A definition and some characteristic properties of pseudo-stopping times

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.