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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: [email protected]
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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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References

Bernyk, V., Dalang, R. C. and Peskir, G. (2008). The law of the supremum of a stable Lévy process with no negative jumps. Ann. Prob. 36, 1777–1789.CrossRefGoogle Scholar
Bernyk, V., Dalang, R. C. and Peskir, G. (2011). Predicting the ultimate supremum of a stable Lévy process with no negative jumps. Ann. Prob. 39, 2385–2423.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes (Cambridge Tracts Math. 121). Cambridge University Press, Cambridge.Google Scholar
Blanchet, J. H. and Sigman, K. (2011). On exact sampling of stochastic perpetuities. J. Appl. Prob. 48A, 165–182.CrossRefGoogle Scholar
Buraczewski, D., Damek, E. and Mikosch, T. (2016). Stochastic Models with Power-Law Tails (Springer Series in Operations Research and Financial Engineering). Springer, Cham.Google Scholar
Chi, Z. (2012). On exact sampling of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 44, 842–873.CrossRefGoogle Scholar
Chi, Z. (2012). On exact sampling of the first passage event of Lévy process with infinite Lévy measure and bounded variation. Stoch. Process. Appl. 126, 1124–1144.CrossRefGoogle Scholar
Chi, Z. (2018). Law and exact sampling of the first passage of a spectrally positive strictly stable process. Available at arXiv:1801.06891v1.Google Scholar
Cloud, K. and Huber, M. (2017). Fast perfect simulation of Vervaat perpetuities. J. Complexity 42, 19–30.CrossRefGoogle Scholar
Dassios, A., Lim, J. and Qu, Y. (2019). Exact simulation of generalised Vervaat perpetuities. J. Appl. Prob. 56, 57–75.CrossRefGoogle Scholar
Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer.CrossRefGoogle Scholar
Devroye, L. (2001). Simulating perpetuities. Methodology Comput. Appl. Prob. 3, 97–115.CrossRefGoogle Scholar
Devroye, L. and Fawzi, O. (2010). Simulating the Dickman distribution. Statist. Prob. Lett. 80, 242–247.CrossRefGoogle Scholar
Devroye, L. and James, L. F. (2011). The double CFTP method. ACM Trans. Model. Comput. Simul. 21, 10:1–10:20.Google Scholar
Devroye, L. and James, L. (2014). On simulation and properties of the stable law. Statist. Methods Appl. 23, 307–343.CrossRefGoogle Scholar
Doney, R. A. (2008). A note on the supremum of a stable process. Stochastics 80, 151–155.CrossRefGoogle Scholar
Ensor, K. B. and Glynn, P. W. (2000). Simulating the maximum of a random walk. J. Statist. Planning Infer. 85, 127–135.CrossRefGoogle Scholar
Fill, J. A. and Huber, M. L. (2010). Perfect simulation of Vervaat perpetuities. Electron. J. Prob. 15 (4), 96–109.CrossRefGoogle Scholar
González Cázares, J., Mijatovi, A. and Uribe Bravo, G. (2018). Code for the simulation of the stable supremum. Available at the GitHub repository: https://github.com/jorgeignaciogc/SupStable.jl.Google Scholar
Huber, M. L. (2016). Perfect Simulation (Monogr. Statist. Appl. Prob. 148). CRC Press, Boca Raton, FL.Google Scholar
Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844–865.CrossRefGoogle Scholar
Kuznetsov, A. (2011). On extrema of stable processes. Ann. Prob. 39, 1027–1060.CrossRefGoogle Scholar
Kuznetsov, A. (2013). On the density of the supremum of a stable process. Stoch. Process. Appl. 123, 986–1003.CrossRefGoogle Scholar
Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Michna, Z. (2013). Explicit formula for the supremum distribution of a spectrally negative stable process. Electron. Commun. Prob. 18 (10), 6.CrossRefGoogle Scholar
Murdoch, D. J. and Green, P. J. (1998). Exact sampling from a continuous state space. Scand. J. Statist. 25, 483–502.CrossRefGoogle Scholar
Pitman, J. and Uribe Bravo, G. (2012). The convex minorant of a Lévy process. Ann. Prob. 40, 1636–1674.CrossRefGoogle Scholar
Revuz, D. (1984). Markov Chains, 2nd edn (North-Holland Math. Library 11). North-Holland, Amsterdam.Google Scholar
Song, R. and Vondraek, Z. (2008). On suprema of Lévy processes and application in risk theory. Ann. Inst. H. Poincaré Prob. Statist. 44, 977–986.CrossRefGoogle Scholar
Uchaikin, V. V. and Zolotarev, V. M. (1999). Chance and Stability (Modern Probab. Statist.). VSP, Utrecht.Google Scholar