The Annals of Probability

Optimal Stopping and Best Constants for Doob-like Inequalities I: The Case $p = 1$

S. D. Jacka

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This paper establishes the best constant $c_q$ appearing in inequalities of the form $\mathbb{E}S_\infty \leq c_q\sup_{t\geq 0}\|M_t\|_q,$ where $M$ is an arbitrary nonnegative submartingale and $S_t = \sup_{s\leq t}M_s.$ The method of proof is via the Lagrangian for a version of the problem $\sup_\tau\mathbb{E}\{\lambda S_t - \lambda^qM^q_t\},$ where $M \equiv |B|, B$ a Brownian motion. More general inequalities of the form $\mathbb{E}S_\infty \leq C_\Phi\sup_{t\geq 0}\|M_t\|_\Phi$ and $\mathbb{E}S_\infty \leq C_\Phi\sup_{t\geq 0}\||M_t\||_\Phi$ (where $\|\cdot\|_\Phi$ and $\||\cdot\||_\Phi$ are, respectively, the Luxemburg norm and its dual, the Orlicz norm, associated with a Young function $\Phi$) are established under suitable conditions on $\Phi$. A simple proof of the John-Nirenberg inequality for martingales is given as an application.

Article information

Ann. Probab., Volume 19, Number 4 (1991), 1798-1821.

First available in Project Euclid: 19 April 2007

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Primary: 60G07: General theory of processes
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J65: Brownian motion [See also 58J65] 60G42: Martingales with discrete parameter 42B25: Maximal functions, Littlewood-Paley theory 42B30: $H^p$-spaces

Brownian motion martingale inequalities optimal stopping Lagrangian Luxemburg norm BMO convex closure greatest convex minorant


Jacka, S. D. Optimal Stopping and Best Constants for Doob-like Inequalities I: The Case $p = 1$. Ann. Probab. 19 (1991), no. 4, 1798--1821. doi:10.1214/aop/1176990237.

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