From Doob's maximal identity to Azema supermartingale

We characterize the random times $\rho$ whose Azema supermartingales $Z^{\rho }$ take the form $Z^{\rho }=U\mathrm{/}{U}^{\ast }$ for some non negative local martingales $U$ starting from 1 vanishing at infinity, where $U^{\ast }$ denotes the running maximum process of $U$.