarly as in Corollary 3.8, in the semimartingales setting, Theorem 3.13 reads as follows.

Finally, we conclude this section with some illustrative examples of path functionals for which Theorem 3.13 holds true.

(ii) Let $Y$ be the solution of the Marcus-type RDE

Therefore, solutions of Marcus-type RDEs can be approximated uniformly in time by linear functionals of the signature of the driving signal By Theorem 5.3 in Friz and Zhang 2017, the same type of argument applies to solutions of RDEs of forward-type. Notice in particular that this concept of RDE is in line with the Itô theory of càdlàg semimartingales (see e.g. Proposition 6.9 in Friz and Zhang 2017)..

Lévy-type signature models

In this section we introduce the class of Lévy-type signature models. The key object is a multidimensional process, which will have the role of encoding all the randomness of the model. Because of this, it will be called market’s primary process.

The next proposition states that the market’s primary process is a Lévy process.

where $h(x):=(0,0,x,x^{2},x^{3},\dots,x^{N})$.

Using the same numbering for the components, we get that $Y$ is a Lévy process with triplet $(b^{Y},C^{Y},0)$, where $b^{Y}=(1,0,\dots,0)$, $C^{Y}_{i,j}=0$ for $i\neq 0$ or $j\neq 0$ and $C^{Y}_{0,0}=1$.

Moreover, since $(h_{*}F)(\{0\})=F(\{0\})=0,$ and

for a each multi-index $I$ with entries in $\{-1,0,1,\dots,N\}$.

The proof of the following result via polynomial technology can be found in Appendix B.1.

Let us restate expression (4.5) in a more explicit form. Observe that since

Applying once again the definition of the exponential function (4.3), we obtain that for $|I|>0$ it holds

for each $J=(j_{1},\ldots,j_{m})$ for $m\geq 3$. For example, setting $I=(1,3)$ we get that

Observe that following the argument of Lemma A.2 in Cuchiero et al. 2023b, we can use the i.i.d. increments of a Lévy process to get

This technical condition is needed in order to achieve an equivalent representation of the model, called sig-model representation, that will be derived in the next section. We refer to Remark 4.11 for a precise discussion on this.

2 Sig-model representation

For each multi-index $I$ and each $j\in\{-1,\ldots,d\}$ we consider the following multi-index transformation

The proof of the next proposition can be found in Appendix B.2.

Fix $j\geq 1$ and observe that by Proposition 4.10 it holds

The sig-model representation is a direct application of Proposition 4.10. We thus omit the proof of the next corollary.

The Lévy-type signature model defined in (4.6) admits the following sig-model representation

In Remark 4.11 we have explained why in order to achieve the sig-model representation one has to include some moments of the jump-measure $\mu$ in the definition of the primary process $X$ (see Definition 4.1). We now want to show that this necessary condition is no longer needed if the last component of $X$ consists of a standard Poisson process.

for some $\lambda>0$ where $\delta_{1}$ denotes the Dirac measure. Set then

and observe that for all $k\geq 2$ and $t\in$

Notice that to simplify the notation we do not compensate the sum of jumps in the jump-component of $X$. As a consequence, for $j=1$ the multi-index transformation introduced in Definition 4.9 simplifies to

It is important to notice that the improvement of tractability achieved in this setting comes with a loss in generality. In particular, if a jump occurs at time $t$ its size has to be

3 Pricing of sig-payoffs

In this section we tackle the problem of pricing a contingent claim in Lévy-type signature models. As an application of Proposition 3.6, we consider contingent claims whose payoff is represented as a linear functional of the signature of the time-extended price process (extended price process henceforth), i.e. so-called sig-payoffs, already analyzed in Lyons et al. 2020; Arribas et al. 2020; Cuchiero et al. 2023b and the references therein. In order to give a clear treatment of the problem, we start by computing the signature of the time-extended price process

