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

Abstract

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.