Exact simulation of generalised Vervaat perpetuities. (March 2019)

• Record Type: Journal Article
• Authors: Dassios, Angelos; Qu, Yan; Lim, Jia Wei
• Is Part Of: Journal of applied probability. Volume 56:Number 1(2019)
• Page Start: 57
• Page End: 75
• Publication Date: 2019-03
• Subjects: Vervaat perpetuity, -- Dickman distribution, -- truncated gamma process, -- exact simulation, -- marked renewal process; Primary 60G51, -- Secondary 60K15, -- 65C05; 519.2
• DOI: 10.1017/jpr.2019.6
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: We consider a generalised Vervaat perpetuity of the form X = Y 1 W 1 + Y 2 W 1 W 2 + · · ·, where $W_i \sim {\cal U}^{1/t}$ and ( Yi ) i ≥0 is an independent and identically distributed sequence of random variables independent from ( Wi ) i ≥0 . Based on a distributional decomposition technique, we propose a novel method for exactly simulating the generalised Vervaat perpetuity. The general framework relies on the exact simulation of the truncated gamma process, which we develop using a marked renewal representation for its paths. Furthermore, a special case arises when Yi = 1, and X has the generalised Dickman distribution, for which we present an exact simulation algorithm using the marked renewal approach. In particular, this new algorithm is much faster than existing algorithms illustrated in Chi (2012), Cloud and Huber (2017), Devroye and Fawzi (2010), and Fill and Huber (2010), as well as being applicable to the general payments case. Examples and numerical analysis are provided to demonstrate the accuracy and effectiveness of our method.