Best Bounds in Doob's Maximal Inequality for Bessel Processes

Abstract

Let ((Zt), Pz) be a Bessel process of dimension α>0 started at z under Pz for z⩾0. Then the maximal inequalityEz(max0⩽t⩽τZpt)⩽pp−(2−αp/(2−α)Ez(Zpτ)−pp−(2−α)zpis shown to be satisfied for all stopping times τ for (Zt) with Ez(τp/2)<∞, and all p>(2−α)∨0. The constants (p/(p−(2−α)))p/(2−α) and p/(p−(2−α)) are the best possible. If λ is the greater root of the equation λ1−(2−α)/p−λ=(2−α)/(cp−c(2−α)), the equality is attained in the limit through the stopping timesτλ, p=inf{t>0:Zpt⩽λmax0⩽r⩽tZpr}when c tends to the best constant (p/(p−(2−α)))p/(2−α) from above. Moreover we show that Ez(τq/2λ, p)<∞ if and only if λ>((1−(2−α)/q)∨0)p/(2−α). The proof of the inequality is based upon solving the optimal stopping problemV∗(z)=supτEz(max0⩽t⩽τZpt−cZpτ)by applying the principle of smooth fit and the maximality principle. In addition, the exact formula for the expected waiting time of the optimal strategy is derived by applying the minimality principle. The main emphasis of the paper is on the explicit expressions obtained.