Global universality via discrete-time signatures

TL;DR

This paper proves that signatures of piecewise linear paths can densely approximate a wide class of path-dependent functionals, including solutions to stochastic differential equations driven by Brownian motion, under certain integrability conditions.

Contribution

It establishes universal approximation theorems for signatures of piecewise linear paths on path spaces, extending the applicability of signature methods in stochastic analysis.

Findings

Signatures of piecewise linear paths are dense in relevant function spaces.

Piecewise linear interpolations of Brownian motion satisfy the integrability condition.

The results enable $L^p$-approximation of path-dependent functionals and SDE solutions.

Abstract

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.