Let X = (Xt) t≥0 be a stable Lévy process of index α ∈ (1, 2) with no negative jumps and let St = sup 0≤s≤t Xs denote its running supremum for t > 0. We show that the density function ft of St can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann-Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for ft. Recalling the familiar relation between St and the first entry time τx of X into [x, ∞), this further translates into an explicit series representation for the density function of τx.1. Introduction. In our study [3] of optimal prediction for a stable Lévy process X = (X t ) t≥0 , we encountered the question of computing the distribution function of S t = sup 0≤s≤t X s for t > 0. In the existing literature, such expressions seem to be available only when X has no positive jumps and the purpose of the present paper is to seek similar expressions when X has no negative jumps. We note that the latter problem dates back to [5], page 282.Our main result (Theorem 1) characterizes the density function f of S 1 as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first order RiemannLiouville fractional differential equation satisfying a boundary condition at
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 problemwhere 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 optimalwhere 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).
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