@article{ECP3151,
author = {Adam Osekowski},
title = {Maximal weak-type inequality for stochastic integrals},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {19},
year = {2014},
keywords = {Martingale;maximal;weak type inequality;best constant},
abstract = {Assume that $X$ is a real-valued martingale starting from $0$, $H$ is a predictable process with values in $[-1,1]$ and $Y$ is the stochastic integral of $H$ with respect to $X$. The paper contains the proofs of the following sharp weak-type estimates. (i) If $X$ has continuous paths, then $$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 2\mathbb{E} \sup_{t\geq 0}X_t.$$
(ii) If $X$ is arbitrary, then$$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 3.477977\ldots\mathbb{E} \sup_{t\geq 0}X_t.$$The proofs rest on Burkholder's method and exploits the existence of certain special functions possessing appropriate concavity and majorization properties.},
pages = {no. 25, 1-13},
issn = {1083-589X},
doi = {10.1214/ECP.v19-3151},
url = {http://ecp.ejpecp.org/article/view/3151}}