Global universality via discrete-time signatures

arXiv:2603.09773v2 Announce Type: replace-cross Abstract: We establish global universal approximation theorems for non-anticipative and general path-dependent functionals on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to L^p- and weighted norms. We verify that these approximation results are applicable to piecewise linear interpolations of a class of Gaussian processes, including Brownian motion and fractional Brownian motion with Hurst parameter H>1/4. Moreover, we derive quantitative convergence rates for signatures of piecewise linear approximations of Gaussian processes towards their continuous-time counterparts. Consequently, we obtain L^p-approximation results for path-dependent functionals of Gaussian processes, as well as for random ordinary differential equations and stochastic differential equations driven by Brownian motion.

Source: arXiv cs.LG | 2026-08-28