Global universality via discrete-time signatures
🤖 AI Summary
This work investigates the universal approximation of path-dependent functionals and stochastic differential equations in the space of piecewise linear paths. By introducing linear functionals of discrete-time path signatures, the authors establish, for the first time, their density in this path space under both $L^p$ and weighted norms, provided the weight function is integrable—thereby proving a global universal approximation theorem. The approach integrates path signature theory, functional analysis, and interpolation techniques for stochastic processes, and verifies that the piecewise linear interpolation of Brownian motion satisfies the required regularity conditions. This framework enables effective $L^p$ approximation of path-dependent functionals, random ordinary differential equations, and stochastic differential equations driven by Brownian motion.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
universal approximation
piecewise linear paths
Brownian motion
path-dependent functionals
stochastic differential equations
Innovation
Methods, ideas, or system contributions that make the work stand out.
signature
universal approximation
piecewise linear paths
Brownian motion
L^p-approximation
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