2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/208516In 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 $τ$ is an arbitrary random (default) time such that its Azéma's supermartingale $Z_t^τ=¶(τ>t|\F_t)$ is continuous, then $τ$ 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 $τ$ is an honest time.Risk ManagementProbability60G07, 60G44, 60G99text