Abstract
We determine the smallest number β for which the following holds. If X is a continuous-path martingale and [X, X] denotes its square bracket, then Then, we extend this inequality to (i) the class of martingales with no positive jumps and (ii) the class of discrete-time conditionally symmetric martingales. We also study some generalizations of the inequality involving an extra term depending on the initial value X 0.
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