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.
From: Mihriban Ceylan [view email]
[v1]
Thu, 18 Dec 2025 10:49:20 UTC (25 KB)
[v2]
Mon, 6 Jul 2026 13:31:36 UTC (32 KB)
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