From Doob's maximal identity to Azema supermartingale

TL;DR

This paper characterizes specific random times whose Azema supermartingales can be expressed as a ratio involving a non-negative local martingale and its maximum, linking Doob's maximal identity to Azema supermartingales.

Contribution

It introduces a characterization of random times with Azema supermartingales of a particular form, extending the understanding of their structure and relation to local martingales.

Findings

Identifies conditions under which Azema supermartingales have the form U/U*

Connects Doob's maximal identity with Azema supermartingale characterization

Provides a new perspective on the structure of certain random times

Abstract

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 $U$.