Path-Signatures: Memory and Stationarity
Abstract
We explore the interplay between path-signatures, memory, and stationarity, highlighting their implications for machine learning, representation of stochastic processes and applications in mathematical finance. In a first part, we provide explicit series expansions to certain stochastic path-dependent integral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index H in (0, 1). Our expressions allow to disentangle an infinite dimensional Markovian structure. In addition they open the door to: (i) straightforward and simple approximation schemes that we illustrate numerically, (ii) representations of certain Fourier-Laplace transforms in terms of a non-standard infinite dimensional Riccati equation with important applications for pricing and hedging in quantitative finance. In a second part, we introduce a time-invariant version of the signature: the fading-memory signature, with powerful algebraic, analytic and probabilistic properties and applications to learning stationary relationships in time series. This is based on joint works with Paul Gassiat, Louis-Amand Gérard, Yuxing Huang, Dimitri Sotnikov.
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Eduardo Abi Jaber is a Professor of Applied Mathematics at Ecole Polytechnique. He defended his Habilitation à Diriger des Recherches in 2024 and his PhD in 2018. His research investigates the role of memory in quantitative finance, advancing the mathematical foundations of sophisticated tools such as Volterra processes and path signatures. Beyond theory, his work translates into practical solutions to key challenges in the field, including volatility modeling and portfolio optimization. Positioned at the crossroads of mathematics and finance, his research combines rigorous analysis, advanced modeling, bespoke numerical methods, and systematic validation against real-world data. Author of more than 40 papers, with publications in leading journals in applied probability and quantitative finance, Eduardo's contributions have been recognized with several prestigious awards, including the Amies Prize for the best CIFRE PhD thesis in applied mathematics (2019) and the Junior Scholar Award of the Bachelier Finance Society (2018). He has delivered over 100 invited talks worldwide. He serves as an Associate Editor for Mathematical Finance and the International Journal of Theoretical and Applied Finance, and co-organizes the internationally recognized Bachelier Seminar in Paris. Over the years, he has led a research group comprising more than 10 PhD students and postdoctoral researchers.