Hazard processes and martingale hazard processes

In this paper, we provide a solution to two problems which have been open in default time modeling in credit risk. We first show that if $\tau$ is an arbitrary random (default) time such that its Az\'ema's supermartingale $Z_t^\tau=\P(\tau>t|\F_t)$ is continuous, then $\tau$ avoids stopping times. We then disprove a conjecture about the equality between the hazard process and the martingale hazard process, which first appeared in \cite{jenbrutk1}, and we show how it should be modified to become a theorem. The pseudo-stopping times, introduced in \cite{AshkanYor}, appear as the most general class of random times for which these two processes are equal. We also show that these two processes always differ when $\tau$ is an honest time.