Convergence of the Adapted Smoothed Empirical Measures

Abstract: The adapted Wasserstein 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 adapted Wasserstein distance are bottlenecked by the failure of empirical measures 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 and adapted empirical measures. However, both approaches have limitations. Specifically, smoothed empirical measures lack comprehensive convergence results, whereas adapted empirical measures 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 smoothed empirical measures. We then introduce a smoothed version for adapted empirical measures, which provide as many distinct samples as desired. We refer them as adapted smoothed empirical measures 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 and its bandwidth effects. Both results are novel and their proof techniques could be of independent interest.