A maximal inequality for supermartingales

A tight upper bound is given on the distribution of the maximum of a supermartin-gale. 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.