Unified Signature Cumulants and Generalized Magnus Expansions

TL;DR

This paper introduces a unified framework for signature cumulants and Magnus expansions, providing a general functional relation for expected signatures in semimartingale contexts with applications across finance and physics.

Contribution

It develops a universal functional relation linking expected signature cumulants and Magnus expansions, generalizing previous characteristic exponent results in a broad semimartingale setting.

Findings

Establishes a general functional relation for expected signature cumulants.

Highlights the role of Magnus expansions in computational algorithms.

Provides numerous examples illustrating the theoretical framework.

Abstract

The signature of a path can be described as its full non-commutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions; with motivation ranging from financial mathematics to statistical physics. From an affine process perspective, the functional relation may be interpreted as infinite-dimensional, non-commutative ("Hausdorff") variation of Riccati's equation. Many examples are given.