×

zbMATH — the first resource for mathematics

Bernstein-gamma functions and exponential functionals of Lévy processes. (English) Zbl 1396.30001
Summary: In this work we analyse the solution to the recurrence equation \[ M_\Psi(z+1)=\frac{-z}{\Psi (-z)}M_\Psi (z),\quad M_\Psi(1)=1, \] defined on a subset of the imaginary line and where \(-\Psi\) is any continuous negative definite function. Using the analytic Wiener-Hopf method we solve this equation as a product of functions that extend the gamma function and are in bijection with the Bernstein functions. We call these functions Bernstein-gamma functions. We establish universal Stirling type asymptotic in terms of the constituting Bernstein function. This allows the full understanding of the decay of \(| M_\Psi (z)|\) along imaginary lines and an access to quantities important for many studies in probability and analysis.
This functional equation is a central object in several recent studies ranging from analysis and spectral theory to probability theory. As an application of the results above, we study from a global perspective the exponential functionals of Lévy processes whose Mellin transform satisfies the equation above. Although these variables have been intensively studied, our new approach based on a combination of probabilistic and analytical techniques enables us to derive comprehensive properties and strengthen several results on the law of these random variables for some classes of Lévy processes that could be found in the literature. These encompass smoothness for its density, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions and bounds. We furnish a thorough study of the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. We deliver intertwining relation between members of the class of positive self-similar semigroups.

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30E25 Boundary value problems in the complex plane
60G51 Processes with independent increments; Lévy processes
33E99 Other special functions
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Afanasyev, V.I., Geiger, J., Kersting, G. and Vatutin, V.A.: Criticality for branching processes in random environment. Ann. Probab., 33(2), (2005), 645–673. · Zbl 1075.60107
[2] Alili, L., Jedidi, W. and Rivero, V.: On exponential functionals, harmonic potential measures and undershoots of subordinators. ALEA Lat. Am. J. Probab. Math. Stat., 11(1), (2014), 711–735. · Zbl 1332.60068
[3] Arendt, W., ter Elst, A.F.M. and Kennedy, J.B.: Analytical aspects of isospectral drums. Oper. Matrices, 8(1), (2014), 255–277. · Zbl 1327.58032
[4] Arista, J. and Rivero, V.: Implicit renewal theory for exponential functionals of Lévy processes. (2015), arXiv:1510.01809
[5] Bérard, P.: Variétés riemanniennes isospectrales non isométriques. Astérisque, (177-178):Exp. No. 705, 127–154, 1989. Séminaire Bourbaki, Vol. 1988/89.
[6] Berg, C.: On powers of Stieltjes moment sequences. II. J. Comput. Appl. Math., 199(1), (2007), 23–38. · Zbl 1111.44001
[7] Berg, C. and Durán, A.J.: A transformation from Hausdorff to Stieltjes moment sequences. Ark. Mat., 42(2), (2004), 239–257. · Zbl 1057.44002
[8] Bernyk, V., Dalang, R.C. and Peskir, G.: The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab., 36(5), (2008), 1777–1789. · Zbl 1185.60051
[9] Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge, 1996.
[10] Bertoin, J., Biane, P. and Yor, M.: Poissonian exponential functionals, \(q\)-series, \(q\)-integrals, and the moment problem for log-normal distributions. In Seminar on Stochastic Analysis, Random Fields and Applications IV, volume 58 of Progr. Probab., pages 45–56. Birkhäuser, Basel, 2004. · Zbl 1056.60046
[11] Bertoin, J. and Caballero, M.E.: Entrance from \(0+\) for increasing semi-stable Markov processes. Bernoulli, 8(2), (2002), 195–205. · Zbl 1002.60032
[12] Bertoin, J., Curien, J. and Kortchemski, I.: Random planar maps & growth-fragmentations. Ann. Probab., 46(1), (2018), 207–260. · Zbl 1447.60058
[13] Bertoin, J. and Kortchemski, I.: Self-similar scaling limits of Markov chains on the positive integers. Ann. Appl. Probab.,26(4), (2016), 2556–2595. · Zbl 1352.60103
[14] Bertoin, J., Lindner, A. and Maller, R.: On continuity properties of the law of integrals of Lévy processes. In Séminaire de probabilités XLI, volume 1934 of Lecture Notes in Math., pages 137–159. Springer, Berlin, 2008. · Zbl 1180.60042
[15] Bertoin, J., and Savov, M.: Some applications of duality for Lévy processes in a half-line. Bull. Lond. Math. Soc., 43(1), (2011), 97–110. · Zbl 1213.60090
[16] Bertoin, J., and Yor, M.: The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal., 17(4), (2002), 389–400. · Zbl 1004.60046
[17] Bertoin, J., and Yor, M.: On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Ann. Fac. Sci. Toulouse Math., 11(1), (2002), 19–32. · Zbl 1031.60038
[18] Bertoin, J., and Yor, M.: Exponential functionals of Lévy processes. Probab. Surv., 2, (2005), 191–212. · Zbl 1189.60096
[19] Borodin, A. and Corwin, I.: Macdonald processes. Probab. Theory Related Fields, 158(1-2), (2014), 225–400.
