Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion

Abstract:

Let $B = {({B_t})_{t \geq 0}}$ be a standard Brownian motion. For $c > 0$, $k > 0$, let \[ \begin {gathered} T(c,k) = \inf \{ t \geq 0:{\max _{s \leq t}}{B_s} - c{B_t} \geq k\} , \hfill \\ {T^*}(c,k) = \inf \{ t \geq 0:{\max _{s \leq t}}|{B_s}| - c|{B_t}| \geq k\} . \hfill \\ \end {gathered} .\] We show that for $c > 0$ and $k > 0$, both $T(c,k)$ and ${T^*}(c,k)$ are finite almost everywhere. Moreover, $T(c,k)$ and ${T^*}(c,k) \in {L^{p/2}}$ if and only if $c < p/(p - 1)$ for $p > 1$, and for all $c > 0$ when $p \leq 1$. These results have analogues for simple random walks. As a consequence, if $T$ is any stopping time of ${B_t}$ such that ${({B_{T \wedge t}})_{t \geq 0}}$ is uniformly integrable, then both of the inequalities \[ \begin {array}{*{20}{c}} {||{{\sup }_{s \leq T}}{B_s}|{|_p} \leq \frac {p}{{p - 1}}||{B_T}|{|_p},} \\ {{{\left \| {{{\sup }_{s \leq T}}\left | {{B_s}} \right |} \right \|}_P} \leq \frac {p}{{p - 1}}{{\left \| {{B_T}} \right \|}_p},} \\ \end {array} \] 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.