Convergence of adapted smoothed empirical measures

Abstract

The adapted Wasserstein distance ($\mathrm{AW}$-distance) controls the calibration errors of optimal values in various stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. However, statistical aspects of the $\mathrm{AW}$-distance are bottlenecked by the failure of empirical measures (Emp) to converge under this distance. Kernel smoothing and adapted projection have been introduced to construct converging substitutes of empirical measures, known respectively as smoothed empirical measures ($S$-Emp) and adapted empirical measures ($A$-Emp). However, both approaches have limitations. Specifically, $S$-Emp lack comprehensive convergence results, whereas $A$-Emp in practical applications lead to fewer distinct samples compared to standard empirical measures.

In this work, we address both of the aforementioned issues. First, we develop comprehensive convergence results of $S$-Emp. We then introduce a smoothed version for $A$-Emp, which provide as many distinct samples as desired. We refer them as $\mathrm{AS}$-Emp and establish their convergence in mean, deviation and almost sure convergence. The convergence estimation incorporates two results: the empirical analysis of the smoothed adapted Wasserstein distance (${\mathrm{AW}}^{\left(\sigma \right)}$-distance) and its bandwidth effects. Both results are novel and their proof techniques could be of independent interest.