Global universality via discrete-time signatures

The paper proves global universal approximation theorems for non‑anticipative and general path‑dependent functionals on spaces of piecewise linear paths, showing that linear functionals of the corresponding signatures are dense in $L^p$ and weighted norms. It demonstrates that these results apply to piecewise linear interpolations of Gaussian processes, including Brownian motion and fractional Brownian motion with Hurst parameter $H>1/4$, and provides quantitative convergence rates for signatures of such approximations toward their continuous‑time counterparts. Consequently, the authors obtain $L^p$‑approximation results for path‑dependent functionals of Gaussian processes and for random ordinary and stochastic differential equations driven by Brownian motion.

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