Assume that X is a càdlàg, real-valued martingale starting from zero, H is a predictable process with values in [−1; 1] and Y =∫HdX. This article contains the proofs of the following inequalities:
(i) If X has continuous paths, then
P(supt≥0 Yt≥ 1)≤ 2Esupt≥0Xt, where the constant 2 is the best possible.
(ii) If X is arbitrary, then
P(supt≥0 Yt≥ 1)≤ cEsupt≥0Xt, where c = 3.0446... is the unique positive number satisfying the equation 3c4 − 8c3 − 32 = 0. This constant is the best possible.