On expected signatures and signature cumulants in semimartingale models

TL;DR

This paper explores the computation of expected signatures and signature cumulants in semimartingale models, providing new formulas and demonstrating how log-signatures simplify the analysis of sequential data in stochastic processes.

Contribution

It introduces new formulas for expected signatures in semimartingale models and highlights the use of log-signatures to reduce complexity in data analysis.

Findings

New formulas for expected signatures in semimartingale models

Log-signatures significantly reduce computational complexity

Expected signatures uniquely determine the law of the process

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.