Predicting the ultimate supremum of a stable Lévy process with no negative jumps

Given a stable L\'{e}vy process X=(X_t)_{0\le t\le T} of index \alpha\in(1,2) with no negative jumps, and letting S_t=\sup_{0\le s\le t}X_s denote its running supremum for t\in [0,T], we consider the optimal prediction problem \[V=\inf_{0\le\tau\le T}\mathsf{E}(S_T-X_{\tau})^p,\] where the infimum is taken over all stopping times \tau of X, and the error... Show more