[20] Caballero, M.E. and Chaumont, L.: Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab., 34(3), (2006), 1012–1034. · Zbl 1098.60038
[21] Carmona, Ph. Petit, F. and Yor, M.: Beta-gamma random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoamericana, 14(2), (1998), 311–368. · Zbl 0919.60074
[22] Chhaibi, R.: A note on a Poissonian functional and a \(q\)-deformed Dufresne identity. Electron. Commun. Probab., 21, (2016), 1–13. · Zbl 1339.33018
[23] Diaconis, P. and Fill, J.A.: Strong stationary times via a new form of duality. Ann. Probab., 18(4), (1990), 1483–1522. · Zbl 0723.60083
[24] Doney, R.: Fluctuation Theory for Lévy Processes. Ecole d’Eté de Probabilités de Saint-Flour XXXV-2005. Springer, 2007. first edition. · Zbl 1128.60036
[25] Doney, R. and Savov, M.: The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab., 38(1), (2010), 316–326. · Zbl 1185.60052
[26] Döring, L. and Savov, M.: (Non)differentiability and asymptotics for potential densities of subordinators. Electron. J. Probab., 16(17), (2011), 470–503. · Zbl 1226.60115
[27] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G.: Higher Transcendental Functions, volume 3. McGraw-Hill, New York-Toronto-London, 1955. · Zbl 0064.06302
[28] Euler. L.: Commentarii Academiae scientiarum imperialis Petropolitanae. De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt, volume t.1 (1726-1728). Petropolis Typis Academiae.
[29] Feller, W.E.: An Introduction to Probability Theory and its Applications, volume 2. Wiley, New York, \(2^{nd}\) edition, 1971. · Zbl 0219.60003
[30] Garret, P.: Phragmén-Lindelöf theorems. page http://www.math.umn.edu/ garrett/.
[31] Goldie, C.M. and Grübel, R.: Perpetuities with thin tails. Adv. in Appl. Probab., 28(2), (1996), 463–480. · Zbl 0862.60046
[32] Gradshteyn, I.S. and Ryshik, I.M.: Table of Integrals, Series and Products. Academic Press, San Diego, \(6^{th}\) edition, 2000.
[33] Haas, B. and Rivero, V.: Quasi-stationary distributions and Yaglom limits of self-similar Markov processes. Stochastic Processes and their Applications, 122(12), (2012), 4054–4095. · Zbl 1267.60038
[34] Hackmann, D. and Kuznetsov, A.: Asian options and meromorphic Lévy processes. Finance Stoch., 18(4), (2014), 825–844. · Zbl 1307.60058
[35] Havin, V.P. and Nikolski, N.K. editors. Commutative harmonic analysis. II, volume 25 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1998.
[36] Hirsch, F. and Yor, M.: On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator. Bernoulli, 19(4), (2013), 1350–1377. · Zbl 1287.60096
[37] Jacob, N.: Pseudo Differential Operators and Markov Processes Vol. 1: Fourier Analysis and Semigroups, volume 1. Imperial College Press, 2001. · Zbl 0987.60003
[38] Kallenberg, O.: Foundations of Modern Probability. Springer, \(2^{nd}\) edition, 2002. · Zbl 0996.60001
[39] Kozlov, M.V.: The asymptotic behaviour of the probability of non-extinction of critical branching processes in a random environment. Teor. Verojatnost. i Primenen., 21(4), (1976), 813–825.
[40] Kuznetsov, A.: On extrema of stable processes. Ann. Probab., 39(3), (2011), 1027–1060. · Zbl 1218.60037
[41] Kuznetsov, A.: On the density of the supremum of a stable process. Stochastic Process. Appl., 123(3), (2013), 986–1003. · Zbl 1277.60084
[42] Kuznetsov, A. and Pardo, J.C.: Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math., 123, (2013), 113–139. · Zbl 1268.60060
[43] Lamperti. J.: Semi-stable stochastic processes. Trans. Amer. Math. Soc., 104, (1962), 62–78, 1962. · Zbl 0286.60017
[44] Lebedev, N.N.: Special functions and their applications. Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication. · Zbl 0271.33001
[45] Li, Z. and Xu, W.: Asymptotic results for exponential functionals of Lévy processes. Stochastic Process. Appl., 128(1), (2018), 108–131. · Zbl 1386.60170
[46] Markushevich, A.I.: Entire functions. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. American Elsevier Publishing Co., Inc., New York, 1966.
