Global universal approximation with Brownian signatures

We establish Lp-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 Lp-distance. To that end, we derive global universal approximation theorems for weighted rough path spaces. We demonstrate that these Lp-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.