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

Abstract

Given a stable Lévy process X = (X_t)_0≤t≤T of index α ∈ (1, 2) with no negative jumps, and letting S_t = sup_0≤s≤t  X_s denote its running supremum for t ∈ [0, T], we consider the optimal prediction problem $$V = \inf_{0≤τ≤T}\mathsf E(S_T − X_τ)^p, $$ where the infimum is taken over all stopping times τ of X, and the error parameter p ∈ (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_∗ ∈ (1, 2) (equal to 1.57 approximately) and a strictly increasing function p_∗ : (α_∗, 2) → (1, 2) satisfying p_∗(α_∗+) = 1, p_∗(2−) = 2 and p_∗(α) < α for α ∈ (α_∗, 2) such that for every α ∈ (α_∗, 2) and p ∈ (1, p_∗(α)) the following stopping time is optimal $$τ_∗ = \inf\{t ∈ [0, T] : S_t − X_t ≥ z_∗(T − t)^{1/α}\},$$ where z_∗ ∈ (0, ∞) is the unique root to a transcendental equation (with parameters α and p). Moreover, if either α ∈ (1, α_∗) or p ∈ (p_∗(α), α) then it is not optimal to stop at t ∈ [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).