[47] Maulik, K. and Zwart, B.: Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl., 116, (2006), 156–177. · Zbl 1090.60046
[48] Miclo, L.: On the Markovian similarity. https://hal-univ-tlse3.archives-ouvertes.fr/hal-01281029v1, 2016. · Zbl 1334.60162
[49] Olver, F.W.J.: Introduction to Asymptotics and Special Functions. Academic Press, 1974. · Zbl 0308.41023
[50] Pal, S. and Shkolnikov, M.: Intertwining diffusions and wave equations. (2013), arXiv:1306.0857
[51] Palau, S., Pardo, J.C. and Smadi, C.: Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment. ALEA Lat. Am. J. Probab. Math. Stat., 13(2), (2016), 1235–1258. · Zbl 1355.60061
[52] Paley, R. and Wiener, N.: Fourier Transforms in the Complex Domain American Mathematical Society, Providence, 1934. · Zbl 0011.01601
[53] Pardo, J.C., Patie, P. and Savov, M.: A Wiener-Hopf type factorization for the exponential functional of Lévy processes. J. Lond. Math. Soc. (2), 86(3), (2012), 930–956. · Zbl 1272.60027
[54] Pardo, J.C., Rivero, V. and van Schaik, K.: On the density of exponential functionals of Lévy processes. Bernoulli, 19(5A), (2013), 1938–1964. · Zbl 1305.60035
[55] Patie, P.: Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration. Bull. Sci. Math., 133(4), (2009), 355–382. · Zbl 1171.60009
[56] Patie, P.: Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat., 45(3), (2009), 667–684. · Zbl 1180.31010
[57] Patie, P.: Law of the absorption time of some positive self-similar Markov processes. Ann. Probab., 40(2), (2012), 765–787. · Zbl 1241.60020
[58] Patie, P.: Asian options under one-sided Lévy models. J. Appl. Probab., 50(2), (2013), 359–373. · Zbl 1266.91109
[59] Patie, P. and Savov, M.: Extended factorizations of exponential functionals of Lévy processes. Electron. J. Probab., 17(38), (2012), 1–22. · Zbl 1253.60063
[60] Patie, P. and Savov, M.: Exponential functional of Lévy processes: generalized Weierstrass products and Wiener-Hopf factorization. C. R. Math. Acad. Sci. Paris, 351(9-10), (2013), 393–396. · Zbl 1273.60057
[61] Patie, P. and Savov, M.: Spectral expansion of non-self-adjoint generalized Laguerre semigroups. Submitted, 162 pages. (current version), 2015. arXiv:1506.01625
[62] Patie, P. and Savov, M.: Cauchy problem of the non-self-adjoint Gauss-Laguerre semigroups and uniform bounds of generalized Laguerre polynomials. J. Spectr. Theory, 7, (2017), 797–846. · Zbl 1460.34073
[63] Patie, P., Savov, M., and Zhao, Y.: Intertwining, excursion theory and Krein theory of strings for non-self-adjoint Markov semigroups. 2017, arXiv:1706.08995
[64] Patie, P. and Simon, T.: Intertwining certain fractional derivatives. Potential Anal., 36(4), (2012), 569–587. · Zbl 1259.60040
[65] Patie, P. and Zhao, Y.: Spectral decomposition of fractional operators and a reflected stable semigroup. J. Differential Equations, 262(3), (2017), 1690–1719. · Zbl 1368.47032
[66] Phargmén, E. and Lindelöf, E.: Sur une extension d’un principe classique de l’analyse. Acta Math., 31, (1908), 381–406.
[67] Rivero, V.: Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli, 11(3), (2005), 471–509. · Zbl 1077.60055
[68] Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge, 1999.
[69] Sato, K. and Yamazato, M.: On distribution functions of class L. Z. Wahrscheinlichkeitstheorie, 43, (1978), 273–308. · Zbl 0395.60019
[70] Schilling, R.L., Song, R. and Vondraček, Z.: Bernstein functions, volume 37 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, (2010). Theory and applications.
[71] Shea, D. and Wainger, St. Variants of the Wiener-Lévy Theorem, with Applications to Stability Problems for Some Volterra Integral Equations American Journal of Mathematics2, (1975), 312–343. · Zbl 0318.45016
[72] Soulier, Ph.: Some applications of regular variation in probability and statistics. XXII Escuela Venezolana de Mathematicas, Instituto Venezolano de Investigaciones Cientcas, (2009).
[73] Titchmarsh, E.C.: The theory of functions. Oxford University Press, Oxford, (1939). · JFM 65.0302.01
[74] Titchmarsh, E.C.: The theory of functions. Oxford University Press, Oxford, (1958). Reprint of the second (1939) edition.
[75] Tucker, H.G.: The supports of infinitely divisible distribution functions. Proc. Amer. Math. Soc., 49, (1975), 436–440. · Zbl 0308.60014
[76] Urbanik, K.: Infinite divisibility of some functionals on stochastic processes. Probab. Math. Statist., 15, (1995), 493–513. · Zbl 0852.60015
[77] Webster, R.: Log-convex solutions to the functional equation \(f(x+1)= f(x)g(x)\): \(Γ \)-type functions. Journal of Mathematical Analysis and Applications, 209, (1997), 605–623. · Zbl 0878.39004
[78] Yor, M.: Exponential functionals of Brownian motion and related processes. Springer Finance, Berlin, (2001). · Zbl 0999.60004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.