The Markov-quantile process attached to a family of marginals
[Le processus Markov-quantile attaché à une famille de marges]
Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1-62.

Soit μ=(μ t ) t une famille à un paramètre de mesures de probabilité sur . Son processus quantile (G t ) t :]0,1[ , donné par G t (α)=inf{x:μ t (]-,x])α}, n’est en général pas markovien. Nous le modifions pour construire le processus markovien que nous nommons « Markov-quantile » : il existe un unique processus markovien X de marges μ t qui est limite, pour la convergence fini-dimensionnelle, de processus quantiles dont le passé est rendu indépendant du futur en un nombre fini d’instants (beaucoup de limites non markoviennes existent en général). Il est frappant qu’aucune hypothèse de régularité sur la famille μ n’est nécessaire. En outre, si μ est croissante pour l’ordre stochastique, les trajectoires de X sont croissantes. Ceci est un analogue d’un résultat de Kellerer traitant de l’ordre convexe. Dans un article associé [8] on montre aussi que si μ est absolument continue dans l’espace de Wasserstein 𝒫 2 (), X est solution d’un problème de transport de Benamou–Brenier avec marges μ t , et fournit donc une représentation markovienne de l’équation de continuité, unique dans le sens donné plus haut.

Let μ=(μ t ) t be any 1-parameter family of probability measures on . Its quantile process (G t ) t :]0,1[ , given by G t (α)=inf{x:μ t (]-,x])α}, is not Markov in general. We modify it to build the Markov process we call “Markov-quantile”: there is a unique Markov process X with marginals μ t , being a limit for the finite dimensional topology of quantile processes where the past is made independent of the future at finitely many times (many non-Markovian limits exist in general). Strikingly, no regularity is required for the family μ. Moreover, if μ is increasing for the stochastic order, X has increasing trajectories. This is an analogue of a result of Kellerer dealing with the convex order. In a companion paper [8] it is also proved that if μ is absolutely continuous in the Wasserstein space 𝒫 2 (), X is solution of a Benamou–Brenier transport problem with marginals μ t , providing a Markov representation of the continuity equation, unique in the sense above.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.177
Classification : 60A10, 28A33, 60J25, 35Q35, 60G44, 49J55
Keywords: Markov process, quantile process, optimal transport, continuity equation, increasing process, Kellerer’s theorem, martingale optimal transport, peacocks, copula
Mot clés : Processus markovien, processus quantile, transport optimal, équation de continuité, processus croissant, théorème de Kellerer, transport optimal martingale optimal, peacocks, copule
Charles Boubel 1 ; Nicolas Juillet 1

1 Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS 7 rue René Descartes, 67000 Strasbourg, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Charles Boubel; Nicolas Juillet. The Markov-quantile process attached to a family of marginals. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1-62. doi : 10.5802/jep.177. https://jep.centre-mersenne.org/articles/10.5802/jep.177/

[1] J. M. P. Albin - “A continuous non-Brownian motion martingale with Brownian motion marginal distributions”, Statist. Probab. Lett. 78 (2008) no. 6, p. 682-686 | DOI | MR | Zbl

[2] L. Ambrosio, N. Gigli & G. Savaré - Gradient flows in metric spaces and in the space of probability measures, Lectures in Math. ETH Zürich, Birkhäuser Verlag, Basel, 2008 | Zbl

[3] M. Beiglböck, M. Huesmann & F. Stebegg - “Root to Kellerer”, in Séminaire de Probabilités XLVIII, Lect. Notes in Math., vol. 2168, Springer, Cham, 2016, p. 1-12 | DOI | MR | Zbl

[4] M. Beiglböck & N. Juillet - “On a problem of optimal transport under marginal martingale constraints”, Ann. Probab. 44 (2016) no. 1, p. 42-106 | DOI | MR | Zbl

[5] M. Beiglböck & N. Juillet - “Shadow couplings”, Trans. Amer. Math. Soc. 374 (2021) no. 7, p. 4973-5002 | DOI | MR | Zbl

