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November 2011 Predicting the ultimate supremum of a stable Lévy process with no negative jumps
Violetta Bernyk, Robert C. Dalang, Goran Peskir
Ann. Probab. 39(6): 2385-2423 (November 2011). DOI: 10.1214/10-AOP598


Given a stable Lévy process X = (Xt)0≤tT of index α ∈ (1, 2) with no negative jumps, and letting St = sup0≤st Xs 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 StXt 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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Violetta Bernyk. Robert C. Dalang. Goran Peskir. "Predicting the ultimate supremum of a stable Lévy process with no negative jumps." Ann. Probab. 39 (6) 2385 - 2423, November 2011.


Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1235.60036
MathSciNet: MR2932671
Digital Object Identifier: 10.1214/10-AOP598

Primary: 45J05 , 60G40 , 60J75
Secondary: 26A33 , 47G20 , 60G25

Keywords: Caputo fractional derivative , curved boundary , fractional free-boundary problem , infinitesimal generator , optimal prediction , Optimal stopping , polar kernel , Riemann–Liouville fractional derivative , smooth fit , Spectrally positive , Stable Lévy process with no negative jumps , stochastic process reflected at its supremum , ultimate supremum , weakly singular Volterra integral equation

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 6 • November 2011
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