Electronic Journal of Probability

Bernstein-gamma functions and exponential functionals of Lévy processes

Pierre Patie and Mladen Savov

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Abstract

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 $\vert M_\Psi (z)\vert $ 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.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 75, 101 pp.

Dates
Received: 8 October 2016
Accepted: 20 July 2018
First available in Project Euclid: 27 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1532678638

Digital Object Identifier
doi:10.1214/18-EJP202

Subjects
Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 60G51: Processes with independent increments; Lévy processes 60J55: Local time and additive functionals 60E07: Infinitely divisible distributions; stable distributions 44A60: Moment problems 33E99: None of the above, but in this section

Keywords
asymptotic analysis functional equations exponential functional Lévy processes Wiener-Hopf factorizations special functions intertwining Bernstein functions

Rights
Creative Commons Attribution 4.0 International License.

Citation

Patie, Pierre; Savov, Mladen. Bernstein-gamma functions and exponential functionals of Lévy processes. Electron. J. Probab. 23 (2018), paper no. 75, 101 pp. doi:10.1214/18-EJP202. https://projecteuclid.org/euclid.ejp/1532678638


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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.
  • [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.
  • [3] Arendt, W., ter Elst, A.F.M. and Kennedy, J.B.: Analytical aspects of isospectral drums. Oper. Matrices, 8(1), (2014), 255–277.
  • [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.
  • [7] Berg, C. and Durán, A.J.: A transformation from Hausdorff to Stieltjes moment sequences. Ark. Mat., 42(2), (2004), 239–257.
  • [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.
  • [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.
  • [11] Bertoin, J. and Caballero, M.E.: Entrance from $0+$ for increasing semi-stable Markov processes. Bernoulli, 8(2), (2002), 195–205.
  • [12] Bertoin, J., Curien, J. and Kortchemski, I.: Random planar maps & growth-fragmentations. Ann. Probab., 46(1), (2018), 207–260.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [18] Bertoin, J., and Yor, M.: Exponential functionals of Lévy processes. Probab. Surv., 2, (2005), 191–212.
  • [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.
  • [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.
  • [22] Chhaibi, R.: A note on a Poissonian functional and a $q$-deformed Dufresne identity. Electron. Commun. Probab., 21, (2016), 1–13.
  • [23] Diaconis, P. and Fill, J.A.: Strong stationary times via a new form of duality. Ann. Probab., 18(4), (1990), 1483–1522.
  • [24] Doney, R.: Fluctuation Theory for Lévy Processes. Ecole d’Eté de Probabilités de Saint-Flour XXXV-2005. Springer, 2007. first edition.
  • [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.
  • [26] Döring, L. and Savov, M.: (Non)differentiability and asymptotics for potential densities of subordinators. Electron. J. Probab., 16(17), (2011), 470–503.
  • [27] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G.: Higher Transcendental Functions, volume 3. McGraw-Hill, New York-Toronto-London, 1955.
  • [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.
  • [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.
  • [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.
  • [34] Hackmann, D. and Kuznetsov, A.: Asian options and meromorphic Lévy processes. Finance Stoch., 18(4), (2014), 825–844.
  • [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.
  • [37] Jacob, N.: Pseudo Differential Operators and Markov Processes Vol. 1: Fourier Analysis and Semigroups, volume 1. Imperial College Press, 2001.
  • [38] Kallenberg, O.: Foundations of Modern Probability. Springer, $2^{nd}$ edition, 2002.
  • [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.
  • [41] Kuznetsov, A.: On the density of the supremum of a stable process. Stochastic Process. Appl., 123(3), (2013), 986–1003.
  • [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.
  • [43] Lamperti. J.: Semi-stable stochastic processes. Trans. Amer. Math. Soc., 104, (1962), 62–78, 1962.
  • [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.
  • [45] Li, Z. and Xu, W.: Asymptotic results for exponential functionals of Lévy processes. Stochastic Process. Appl., 128(1), (2018), 108–131.
  • [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.
  • [48] Miclo, L.: On the Markovian similarity. https://hal-univ-tlse3.archives-ouvertes.fr/hal-01281029v1, 2016.
  • [49] Olver, F.W.J.: Introduction to Asymptotics and Special Functions. Academic Press, 1974.
  • [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.
  • [52] Paley, R. and Wiener, N.: Fourier Transforms in the Complex Domain American Mathematical Society, Providence, 1934.
  • [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.
  • [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.
  • [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.
  • [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.
  • [57] Patie, P.: Law of the absorption time of some positive self-similar Markov processes. Ann. Probab., 40(2), (2012), 765–787.
  • [58] Patie, P.: Asian options under one-sided Lévy models. J. Appl. Probab., 50(2), (2013), 359–373.
  • [59] Patie, P. and Savov, M.: Extended factorizations of exponential functionals of Lévy processes. Electron. J. Probab., 17(38), (2012), 1–22.
  • [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.
  • [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.
  • [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.
  • [65] Patie, P. and Zhao, Y.: Spectral decomposition of fractional operators and a reflected stable semigroup. J. Differential Equations, 262(3), (2017), 1690–1719.
  • [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.
  • [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.
  • [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 Mathematics 2, (1975), 312–343.
  • [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).
  • [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.
  • [76] Urbanik, K.: Infinite divisibility of some functionals on stochastic processes. Probab. Math. Statist., 15, (1995), 493–513.
  • [77] Webster, R.: Log-convex solutions to the functional equation $f(x+1)= f(x)g(x)$: $\Gamma $-type functions. Journal of Mathematical Analysis and Applications, 209, (1997), 605–623.
  • [78] Yor, M.: Exponential functionals of Brownian motion and related processes. Springer Finance, Berlin, (2001).