ZOOMING IN ON A LEVY PROCESS AT ITS SUPREMUM

• 成果类型: Article
• 署名作者: Ivanovs, Jevgenijs
• 署名单位: Aarhus University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1320
• 发表日期: 2018
• 页码: 912-940
• 关键词: wiener-hopf factorization; random-walks; INVARIANCE-PRINCIPLES; path decompositions; small jumps; times; approximations; simulation; discrete; passage

摘要

Let M and iota be the supremum and its time of a Levy process X on some finite time interval. It is shown that zooming in on X at its supremum, that is, considering ((X tau+t epsilon - M)/a epsilon)(t is an element of R) as e down arrow 0, results in (xi(t))(t is an element of R) constructed from two independent processes having the laws of some self-similar Levy process (X) over cap conditioned to stay positive and negative. This holds when X is in the domain of attraction of (X) over cap under the zooming-in procedure as opposed to the classical zooming out [Trans. Amer. Math. Soc. 104 (1962) 62-78]. As an application of this result, we establish a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result in [Ann. Appl. Probab. 5 (1995) 875-896] for a linear Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a Levy process is provided.