Sharp Maximal Inequalities for Conditionally Symmetric Martingales and Brownian Motion

Let B = {Bt)t>0 be a standard Brownian motion. For c > 0, k > 0 , let T(c, k) = inî{t > 0: maxs<í Bs cBt > k} , T"(c,k)= inf{r>0: max^, \BS\ c\B,\ > k} . We show that for c > 0 and k > 0, both T(c, k) and T*{c, k) axe finite almost everywhere. Moreover, T(c, k) and T*(c, k) e L if and only if c < pKp 1) for p > 1 , and for all c > 0 when p < 1 . These results have analogues for simple random walks. As a consequence, if T is any stopping time of Bt suchthat (BTAl)t>0 is uniformly integrable, then both of the inequalities llsup^iy^^-fPrll,, l«up,<rl*,lll,< jT7ll*rllp. are sharp. This implies that q = p/(p 1) is not only the best constant for Doob's maximal inequality for general martingales but also for conditionally symmetric martingales (in particular, for dyadic martingales), and for Brownian motion.