\(\varepsilon\)-strong simulation of the convex minorants of stable processes and meanders. (English) Zbl 1459.60083

Summary: Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisation to construct \(\varepsilon\)-strong simulation \((\varepsilon\) SS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our \(\varepsilon\) SS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.


60G17 Sample path properties
60G51 Processes with independent increments; Lévy processes
65C05 Monte Carlo methods


GitHub; SupStable.jl
Full Text: DOI arXiv Euclid


[1] [AC01] Larbi Alili and Loïc Chaumont, A new fluctuation identity for Lévy processes and some applications, Bernoulli 7 (2001), no. 3, 557-569. · Zbl 1003.60045
[2] [ACGZ19] L. Alili, L. Chaumont, P. Graczyk, and T. Żak, Space and time inversions of stochastic processes and Kelvin transform, Math. Nachr. 292 (2019), no. 2, 252-272. · Zbl 1415.31004
[3] [AHUB19] Osvaldo Angtuncio Hernández and Gerónimo Uribe Bravo, Dini derivatives for Exchangeable Increment processes and applications, arXiv e-prints (2019), arXiv:1903.04745. · Zbl 1469.60105
[4] [AI18] Søren Asmussen and Jevgenijs Ivanovs, A factorization of a Lévy process over a phase-type horizon, Stoch. Models 34 (2018), no. 4, 397-408. · Zbl 1411.60068
[5] [AP11] Josh Abramson and Jim Pitman, Concave majorants of random walks and related Poisson processes, Combin. Probab. Comput. 20 (2011), no. 5, 651-682. · Zbl 1229.60056
[6] [BC15] Jose Blanchet and Xinyun Chen, Steady-state simulation of reflected Brownian motion and related stochastic networks, Ann. Appl. Probab. 25 (2015), no. 6, 3209-3250. · Zbl 1332.60120
[7] [BCD17] Jose Blanchet, Xinyun Chen, and Jing Dong, \( \varepsilon \)-strong simulation for multidimensional stochastic differential equations via rough path analysis, Ann. Appl. Probab. 27 (2017), no. 1, 275-336. · Zbl 1436.65012
[8] [BDM16] Dariusz Buraczewski, Ewa Damek, and Thomas Mikosch, Stochastic models with power-law tails, Springer Series in Operations Research and Financial Engineering, Springer, [Cham], 2016. · Zbl 1357.60004
[9] [Ber96] Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. · Zbl 0861.60003
[10] [Bil99] Patrick Billingsley, Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication.
[11] [Bin73] N. H. Bingham, Maxima of sums of random variables and suprema of stable processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), no. 4, 273-296. · Zbl 0238.60036
[12] [BM18] Jose Blanchet and Karthyek Murthy, Exact simulation of multidimensional reflected Brownian motion, J. Appl. Probab. 55 (2018), no. 1, 137-156. · Zbl 1401.65006
[13] [BPR12] Alexandros Beskos, Stefano Peluchetti, and Gareth Roberts, \( \epsilon \)-strong simulation of the Brownian path, Bernoulli 18 (2012), no. 4, 1223-1248. · Zbl 1263.65007
[14] [BS02] Andrei N. Borodin and Paavo Salminen, Handbook of Brownian motion—facts and formulae, second ed., Probability and its Applications, Birkhäuser Verlag, Basel, 2002. · Zbl 1012.60003
[15] [BZ04] Stefano Bonaccorsi and Lorenzo Zambotti, Integration by parts on the Brownian meander, Proc. Amer. Math. Soc. 132 (2004), no. 3, 875-883. · Zbl 1039.60052
[16] [BZ17] Jose Blanchet and Fan Zhang, Exact Simulation for Multivariate Itô Diffusions, arXiv e-prints (2017), arXiv:1706.05124.
[17] [CD05] L. Chaumont and R. A. Doney, On Lévy processes conditioned to stay positive, Electron. J. Probab. 10 (2005), no. 28, 948-961. · Zbl 1109.60039
[18] [CD08] L. Chaumont and R. A. Doney, Corrections to: “On Lévy processes conditioned to stay positive” [Electron J. Probab. 10 (2005), no. 28, 948-961; ], Electron. J. Probab. 13 (2008), no. 1, 1-4. · Zbl 1189.60097
[19] [CDN19] Yi Chen, Jing Dong, and Hao Ni, \( \epsilon \)-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations, arXiv e-prints (2019), arXiv:1902.07824.
[20] \([CGH^+96]\) R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert \(W\) function, Adv. Comput. Math. 5 (1996), no. 4, 329-359. · Zbl 0863.65008
[21] [CH12] Nan Chen and Zhengyu Huang, Brownian meanders, importance sampling and unbiased simulation of diffusion extremes, Oper. Res. Lett. 40 (2012), no. 6, 554-563. · Zbl 1257.62086
[22] [CH13] Nan Chen and Zhengyu Huang, Localization and exact simulation of Brownian motion-driven stochastic differential equations, Math. Oper. Res. 38 (2013), no. 3, 591-616. · Zbl 1291.65011
[23] [Cha97] Loïc Chaumont, Excursion normalisée, méandre et pont pour les processus de Lévy stables, Bull. Sci. Math. 121 (1997), no. 5, 377-403. · Zbl 0882.60074
