On Doob's maximal inequality for Brownian motion

If B = (Bt)t [greater-or-equal, slanted] 0 is a standard Brownian motion started at x under Px for x [greater-or-equal, slanted] 0, and [tau] is any stopping time for B with Ex([tau]) < [infinity], then for each p > 1 the following inequality is shown to be sharp: The sharpness is realized through the stopping times of the form for which it is computed: whenever [var epsilon] > 0 and 0 < [lambda] < 2. Hence, for the stopping time which is shown to be a convolution of [tau][lambda], [lambda][var epsilon] with the first hitting time of [var epsilon] by B = (Bt)t [greater-or-equal, slanted] 0, we have for all [var epsilon] > 0 and all 0 < [lambda] < 2. The method of proof relies upon the principle of smooth fit and the maximality principle for a Stephan problem with moving (free) boundary, and Itô-Tanaka's formula (being applied two dimensionally). The main emphasis is on the explicit formulas obtained throughout.