2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71647Recently, D. Williams \cite{williams} gave an explicit example of a random time $ρ$ associated with Brownian motion such that $ρ$ is not a stopping time but $\mathbb{E}M_ρ=\mathbb{E}M_{0}$ for every bounded martingale $M$. The aim of this paper is to give some characterizations for such random times, which we call pseudo-stopping times, and to construct further examples, using techniques of progressive enlargements of filtrations.30 pages; to appear in Annals of ProbabilityProbability60G07; 60G40; 60G44text