Exponentially Fading Memory Signature

TL;DR

The paper introduces the exponentially fading memory (EFM) signature, a time-invariant, mean-reverting transformation of paths that enhances time-series analysis by providing a stationary, well-behaved alternative to classical signatures.

Contribution

It develops the EFM-signature using rough path theory, establishing its algebraic, analytical, and probabilistic properties, including stationarity, ergodicity, and explicit formulas for expectations.

Findings

EFM-signature is time-invariant and mean-reverting.

Evolved as a group-valued Ornstein-Uhlenbeck process.

Proved stationarity, Markov property, and exponential ergodicity.

Abstract

We introduce the exponentially fading memory (EFM) signature, a time-invariant transformation of an infinite (possibly rough) path that serves as a mean-reverting analogue of the classical path signature. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity. The EFM-signature retains many of the key algebraic and analytical properties of classical signatures, including a suitably modified Chen identity, the linearization property, path-determinacy, and the universal approximation property. From the probabilistic perspective, the EFM-signature provides a "stationarized" representation, making it particularly well-suited for time-series analysis and signal processing overcoming the shortcomings of the standard signature. In particular, the EFM-signature of time-augmented Brownian motion evolves as a group-valued Ornstein-Uhlenbeck process. We establish its stationarity, Markov property, and exponential ergodicity in the Wasserstein distance, and we derive an explicit formula \`a la Fawcett for its expected value in terms of Magnus expansions. We also study linear combinations of EFM-signature elements and the computation of associated characteristic functions in terms of a mean-reverting infinite dimensional Riccati equation.