A sharp maximal inequality for one-dimensional Dunkl martingales

Let X=(Xt)t≥0X=(Xt)t≥0 be a one-dimensional Dunkl process of parameter k≥0k≥0, starting from 00. For any p≥1p≥1, we find the least constant Cp,k∈(0,∞]Cp,k∈(0,∞] in the Doob-type inequality E(sup0≤t≤τXτ)p≤Cp,kE∣Xτ∣p where ττ runs over all p/2p/2-integrable stopping times of XX. The proof exploits optimal stopping techniques.