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October, 1991 Optimal Stopping and Best Constants for Doob-like Inequalities I: The Case $p = 1$
S. D. Jacka
Ann. Probab. 19(4): 1798-1821 (October, 1991). DOI: 10.1214/aop/1176990237


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.


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S. D. Jacka. "Optimal Stopping and Best Constants for Doob-like Inequalities I: The Case $p = 1$." Ann. Probab. 19 (4) 1798 - 1821, October, 1991.


Published: October, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0796.60050
MathSciNet: MR1127729
Digital Object Identifier: 10.1214/aop/1176990237

Primary: 60G07
Secondary: 42B25 , 42B30 , 60G40 , 60G42 , 60J65

Keywords: BMO , Brownian motion , convex closure , greatest convex minorant , Lagrangian , Luxemburg norm , martingale inequalities , Optimal stopping

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 4 • October, 1991
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