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

Abstract: 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 parameter $p\in(1,\alpha)$ is given and fixed. Reducing the optimal prediction problem to a fractional free-boundary problem of Riemann--Liouville type, and finding an explicit solution to the latter, we show that there exists $\alpha_\in(1,2)$ (equal to 1.57 approximately) and a strictly increasing function $p_:(\alpha_,2)\rightarrow(1,2)$ satisfying $p_(\alpha_+)=1$, $p_(2-)=2$ and $p_(\alpha)<\alpha$ for $\alpha\in(\alpha_,2)$ such that for every $\alpha\in (\alpha_,2)$ and $p\in(1,p_(\alpha))$ the following stopping time is optimal [\tau_=\inf{t\in[0,T]:S_t-X_t\ge z_(T-t)^{1/\alpha}},] where $z_\in(0,\infty)$ is the unique root to a transcendental equation (with parameters $\alpha$ and $p$). Moreover, if either $\alpha\in(1,\alpha_)$ or $p\in(p_(\alpha),\alpha)$ then it is not optimal to stop at $t\in[0,T)$ when $S_t-X_t$ is sufficiently large. The existence of the breakdown points $\alpha_$ and $p_*(\alpha)$ stands in sharp contrast with the Brownian motion case (formally corresponding to $\alpha=2$), and the phenomenon itself may be attributed to the interplay between the jump structure (admitting a transition from lighter to heavier tails) and the individual preferences (represented by the error parameter $p$).