PREDICTING THE ULTIMATE SUPREMUM OF A STABLE LEVY PROCESS WITH NO NEGATIVE JUMPS
成果类型：
Article
署名作者：
Bernyk, Violetta; Dalang, Robert C.; Peskir, Goran
署名单位：
Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; University of Manchester
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/10-AOP598
发表日期：
2011
页码：
2385-2423
关键词：
摘要：
Given a stable Levy process X = (X-t)(0 <= t <= T) of index alpha is an element of (1, 2) with no negative jumps, and letting S-t = sup(0 <= s <= t) X-s denote its running supremum for t is an element of [0, T], we consider the optimal prediction problem V = inf(0 <=tau <= T) E(S-T - X-tau)(p), where the infimum is taken over all stopping times tau of X, and the error parameter p is an element of(1, alpha) 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 alpha(*) is an element of (1, 2) (equal to 1.57 approximately) and a strictly increasing function p(*) : (alpha(*), 2) -> (1, 2) satisfying p(*)(alpha(*)+) = 1, p(*)(2-) = 2 and p(*)(alpha) < alpha for (alpha*, 2) such that for every alpha is an element of (alpha(*), 2) and p is an element of (1, p(*)(alpha)) the following stopping time is optimal tau(*) = inf{t is an element of [0, T]: S-t - X-t >= z(*) (T - t)(1/alpha)}, where z(*) is an element of (0,infinity) is the unique root to a transcendental equation (with parameters alpha and p). Moreover, if either alpha is an element of (1, alpha(*)) or p is an element of (p(*)(alpha), alpha) then it is not optimal to stop at t is an element of [0, T) when St - Xt is sufficiently large. The existence of the breakdown points alpha(*) and p(*)(alpha) stands in sharp contrast with the Brownian motion case (formally corresponding to alpha = 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).