Consider the price process defined as in equation (4.6). Let $M:=\sum_{i=0}^{n}(d+2)^{i}$ be the number of possible multi-indices with length at most $n$ and indices in $\{-1,0,1,\dots,d\}$. Observe that we are considering also the empty index $\emptyset$. Define

to be an ordered version of $\{\epsilon_{J},|J|\leq n\}$. For instance, we may consider the order with respect to the length first and alphabetically next. Then,

We are now ready to state the announced result. It will become clear later that this is the key result to solve analytically the pricing problem for sig-payoffs. The proof can be found in Appendix B.3, where a discussion on the necessity of the lower bound on $N$ is also provided.

Finally, we recall the notion of sig-payoffs as introduced in Lyons et al. 2020 and address the problem of computing their arbitrage free prices.

Observe that by Theorem 4.15, Proposition 4.2, and Proposition 4.4 the random variable $C^{S}$ given in (4.15) is square integrable.

Applying Theorem 4.15, we are now ready to provide a pricing formula for sig-payoffs.

be a sig-payoff with maturity $T>0$. If $N\geq m(nd+1)$, then, its price can be expressed by

4 Hedging of sig-payoffs

In the Lévy-type signature model the market is not complete. One risk management criterion is to minimize the hedging error in a mean square sense, i.e. to use a quadratic loss function. In this context losses and gains are treated in a symmetric manner and the criterion to be minimized can be either the squared hedging error at maturity or measured locally in time (see e.g. Schweizer 1991; Schweizer 1990; Schweizer 1995; Schweizer 2001, Pham 2000 and the references therein for an extensive analysis of these problems). In what follows, we focus on the first approach which leads to the construction of the so-called mean variance hedging strategy. It specifically consists in solving the optimization problem

where $[\,\cdot\,]^{pred}$ and $[\cdot,\cdot]^{pred}$ stand for predictable quadratic variation and covariation, respectively.

We report here some further observations about Theorem 4.20.

The above result gives a fully explicit formula for the mean variance strategy that hinges solely on the option and the model’s parameters as well as the signature of the market’s primary process $X$.

A contingent claim $H^{S}$ is attainable if it almost surely admits the representation

5 Measure transformations for Lévy-type signature models

Moreover, since we are dealing with the augmented filtration generated by $W$ and $\mu$, the martingale $Z$ is the solution of the following SDE

The proof of the following proposition can be found in Appendix B.5

Appendix A Proofs of Section 3

For notational convenience, in the proofs of Section 3 we focus only on the set of time-extended w.g. càdlàg $p$-rough paths, for $p\in[2,3)$. However, as pointed out in Remark A.1 (i) below, the exact same reasoning can also be applied when working with time-extended w.g. càdlàg $p$-rough paths, for $p\in[1,2)$.

This result follows by the Stone-Weierstrass theorem applied to the set $A$ given by

Therefore, we must prove that $A$ satisfies the following conditions:

it is a sub-algebra and contains a non-zero constant function,

Next, observe that for a generic metric space $(E,d)$, the evaluation map

is continuous. More precisely, let $X\in D(,E)$ and let $(X^{n})_{n}\subset D(,E)$ be a sequence such that $X^{n}\rightarrow X$ as $n\rightarrow\infty$ with respect to the $J_{1}$-topology. Then, there exists $(\lambda^{n})_{n}\subset\Lambda$ such that $|\lambda^{n}|\rightarrow 0$ and $\sup_{s\in}d(X^{n}_{\lambda^{n}(s)},X_{s})\rightarrow 0$ as $n\rightarrow\infty$ (see e.g. Chapter 3 in Billingsley 1999), and in particular $d(X^{n}_{\lambda^{n}(1)},X_{1})=d(X^{n}_{1},X_{1})\rightarrow 0$, as $n\rightarrow\infty$.