[6] J.-D. Benamou & Y. Brenier - “A numerical method for the optimal time-continuous mass transport problem and related problems”, in Monge Ampère equation: applications to geometry and optimization (Deerfield Beach, FL, 1997), Contemp. Math., vol. 226, American Mathematical Society, Providence, RI, 1999, p. 1-11 | DOI | MR | Zbl

[7] P. Billingsley - Convergence of probability measures, Wiley Series in Probability and Statistics, John Wiley & Sons Inc., New York, 1999 | DOI | Zbl

[8] C. Boubel & N. Juillet - “On absolutely continuous curves in the Wasserstein space over and their representation by an optimal Markov process”, 2021 | arXiv

[9] G. Brunick & S. Shreve - “Mimicking an Itô process by a solution of a stochastic differential equation”, Ann. Appl. Probab. 23 (2013) no. 4, p. 1584-1628 | DOI | Zbl

[10] W. Doeblin - “Sur certains mouvements aléatoires discontinus”, Skand. Aktuarietidskr. 22 (1939), p. 211-222 | DOI | MR | Zbl

[11] B. Dupire et al. - “Pricing with a smile”, Risk 7 (1994) no. 1, p. 18-20

[12] H. Federer - Geometric measure theory, Grundlehren Math. Wiss., vol. 153, Springer-Verlag New York Inc., New York, 1969 | MR | Zbl

[13] W. Feller - An introduction to probability theory and its applications. Vol. I, John Wiley & Sons, Inc., New York-London-Sydney, 1968

[14] I. Gyöngy - “Mimicking the one-dimensional marginal distributions of processes having an Itô differential”, Probab. Theory Relat. Fields 71 (1986) no. 4, p. 501-516 | DOI | Zbl

[15] K. Hamza & F. C. Klebaner - “A family of non-Gaussian martingales with Gaussian marginals”, J. Appl. Math. Stochastic Anal. (2007), article ID 92723, 19 pages | DOI | MR | Zbl

[16] P. Henry-Labordère, X. Tan & N. Touzi - “An explicit martingale version of the one-dimensional Brenier’s theorem with full marginals constraint”, Stochastic Process. Appl. 126 (2016) no. 9, p. 2800-2834 | DOI | MR | Zbl

[17] E. Hillion - “W 1,+ -interpolation of probability measures on graphs”, Electron. J. Probab. 19 (2014), article ID 92, 29 pages | DOI | MR | Zbl

[18] F. Hirsch, C. Profeta, B. Roynette & M. Yor - Peacocks and associated martingales, with explicit constructions, Bocconi & Springer Series, vol. 3, Springer, Milan, 2011 | DOI | MR | Zbl

[19] F. Hirsch & B. Roynette - “On d -valued peacocks”, ESAIM Probab. Statist. 17 (2013), p. 444-454 | DOI | MR | Zbl

[20] F. Hirsch, B. Roynette & M. Yor - “Kellerer’s theorem revisited”, in Asymptotic Laws and Methods in Stochastics. Volume in Honour of Miklos Csorgo, Fields Inst. Commun. Series, vol. 76, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, p. 347-363 | DOI | MR | Zbl

[21] D. G. Hobson - “Fake exponential Brownian motion”, Statist. Probab. Lett. 83 (2013) no. 10, p. 2386-2390 | DOI | MR | Zbl

[22] D. G. Hobson - “Mimicking martingales”, Ann. Appl. Probab. 26 (2016) no. 4, p. 2273-2303 | DOI | MR | Zbl

[23] N. Juillet - “Peacocks parametrised by a partially ordered set”, in Séminaire de Probabilités XLVIII, Lect. Notes in Math., vol. 2168, Springer, Cham, 2016, p. 13-32 | DOI | MR | Zbl

[24] N. Juillet - “Martingales associated to peacocks using the curtain coupling”, Electron. J. Probab. 23 (2018), article ID 8, 29 pages | DOI | MR | Zbl

