2022/02/01 by Chad Giusti, Darrick Lee, Giusti, Chad +5 · 2 citations
Economics, Econometrics and Finance · Mathematics · #28C20 #55U15 #60L10 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2202.00491
openalex publication_date 2022/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A common approach for describing classes of functions and probability measures on a topological space X is to construct a suitable map Φ from X into a vector space, where linear methods can be applied to address both problems. The case where X is a space of paths [0,1] → ℝn and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where X is a space of maps [0,1]d → ℝn for any d ∈ ℕ, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to d ≥ 2. The key ingredient to our approach is topological; in particular, our starting point is a generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.