On Doobʹs maximal inequality for Brownian motion

If B = (Bt)t ⩾ 0 is a standard Brownian motion started at x under Px for x ⩾ 0, and τ is any stopping time for B with Ex(τ) < ∞, then for each p > 1 the following inequality is shown to be sharp: Exmax0⩽t⩽τ |Bt|p⩽pp−1p Ex|Bt|p−pp−1xp. The sharpness is realized through the stopping times of the form τλ,ϵ=inft >0| max0⩽s⩽t Bs−λBt⩾ϵ ich it is computed: E0(τλ,ϵ=ϵ2λ(2−λ) er ε > 0 and 0 < λ < 2. Hence, for the stopping time σλ,ϵ=inft>0max0⩽s⩽t|Bs|−λ|Bt|⩾ϵ is shown to be a convolution of τλ, λε with the first hitting time of ε by |B| = (|Bt|)t ⩾ 0, we have E0(σλ,ϵ)=2ϵ2(2−λ) l ε > 0 and all 0 < λ < 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-Tanakaʹs formula (being applied two dimensionally). The main emphasis is on the explicit formulas obtained throughout.