In the present setting this result yields that continuity of (A.2) implies continuity of

both starting in $0$ by Assumption 2.2, we then deduce that $\langle\epsilon_{i},\widehat{\mathbf{X}}\rangle=\langle\epsilon_{i},\widehat{\mathbf{Y}}\rangle$.

Similarly, choosing $i=-1$, equation (A.3) yields that

The same reasoning as above yields then that $\langle\epsilon_{(k,j)},\widehat{\mathbf{X}}\rangle=\langle\epsilon_{(k,j)},\widehat{\mathbf{Y}}\rangle$ and thus $\widehat{\mathbf{X}}=\widehat{\mathbf{Y}}$.

We presented the proof only for $p\in[2,3)$. However, an analogous and simpler result can be given for continuous functionals of w.g. càdlàg $p$-rough path, with $p\in[1,2)$. Indeed, the only difference in this setting concerns the interpretation of the Marcus RDE (2.10) as the differential equation (C.4).

On the subset of continuous paths the continuity of the solution map (A.2) can be deduced by standard theorems in rough paths theory. We refer to e.g. Theorem 10.26 and Corollary 10.28 in Friz and Victoir 2010, where continuity with respect to the variation metric $d_{p}$ (2.6) is also provided.

A.2 Proof of Theorem 3.13

We start by summarizing the reasoning used in the proof of Theorem 3.13. Recall that we analyze here only the case of $p\in[2,3)$ but as explained in Remark A.1 (i), the proof can be easily adapted to $p\in[1,2)$.

Consider the set of stopped w.g. rough paths given by

Throughout the proof, for all $t\in$, $\widehat{\mathbf{X}}\in K$, we set

By definition of $\widehat{H}$ and since $\|\widecheck{\mathbf{X}}^{t}_{t,s}\|_{CC}=(s-t)$ for each $s\in[t,1]$, we can then compute

This shows that (A.4) is satisfied by $\widehat{H}$. Similarly, observe that $d_{CC}(\widecheck{{\mathbf{X}}}^{t}_{s},\widecheck{{\mathbf{X}}}^{t}_{u})=(u-s)$ for each $t\leq s\leq u$ and

for each $s\leq t\leq u$. We can then compute

proving that $\widehat{H}$ satisfies (A.5) too and is thus relatively compact.

Let then $\mathcal{D}$ be a fixed partition of $$. Observe that since by assumption$K$is bounded with respect to$\|\,\cdot\,\|_{p-var}$,$\widehat{H}^{1}$and thus$\widehat{H}$ are bounded as well. We can thus compute

where the second equality follows by the invariance of the $p$-variation norm with respect to time-reparametrization. Taking the sup over all the partition we get that $cl(\widehat{H})$ is bounded with respect to the $p$-variation norm.

implying in particular that for all $|I|\leq 2$

Finally, observe that for fixed $s\in$ and for all continuity points $u\in\setminus\{s\}$ of $\mathbf{Y}$,

Thus, the dominated convergence theorem yields

By definition of the path functional $\widehat{F}$ and Notation A.3, we then conclude that

Appendix B Proofs of Section 4

The operator $\mathcal{A}$ is a jump-diffusion operator whose coefficients

By Theorem 2.5 in Filipović and Larsson 2020 we can then conclude that

B.2 Proof of Proposition 4.10

The claim is clear for $I=\emptyset$. If $I\neq\emptyset$, recall from equation (4.2) that

In particular, observe that for each $j>0$ it holds

We already showed that the claim holds for $r=0$. Supposing that it is true for $r-1$, we get

where in $(\star)$ we set $\epsilon_{I_{122}}=\epsilon_{I_{12}}\otimes\epsilon_{I_{2}}$. Since $\alpha(r,k)=0$ for $k>r$, we can conclude that

B.3 Proof of Theorem 4.15

Therefore, as a direct application of Proposition 4.10 and the induction hypothesis, with the notation of (B.2) we can write

The necessity of the condition $N\geq|I|(nd+1)$ follows from the fact that

B.4 Proof of Theorem 4.20

Moreover using that $X$ is a Lévy process by Proposition 4.2 and Remark 4.6 we also get that

Observe that in the first step we used the result of Theorem 4.15. Since for each $|H|\geq 1$ by Remark 4.5 it holds

B.5 Proof of Proposition 4.23

First of all, observe that by Theorem 9 in Protter and Shimbo 2006, the pair $(f(t),g(x))$ is such that the solution of the SDE (4.20) is a true martingale.

