@article{ECP1582,
author = {Adam Osekowski},
title = {Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {15},
year = {2010},
keywords = {Submartingale; Weak-type inequality; Strong differential subordination},
abstract = {Let $f=(f_n)_{n\geq 0}$ be a nonnegative submartingale starting from $x$ and let $g=(g_n)_{n\geq 0}$ be a sequence starting from $y$ and satisfying $$|dg_n|\leq |df_n|,\quad |\mathbb{E}(dg_n|\mathcal{F}_{n-1})|\leq \mathbb{E}(df_n|\mathcal{F}_{n-1})$$ for $n\geq 1$. We determine the best universal constant $U(x,y)$ such that $$\mathbb{P}(\sup_ng_n\geq 0)\leq ||f||_1+U(x,y).$$ As an application, we deduce a sharp weak type $(1,1)$ inequality for the one-sided maximal function of $g$ and determine, for any $t\in [0,1]$ and $\beta\in\mathbb{R}$, the number $$ L(x,y,t,\beta)=\inf\{||f||_1: \mathbb{P}(\sup_ng_n\geq \beta)\geq t\}.$$ The estimates above yield analogous statements for stochastic integrals in which the integrator is a nonnegative submartingale. The results extend some earlier work of Burkholder and Choi in the martingale setting.},
pages = {no. 46, 508-521},
issn = {1083-589X},
doi = {10.1214/ECP.v15-1582},
url = {http://ecp.ejpecp.org/article/view/1582}}