Global universal approximation with Brownian signatures

Abstract:We establish $L^p$-universal approximation theorems for general path-dependent and non-anticipative functionals on suitable rough path spaces, showing that linear functionals acting on signatures of time-extended rough paths are dense with respect to the $L^p$-distance. To that end, we derive global universal approximation theorems for weighted rough path spaces. We demonstrate that these $L^p$-universal approximation theorems apply to Gaussian processes, in particular, to fractional Brownian motion. As a consequence, linear functionals on the signature of the time-extended Brownian motion can approximate any $p$-integrable stochastic process adapted to the Brownian filtration, including solutions to stochastic differential equations.

Probability (math.PR); Machine Learning (cs.LG); Mathematical Finance (q-fin.MF)

arXiv:2512.16396 [math.PR]

(or arXiv:2512.16396v3 [math.PR] for this version)

https://doi.org/10.48550/arXiv.2512.16396 Focus to learn more arXiv-issued DOI via DataCite

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