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

Given a stable Lévy process $X=(X_t)_{0\le t\le T}$ of index $α\in(1,2)$ 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 $V=\inf_{0\leτ\le T}\mathsf{E}(S_T-X_τ)^p,$ where the infimum is taken over all stopping times $τ$ of $X$, and the error parameter $p\in(1,α)$ 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 $α_*\in(1,2)$ (equal to 1.57 approximately) and a strictly increasing function $p_*:(α_*,2)\rightarrow(1,2)$ satisfying $p_*(α_*+)=1$, $p_*(2-)=2$ and $p_*(α)<α$ for $α\in(α_*,2)$ such that for every $α\in (α_*,2)$ and $p\in(1,p_*(α))$ the following stopping time is optimal $τ_*=\inf\{t\in[0,T]:S_t-X_t\ge z_*(T-t)^{1/α}\},$ where $z_*\in(0,\infty)$ is the unique root to a transcendental equation (with parameters $α$ and $p$). Moreover, if either $α\in(1,α_*)$ or $p\in(p_*(α),α)$ 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 $α_*$ and $p_*(α)$ stands in sharp contrast with the Brownian motion case (formally corresponding to $α=2$), 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 $p$).

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