Year: 2026
Type: preprint
Abstract: We establish functional quantization and empirical Wasserstein rates for continuous Itô processes under the supremum norm. We assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $ρ$-moment for some $ρ>1$. Under this assumption, the $n$-point $L^q$ quantization error is at most $C(\log n)^{-1/2}$ for every $1\leq q<ρ$. The known Brownian lower bound shows that the exponent $1/2$ is optimal over this class. Our proof combines an adaptive dyadic time partition with localization according to the size of the integrands. A general transfer principle yields the sharp mean rate $(\log N)^{-1/2}$ for the $p$-Wasserstein distance between the empirical law of $N$ independent copies and their common path law, together with nonasymptotic deviation bounds, whenever $1\leq p<ρ$. Applications include empirical path-law estimates for path-dependent SDEs and a path-space limit-theory estimate for path-dependent McKean--Vlasov interacting particle systems with common noise.
Source: arXiv (Cornell University)
Authors: He, Xihao, Yuan, Fengyi
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7202156610