THE FIRST PASSAGE EVENT FOR SUMS OF DEPENDENT LEVY PROCESSES WITH APPLICATIONS TO INSURANCE RISK

• 成果类型: Article
• 署名作者: Eder, Irmingard; Klueppelberg, Claudia
• 署名单位: Technical University of Munich; Technical University of Munich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/09-AAP601
• 发表日期: 2009
• 页码: 2047-2079
• 关键词: ruin probabilities; overshoots

摘要

For the sum process X = X-1 + X-2 of a bivariate Levy process (X-1, X-2) with possibly dependent components, we derive a quintuple law describing the first upwards passage event of X over a fixed barrier, caused by a jump, by the joint distribution of five quantities: the time relative to the time of the previous maximum, the time of the previous maximum, the overshoot, the undershoot and the undershoot of the previous maximum. The dependence between the jumps of X-1 and X-2 is modeled by a Levy copula. We calculate these quantities for some examples, where we pay particular attention to the influence of the dependence structure. We apply our findings to the ruin event of an insurance risk process.