Global universality via discrete-time signatures

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.

Probability (math.PR); Machine Learning (cs.LG); Mathematical Finance (q-fin.MF)

arXiv:2603.09773 [math.PR]

(or arXiv:2603.09773v2 [math.PR] for this version)

https://doi.org/10.48550/arXiv.2603.09773 Focus to learn more arXiv-issued DOI via DataCite

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