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 à 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.
| Subjects: | Probability (math.PR) |
| Cite as: | arXiv:2507.03700 [math.PR] |
| (or arXiv:2507.03700v2 [math.PR] for this version) | |
| https://doi.org/10.48550/arXiv.2507.03700 arXiv-issued DOI via DataCite |
From: Dimitri Sotnikov [view email]
[v1]
Fri, 4 Jul 2025 16:30:59 UTC (481 KB)
[v2]
Fri, 22 May 2026 13:15:09 UTC (479 KB)
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