Exact simulation of generalised Vervaat perpetuities

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 ( Y i ) i ≥0 is an independent and identically distributed sequence of random variables independent from ( W i ) 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 Y i = 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.