Global universality via discrete-time signatures
This provides foundational mathematical tools for approximating complex path-dependent systems in stochastic analysis and machine learning.
The paper establishes global universal approximation theorems for linear functionals of signatures on piecewise linear paths, showing density under integrability conditions, and applies this to prove L^p-approximation results for path-dependent functionals and stochastic equations driven by Brownian motion.
We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition on the underlying weight function. As an application, we show that piecewise linear interpolations of Brownian motion satisfies this integrability condition. Consequently, we obtain $L^p$-approximation results for path-dependent functionals, random ordinary differential equations, and stochastic differential equations driven by Brownian motion.