[25] O. Kallenberg - Foundations of modern probability, Probability and its Appl. (New York), Springer-Verlag, New York, 2002 | DOI | Zbl

[26] T. Kamae & U. Krengel - “Stochastic partial ordering”, Ann. Probab. 6 (1978) no. 6, p. 1044-1049 | DOI | MR | Zbl

[27] K. S. Kaminsky, E. M. Luks & P. I. Nelson - “Strategy, nontransitive dominance and the exponential distribution”, Austral. J. Statist. 26 (1984) no. 2, p. 111-118 | DOI | MR | Zbl

[28] H. G. Kellerer - “Markov-Komposition und eine Anwendung auf Martingale”, Math. Ann. 198 (1972), p. 99-122 | DOI | MR | Zbl

[29] H. G. Kellerer - “Integraldarstellung von Dilationen”, in Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Tech. Univ., Prague, 1971; dedicated to the memory of Antonín Špaček), Academia, Prague, 1973, p. 341-374 | MR | Zbl

[30] H. G. Kellerer - “Order conditioned independence of real random variables”, Math. Ann. 273 (1986), p. 507-528 | DOI | MR | Zbl

[31] H. G. Kellerer - “Markov property of point processes”, Probab. Theory Relat. Fields 76 (1987), p. 71-80 | DOI | MR | Zbl

[32] C. Léonard - “Lazy random walks and optimal transport on graphs”, Ann. Probab. 44 (2016) no. 3, p. 1864-1915 | DOI | MR | Zbl

[33] S. Lisini - “Characterization of absolutely continuous curves in Wasserstein spaces”, Calc. Var. Partial Differential Equations 28 (2007) no. 1, p. 85-120 | DOI | MR | Zbl

[34] G. Lowther - “Fitting martingales to given marginals”, 2008 | arXiv

[35] G. Lowther - “Limits of one-dimensional diffusions”, Ann. Probab. 37 (2009) no. 1, p. 78-106 | DOI | MR | Zbl

[36] D. B. Madan & M. Yor - “Making Markov martingales meet marginals: with explicit constructions”, Bernoulli 8 (2002) no. 4, p. 509-536 | MR | Zbl

[37] M. Nagasawa - Schrödinger equations and diffusion theory, Monographs in Math., vol. 86, Birkhäuser Verlag, Basel, 1993 | DOI | MR | Zbl

[38] Y. Ollivier - “Ricci curvature of Markov chains on metric spaces”, J. Funct. Anal. 256 (2009) no. 3, p. 810-864 | DOI | MR | Zbl

[39] G. Pagès - “Functional co-monotony of processes with applications to peacocks and barrier options”, in Séminaire de Probabilités XLV, Lect. Notes in Math., vol. 2078, Springer, Cham, 2013, p. 365-400 | DOI | MR | Zbl

[40] B. Pass - “On a class of optimal transportation problems with infinitely many marginals”, SIAM J. Appl. Math. 45 (2013) no. 4, p. 2557-2575 | DOI | MR | Zbl

[41] S. T. Rachev & L. Rüschendorf - Mass transportation problems. Vol. I & II, Probability and its Appl. (New York), Springer-Verlag, New York, 1998 | Zbl

[42] Y. Rinott, M. Scarsini & Y. Yu - “A Colonel Blotto gladiator game”, Math. Oper. Res. 37 (2012) no. 4, p. 574-590 | DOI | MR | Zbl

[43] M. Shaked & J. G. Shanthikumar - Stochastic orders, Springer Series in Statistics, Springer, New York, 2007 | DOI | Zbl

[44] V. Strassen - “The existence of probability measures with given marginals”, Ann. Math. Statist. 36 (1965), p. 423-439 | DOI | MR | Zbl

[45] C. Villani - Topics in optimal transportation, Graduate Studies in Math., vol. 58, American Mathematical Society, Providence, RI, 2003 | DOI | MR | Zbl

[46] C. Villani - Optimal transport, Grundlehren Math. Wiss., vol. 338, Springer-Verlag, 2009 | DOI | MR

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