The law of the 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: University of Cambridge; 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: 10.1214/07-AOP376
• 发表日期: 2008
• 页码: 1777-1789
• 关键词: random-variables; sums

摘要

Let X = (X-t)(t >= 0) be a stable Levy process of index alpha is an element of (1.2) with no negative jumps and let S-t = sup(0 <= s <= 1) X-s denote its running spremum for t > 0. We show that the density funtion f(1) of S-1 can be characterized as the unique solution to a weakly singular Voleterra integral equation of the first kind or equivalently, as the unique solution to a first-order Riemann-Liouville fractional differential equation satisfying a boundry condition at zero. This yeilds an explicit series representation for f(1). Recalling the familiar relation between S-1 and the first entry time iota(x) of X into (x, infinity), this further translates into an explicit series representation for the density function tau(x).