A temporal factorization at the maximum for certain positive self-similar Markov processes. (December 2020)

• Record Type: Journal Article
• Authors: Vidmar, Matija
• Is Part Of: Journal of applied probability. Volume 57:Number 4(2020)
• Page Start: 1045
• Page End: 1069
• Publication Date: 2020-12
• Subjects: spectrally negative Lévy processes, -- positive self-similar Markov processes, -- splitting at the maximum, -- Wiener–Hopf factorization, -- martingales, -- time-changes; 60G51, -- 60G18, -- 60G44; 519.2
• DOI: 10.1017/jpr.2020.62
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: For a spectrally negative self-similar Markov process on $[0, \infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.