Characterization of max-continuous local martingales vanishing at infinity

TL;DR

This paper characterizes non-negative local martingales with continuous supremum that vanish at infinity, identifying the random times of their maxima, extending previous results to general filtrations.

Contribution

It generalizes existing characterizations of local martingales to broader filtrations, providing new insights into their maximum times.

Findings

Characterization of non-negative local martingales with continuous supremum

Identification of maximum times via random times

Extension of previous results to general filtrations

Abstract

We provide a characterization of the family of non-negative local martingales that have continuous running supremum and vanish at infinity. This is done by describing the class of random times that identify the times of maximum of such processes. In this way we extend to the case of general filtrations a result proved by Nikeghbali and Yor [NY06] for continuous filtrations. Our generalization is complementary to the one presented by Kardaras [Kar14], and is obtained by means of similar tools.