Global universal approximation with Brownian signatures
This work provides a foundational mathematical framework for approximating stochastic processes, which is incremental in extending universal approximation to rough path spaces.
The paper tackles the problem of approximating general non-anticipative functionals on rough path spaces, establishing L^p-type universal approximation theorems that show linear functionals on signatures of time-extended rough paths are dense. As a result, it proves that linear functionals on the signature of time-extended Brownian motion can approximate any p-integrable stochastic process adapted to the Brownian filtration, including solutions to stochastic differential equations.
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.