A maximal inequality for stochastic integrals

Abstrakt

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
Psup_t≥0 Y_t≥ 1≤ 2Esup_t≥0X_t, where the constant 2 is the best possible.
ii If X is arbitrary, then
Psup_t≥0 Y_t≥ 1≤ cEsup_t≥0X_t, where c = 3.0446... is the unique positive number satisfying the equation 3c⁴ − 8c³ − 32 = 0. This constant is the best possible.