Annals of Probability

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

Violetta Bernyk, Robert C. Dalang, and Goran Peskir

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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).

Article information

Ann. Probab., Volume 39, Number 6 (2011), 2385-2423.

First available in Project Euclid: 17 November 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J75: Jump processes 45J05: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20]
Secondary: 60G25: Prediction theory [See also 62M20] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 26A33: Fractional derivatives and integrals

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


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

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