From Doob's maximal identity to Azema supermartingale
  We characterize the random times $\rho$ whose Azema supermartingales $Z^\rho$ take the form $Z^\rho=U/U^*$ for some non negative local martingales $U$ starting from 1 vanishing at infinity, where $U^*$ denotes the running maximum process of $U$. 
Shiqi Song
math.PR