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Exact simulation of generalised Vervaat perpetuities

Part of: Special processes Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  12 July 2019

Angelos Dassios ,
Yan Qu  and
Jia Wei Lim
Angelos Dassios*
Affiliation:
London School of Economics and Political Science
Yan Qu*
Affiliation:
London School of Economics and Political Science
Jia Wei Lim*
Affiliation:
University of Bristol
*
*Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK.
*Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK.
***Postal address: Department of Mathematics, University of Bristol, Senate House, Tyndall Avenue, Bristol BS8 1TH, UK.
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Abstract

We consider a generalised Vervaat perpetuity of the form X = Y1W1 +Y2W1W2 + · · ·, 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.

Keywords

Vervaat perpetuityDickman distributiontruncated gamma processexact simulationmarked renewal process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60K15: Markov renewal processes, semi-Markov processes 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 57 - 75
Copyright
© Applied Probability Trust 2019 

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