A sharp maximal inequality for one-dimensional Dunkl martingales

Abstract

Let X=(Xt)t≥0 be a one-dimensional Dunkl process of parameter k≥0, starting from 0. For any p≥1, we find the least constant Cp,k∈(0,∞] in the Doob-type inequality E(sup0≤t≤τXτ)p≤Cp,kE∣Xτ∣p where τ runs over all p/2-integrable stopping times of X. The proof exploits optimal stopping techniques.