  Recently, D. Williams \cite{williams} gave an explicit example of a random
time $\rho $ associated with Brownian motion such that $\rho $ is not a
stopping time but $\mathbb{E}M_{\rho}=\mathbb{E}M_{0}$ 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.
