What is the point of using rough path expected signature to characterize the law of а stochastic process when the cumulant generating function is known ($\log\mathbb{E}[e^{i\theta X(t)}]$)?
Since an average of group-like elements is not a group-like element, an expected signature (of a sample of paths) is not a signature, which seems to imply that expected signatures do not enjoy a nice geometric interpretation as signatures do. It seems that two main applications of expected signatures are the characterization of the law of a process (if moments completely describe the distribution) and the approximation of continuous functions on a collection of paths as linear functions on the expected signature of that collection of paths. If one doesn't need the latter use case, how is the description of a stochastic processes in terms of expected signatures qualitatively different from the classic characteristic function description? Is there something else that expected signatures can tell besides the moments?
For example, in case of Levy processes, characteristic function of a process has some very useful information about the process besides moments in light of Levy–Khintchine representation. It tells what parts this Levy process is composed of, namely its linear drift, Brownian motion, and Lévy jump process.
By expected signature I mean $ \Big( \mathbb{E} \Big[ \int dX^{\otimes n} \Big] \Big)_{n\geq 0} = \Big( \mathbb{E} \Big[\int_0^T \int_0^{r_n} \cdots \int_0^{r_2} dX_{r_1} \otimes \cdots \otimes dX_{r_n} \Big]\Big)_{n\geq 0} $ described in [LM22] (Chapters 6 and 13), [CO22], [LSDBL21], [BDMN20].
END OF THE ORIGINAL POST
EDIT: Reply to JeremyFR
Thank you, @JeremyFR, for your response! However, I still struggle to see advantages of expected signatures vis-à-vis CGF when applied to common 1-D Levy processes.
Let us start with Brownian motion.
Let B be a standard one-dimensional Brownian motion calculated up to some fixed time T > 0. Then B has the following expected signature (Fawcett’s formula):
\begin{align*} \mathbb{E}[S(B_{[0,T]})] &= \exp(\frac{T}{2} e_i \otimes e_i ) \\ &= 1+ \frac{T}{2} e_i ^ {\otimes 2} + \frac{1}{2!} \frac{T^2}{4} e_i ^ {\otimes 4} + \dots \end{align*}
It is also known that it’s moment-generating function is \begin{align*} M_{B_T}(\theta) &= \exp( \frac{T}{2} \theta^2) \\ &= 1 + \frac{T}{2} \theta^2 + \frac{1}{2!} \frac{T^2}{4} \theta^4 + \dots \end{align*} It seems that both expressions carry identical information once we take apart the notation. In the case of Brownian motion, it seems that one hardly can argue that one approach is better than the other because the results are semantically equal.
Now let’s move on to a more general case of d-dimensional Lévy process with triplet (a, b, K). It is shown in [FS14] (Theorem 43), that the expected signature of such process is awfully similar to Lévy–Kintchine formula, \begin{align*} \mathbb{E}[S(X)_{[0,T]}] =\exp\Big [ T \Big (b + \frac{1}{2} a + \int (exp(y) – 1 – y 1 _{|y| < 1}) K(dy) \Big ) \Big] \in T((\mathbb{R}^d)). \end{align*} In light of this result, one cannot help but wonder what is the point of using expected signatures for time series analysis from the practitioner’s perspective in applications involving common types of time series, which are usually 1D Levy Processes, that frequently occur in finance and energy / load research, given that the information in expected signature is essentially identical to the one provided by characteristic function? Granted, there are more exotic cases that require more general expression for expected signature [FHT21] but that hardly looks like something arising frequently in day-to-day applications.
Here are differences between expected signatures and cumulant generating functions (/ characteristic functions), by which you mean the CGF at every t.
The CGF is determined by the marginal laws of the process at each time. So two processes whose laws are different but which have the same marginals will agree on the CGF but not on the expected signature.
The expected signature is invariant to time warping but the CGF is not.
Signatures correspond to paths. Expected signatures correspond to distributions on paths. It is therefore very natural that expected signatures take values in a much larger set than signatures. If an expected signature is grouplike (i.e. is a signature) then it corresponds to a deterministic law, i.e. the one whose value is the corresponding path with probability 1.