Note that since $S(J_{2})>2$ we have that $\Delta X_{s}^{S(J_{2})}=\Delta Y_{s}^{S(J_{2})}$. Moreover,

For notational convenience, the function $f$ in (4.22) has been defined via multi-indices $I$ with indices in the set $\{-1,0,1,2,\dots,N\}\setminus\{0,1\}$. However, observe that the result of Proposition 4.23 still holds if we consider the larger set of multi-indices with indices in $\{-1,0,1,2,\dots,N\}\setminus\{0\}$.

Appendix C Auxiliary results on the signature of càdlàg rough paths

In this section, we present some auxiliary results on the signature of w.g. càdlàg rough paths. The main purpose is to give a more comprehensive account of the Marcus rough differential equation (2.10) solved by the minimal jump extension (truncated signature) of a w.g. càdlàg rough path. To this end, we first elaborate on the different definitions of w.g. rough paths often used in literature, then after some brief reminders about the rough integral, we conclude by providing an explicit form of equation (2.10) along with some properties of the signature of time-extended w.g. càdlàg rough paths.

In Definition 2.1, we introduced the notion of w.g. càdlàg $p$-rough path as a group-valued path with a predetermined regularity. More often, however, especially when the main interest is in rough integration and differential equations, it is presented following the line below.

C.2 Rough integration

The rough integral of a controlled rough path $(Y,Y^{\prime})$ with respect to ${X}$ is defined as follows: for $t\in$

where $\mathcal{D}$ denotes a partition of $[0,t]$ and the limit is understood in Refinement Riemann-Stieltjes (RRS) sense, as introduced in Definition 1 in Friz and Shekhar 2017.

The convergence in equation (C.2) has been proved to hold also in Mesh Riemann-Stieltjes sense (see Definition 1.1 and Proposition 2.6 in Friz and Zhang 2017).

C.3 Marcus rough differential equation

We are now ready to analyze in detail the explicit form of equation (2.10).

for each multi-index $I=(i_{1},\dots,i_{|I|})\in\{1,\dots,d\}^{|I|}$ with $3\leq|I|\leq N$. Finally, notice that, if $\mathbf{X}$ is additionally Marcus-like, using (2.3) equation (2.11) simplifies to

and the summation term can be expressed only by means of jumps at the first level of $\mathbf{X}$.

for each multi-index $I=(i_{1},\dots,i_{|I|})\in\{1,\dots,d\}^{|I|}$ with $2\leq|I|\leq N$.

Finally, we conclude this section by highlighting a fundamental property of the signature of time-extended w.g. rough paths. It is based on the following key lemma concerning the consistency of the Young and the rough integral.

where the integral on the right hand side is a Young integral.

By Proposition 25 in Friz and Shekhar 2017

Therefore, the claim follows if we show that

This is obtained by a slight modification of the proof of Theorem 35 in Friz and Shekhar 2017. Using their notation, observe that in this case $[2,3)\ni p\neq q=1$. Therefore, one needs to choose $2\leq p<p^{\prime}<3$ and $1<q^{\prime}$ such that $\frac{1}{p^{\prime}}+\frac{1}{q^{\prime}}>1$, and the Young integral is still well defined. Then, the result follows by applying the same reasoning. ∎

The proof follows by the definition of time-extended w.g. càdlàg rough path, equation (C.3) and Lemma C.4. ∎

References