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Exact simulation of the extrema of stable processes

Part of: Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  15 November 2019

Jorge I. González Cázares ,
Aleksandar Mijatović  and
Gerónimo Uribe Bravo
Jorge I. González Cázares*
Affiliation:
University of Warwick and The Alan Turing Institute Universidad Nacional Autónoma de México
Aleksandar Mijatović*
Affiliation:
University of Warwick and The Alan Turing Institute
Gerónimo Uribe Bravo*
Affiliation:
Universidad Nacional Autónoma de México
*
*Postal address: University of Warwick, Coventry CV4 7AL, UK.
*Postal address: University of Warwick, Coventry CV4 7AL, UK.
**** Postal address: Instituto de Matemáticas, Ciudad Universitaria, Coyoacán, Ciudad de México, 04510, México. Email address: geronimo@matem.unam.mx
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Abstract

We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Lévy processes [27]) and use it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse the performance of the algorithm numerically (the code, written in Julia 1.0, is available on GitHub).

Keywords

Random variate generationperpetuitiessimulationperfect simulationdominated coupling from the paststable process

MSC classification

Primary: 65C10: Random number generation
Secondary: 65C05: Monte Carlo methods
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 4 , December 2019 , pp. 967 - 993
Copyright
© Applied Probability Trust 2019 

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