From Doob's maximal identity to Azema supermartingale

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

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