A generalization of Doob's maximal identity

Ashkan Nikeghbali

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arXiv · 0802.1317

A generalization of Doob's maximal identity

Ashkan Nikeghbali

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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 Azéma 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 $M_\infty=ψ(\sup_{t\geq0}M_t)$ for some measurable function $ψ$ as well as the law of the last time this maximum is reached.

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BibTeX RIS

Ashkan Nikeghbali. 2008-02-10. A generalization of Doob's maximal identity. https://arxiv.org/abs/0802.1317

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