[24] [Cha13] Loïc Chaumont, On the law of the supremum of Lévy processes, Ann. Probab. 41 (2013), no. 3A, 1191-1217. · Zbl 1277.60081
[25] [CM16] Loïc Chaumont and Jacek Małecki, On the asymptotic behavior of the density of the supremum of Lévy processes, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 3, 1178-1195. · Zbl 1350.60042
[26] [CPR13] Loïc Chaumont, Henry Pantí, and Víctor Rivero, The Lamperti representation of real-valued self-similar Markov processes, Bernoulli 19 (2013), no. 5B, 2494-2523. · Zbl 1284.60077
[27] [Dev10] Luc Devroye, On exact simulation algorithms for some distributions related to Brownian motion and Brownian meanders, Recent developments in applied probability and statistics, Physica, Heidelberg, 2010, pp. 1-35. · Zbl 1204.65005
[28] [DI77] Richard T. Durrett and Donald L. Iglehart, Functionals of Brownian meander and Brownian excursion, Ann. Probability 5 (1977), no. 1, 130-135. · Zbl 0356.60035
[29] [dLF15] Arnaud de La Fortelle, A study on generalized inverses and increasing functions part I: generalized inverses, 09 2015.
[30] [DS10] R. A. Doney and M. S. Savov, The asymptotic behavior of densities related to the supremum of a stable process, Ann. Probab. 38 (2010), no. 1, 316-326. · Zbl 1185.60052
[31] [EG00] Katherine Bennett Ensor and Peter W. Glynn, Simulating the maximum of a random walk, J. Statist. Plann. Inference 85 (2000), no. 1-2, 127-135, 2nd St. Petersburg Workshop on Simulation (1996). · Zbl 0974.60030
[32] [FKY14] Takahiko Fujita, Yasuhiro Kawanishi, and Marc Yor, On the one-sided maximum of Brownian and random walk fragments and its applications to new exotic options called “meander option”, Pac. J. Math. Ind. 6 (2014), Art. 2, 7. · Zbl 1386.91140
[33] [GCMUB19] Jorge I. González Cázares, Aleksandar Mijatović, and Gerónimo Uribe Bravo, Exact simulation of the extrema of stable processes, Adv. in Appl. Probab. 51 (2019), no. 4, 967-993. · Zbl 07141464
[34] [GCMUB18a] Jorge I. González Cázares, Aleksandar Mijatović, and Gerónimo Uribe Bravo, Geometrically Convergent Simulation of the Extrema of Lévy Processes, arXiv e-prints (2019), arXiv:1810.11039. · Zbl 07141464
[35] [GCMUB18b] Jorge I. González Cázares, Aleksandar Mijatović, and Gerónimo Uribe Bravo, Code for the \(\epsilon \)-strong simulation of stable meanders, https://github.com/jorgeignaciogc/StableMeander.jl, 2018, GitHub repository. · Zbl 07141464
[36] [GY05] Léonard Gallardo and Marc Yor, Some new examples of Markov processes which enjoy the time-inversion property, Probab. Theory Related Fields 132 (2005), no. 1, 150-162. · Zbl 1087.60058
[37] [IO19] F. Iafrate and E. Orsingher, Some results on the brownian meander with drift, Journal of Theoretical Probability (2019). · Zbl 1456.60090
[38] [Kal81] Olav Kallenberg, Splitting at backward times in regenerative sets, Ann. Probab. 9 (1981), no. 5, 781-799. · Zbl 0526.60061
[39] [Kal02] Olav Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002.
[40] [LBDM18] Zhipeng Liu, Jose H. Blanchet, A. B. Dieker, and Thomas Mikosch, On Optimal Exact Simulation of Max-Stable and Related Random Fields, to appear in Bernoulli (2018), arXiv:1609.06001. · Zbl 1428.62426
[41] [MT09] Sean Meyn and Richard L. Tweedie, Markov chains and stochastic stability, second ed., Cambridge University Press, Cambridge, 2009, With a prologue by Peter W. Glynn. · Zbl 1165.60001
[42] [Pit06] J. Pitman, Combinatorial stochastic processes, Lecture Notes in Mathematics, vol. 1875, Springer-Verlag, Berlin, 2006, Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002, With a foreword by Jean Picard.
[43] [PJR16] Murray Pollock, Adam M. Johansen, and Gareth O. Roberts, On the exact and \(\varepsilon \)-strong simulation of (jump) diffusions, Bernoulli 22 (2016), no. 2, 794-856. · Zbl 1343.60099
[44] [PR12] Jim Pitman and Nathan Ross, The greatest convex minorant of Brownian motion, meander, and bridge, Probab. Theory Related Fields 153 (2012), no. 3-4, 771-807. · Zbl 1255.60143
[45] [PUB12] Jim Pitman and Gerónimo Uribe Bravo, The convex minorant of a Lévy process, Ann. Probab. 40 (2012), no. 4, 1636-1674. · Zbl 1248.60053
[46] [Sat13] Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 2013, Translated from the 1990 Japanese original, Revised edition of the 1999 English translation. · Zbl 1287.60003
[47] [Set94] Jayaram Sethuraman, A constructive definition of Dirichlet priors, Statist. Sinica 4 (1994), no. 2, 639-650.
[48] [UB14] Gerónimo Uribe Bravo, Bridges of Lévy processes conditioned to stay positive, Bernoulli 20 (2014), no. 1, 190-206. · Zbl 1296.60121
[49] [UZ99] Vladimir V.
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