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.
"A maximal inequality for supermartingales." Electron. Commun. Probab. 19 1 - 10, 2014. https://doi.org/10.1214/ECP.v19-3237