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Sharp maximal inequality for martingales and stochastic integrals

   
@article{ECP1438,
	author = {Adam Osekowski},
	title = {Sharp maximal inequality for martingales and stochastic integrals},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {14},
	year = {2009},
	keywords = {Martingale; stochastic integral; maximal function},
	abstract = {Let $X=(X_t)_{t\geq 0}$ be a martingale and $H=(H_t)_{t\geq 0}$ be a predictable process taking values in $[-1,1]$. Let $Y$ denote the stochastic integral of $H$ with respect to $X$. We show that  $$ ||\sup_{t\geq 0}Y_t||_1 \leq \beta_0 ||\sup_{t\geq 0}|X_t|||_1,$$ where $\beta_0=2,0856\ldots$ is the best possible. Furthermore, if, in addition, $X$ is nonnegative, then $$ ||\sup_{t\geq 0}Y_t||_1 \leq \beta_0^+ ||\sup_{t\geq 0}X_t||_1,$$ where $\beta_0^+=\frac{14}{9}$ is the best possible.},
	pages = {no. 2, 17-30},
	issn = {1083-589X},
	doi = {10.1214/ECP.v14-1438},    
        url = {http://ecp.ejpecp.org/article/view/1438}}