A generalization of Doob's maximal identity

TL;DR

This paper extends Doob's maximal identity to a broader class of continuous nonnegative local submartingales, providing new decompositions and laws related to their maxima and last passage times.

Contribution

It introduces a generalized form of Doob's maximal identity for specific submartingales and derives new laws for maxima and last passage times of continuous local martingales.

Findings

Derived a multiplicative decomposition for the Azéma supermartingale.

Established the law of the maximum for certain continuous nonnegative local martingales.

Analyzed the informational content of non-stopping times related to these processes.

Abstract

In this paper, using martingale techniques, we prove a generalization of Doob's maximal identity in the setting of continuous nonnegative local submartingales $(X_{t})$ of the form: $X_{t}=N_{t}+A_{t}$, where the measure $(dA_{t})$ is carried by the set $\left\{t: X_{t}=0\right\}$. In particular, we give a multiplicative decomposition for the Az\'ema supermartingale associated with some last passage times related to such processes and we prove that these non-stopping times contain very useful information. As a consequence, we obtain the law of the maximum of a continuous nonnegative local martingale $(M_t)$ which satisfies $M_\infty=\psi(\sup_{t\geq0}M_t)$ for some measurable function $\psi$ as well as the law of the last time this maximum is reached.