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A maximal inequality for supermartingales

   
@article{ECP3237,
	author = {Bruce Hajek},
	title = {A maximal inequality for supermartingales},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {19},
	year = {2014},
	keywords = {maximal inequality; semimartingale calculus; drift; martingale},
	abstract = {A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a supermartingale, then the probability the maximum of $Y$ is greater than or equal to a positive constant $a$ is less than or equal to$1/(1+a).$ The proof makes use of the semimartingale calculus and is inspired by dynamic programming.},
	pages = {no. 55, 1-10},
	issn = {1083-589X},
	doi = {10.1214/ECP.v19-3237},    
        url = {http://ecp.ejpecp.org/article/view/3237}}