Global universal approximation with Brownian signatures

• URL: http://arxiv.org/abs/2512.16396v1
• Date: Thu, 18 Dec 2025 10:49:20 GMT
• Title: Global universal approximation with Brownian signatures
• Abstract summary: We show that linear functionals acting on signatures of time-extended rough paths are dense with respect to an $Lp$-distance.<n>We demonstrate that these $Lp$-type universal approximation theorems apply in particular to Brownian motion.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We establish $L^p$-type universal approximation theorems for general 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 an $L^p$-distance. To that end, we derive global universal approximation theorems for weighted rough path spaces. We demonstrate that these $L^p$-type universal approximation theorems apply in particular to 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.

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