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On expected signatures and signature cumulants in semimartingale models

Peter K. Friz, Paul P. Hager, Nikolas Tapia

TL;DR

The paper develops a unified functional-analytic framework for expected signatures and their cumulants of semimartingales, encompassing both discrete and continuous settings. By deriving central backward functional equations for the conditional expected signature $\pmb{\mu}$ and the cumulants $\pmb{\kappa}$, it provides recursive tensor-level formulas for signature moments and cumulants, organized via diamond products. The multivariate extension uses the symmetric algebra to yield multivariate moments and cumulants, with corresponding continuous and discrete diamond relations. Applications to time-inhomogeneous Lévy processes and Brownian rough paths illustrate the practical computability and flexibility of the approach, including explicit formulas like backward equations for $\mathfrak{y}(t)$ and closed-form expressions for Brownian rough paths. Overall, the framework offers computationally tractable paths to characterize the law of the signature, with potential impact on stochastic modeling, numerical signatures in ML, and rough-path analysis.

Abstract

The concept of signatures and expected signatures is vital in data science, especially for sequential data analysis. The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors [Unified signature cumulants and generalized Magnus expansions, FoM Sigma '22] we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity.

On expected signatures and signature cumulants in semimartingale models

TL;DR

The paper develops a unified functional-analytic framework for expected signatures and their cumulants of semimartingales, encompassing both discrete and continuous settings. By deriving central backward functional equations for the conditional expected signature and the cumulants , it provides recursive tensor-level formulas for signature moments and cumulants, organized via diamond products. The multivariate extension uses the symmetric algebra to yield multivariate moments and cumulants, with corresponding continuous and discrete diamond relations. Applications to time-inhomogeneous Lévy processes and Brownian rough paths illustrate the practical computability and flexibility of the approach, including explicit formulas like backward equations for and closed-form expressions for Brownian rough paths. Overall, the framework offers computationally tractable paths to characterize the law of the signature, with potential impact on stochastic modeling, numerical signatures in ML, and rough-path analysis.

Abstract

The concept of signatures and expected signatures is vital in data science, especially for sequential data analysis. The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors [Unified signature cumulants and generalized Magnus expansions, FoM Sigma '22] we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity.
Paper Structure (24 sections, 18 theorems, 166 equations, 2 figures)

This paper contains 24 sections, 18 theorems, 166 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The tensor algebra and tensor series
  4. Semimartingales
  5. Generalized signatures
  6. Expected signatures and signature cumulants
  7. Functional equations for the expected signature
  8. Discrete processes
  9. The continuous case
  10. Recursive formulas and diamond products
  11. Recursive formula for signature moments
  12. Recursive formula for signature cumulants
  13. Diamond Products
  14. Remark on tree representation
  15. Multivariate Moments and Cumulants
  16. ...and 9 more sections

Key Result

Proposition 2.1

Suppose $\mathbf{X}\in\mathscr{S}^c(\mathcal{T}_0)$. For every $s \in [0,T]$ and $\mathbf{s} \in \mathcal{T}_1$, equation eq:stratonovich_sig has a unique global solution on $\mathcal{T}_1$ starting from $\mathbf{S}_s=\mathbf{s}$.

Figures (2)

  • Figure 1: An overview of the different formulas and corresponding recursions (in Figure \ref{['fig:cube.recur']}) with the relation to recent and classical results from the literature. Recall that $\mathscr{S}^{c}$ stands for continuous semimartingales and $\mathscr{V}^{c}$ for continuous processes of finite variation. Here, "trivial" refers to the linear equation $\mathrm{d}{\hat{\mathbf{S}}}=\hat{\mathbf{S}}\mathrm{d}{\hat{\mathbf{X}}}$ with $\hat{\mathbf{S}}_0=1\in\mathcal{S}$ in the commutative case $\hat{\mathbf{X}}\in\mathscr{V}^c(\mathcal{S})$, which is solved by $e^{\hat{\mathbf{X}}-\hat{\mathbf{X}}_0}$ with logarithm $\hat{\mathbf{X}}-\hat{\mathbf{X}}_0$.
  • Figure 2: Accompanying recursions to Figure \ref{['fig:theorems']}.

Theorems & Definitions (44)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • Remark 2.7
  • ...and 34 more