Electronic Journal of Probability

Fluctuation theory for Lévy processes with completely monotone jumps

Mateusz Kwaśnicki

Full-text: Open access

Abstract

We study the Wiener–Hopf factorization for Lévy processes $X_{t}$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener–Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_{t}$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_{t}$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi )$ of $X_{t}$, including a peculiar structure of the curve along which $f(\xi )$ takes real values.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 40, 40 pp.

Dates
Received: 25 November 2018
Accepted: 24 March 2019
First available in Project Euclid: 12 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1555034439

Digital Object Identifier
doi:10.1214/19-EJP300

Zentralblatt MATH identifier
1412.60067

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 47A68: Factorization theory (including Wiener-Hopf and spectral factorizations) 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx]

Keywords
complete Bernstein function fluctuation theory Lévy process Nevanlinna–Pick function Wiener–Hopf factorisation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kwaśnicki, Mateusz. Fluctuation theory for Lévy processes with completely monotone jumps. Electron. J. Probab. 24 (2019), paper no. 40, 40 pp. doi:10.1214/19-EJP300. https://projecteuclid.org/euclid.ejp/1555034439


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References

  • [1] D. Applebaum, Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge, 2004.
  • [2] N. Aronszajn, W. F. Donoghue, On exponential representations of analytic functions in the upper half-plane with positive imaginary part. J. Anal. Math. 5 (1956): 321–385.
  • [3] O. E. Barndorff-Nielsen, T. Mikosch, S. I. Resnick (Eds.), Lévy Processes: Theory and Applications. Birkhäuser, Boston, 2001.
  • [4] G. Baxter, M. D. Donsker, On the distribution of the supremum functional for processeswith stationary independent increments. Trans. Amer. Math. Soc. 85 (1957) 73–87.
  • [5] V. Bernyk, R. C. Dalang, G. Peskir, The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36(5) (2008): 1777–1789.
  • [6] J. Bertoin, Lévy Processes. Cambridge Univ. Press, Melbourne, New York, 1996.
  • [7] N. H. Bingham, Maxima of sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 26 (1973): 273–296.
  • [8] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, Z. Vondraček, Potential Analysis of Stable Processes and its Extensions. Lecture Notes in Mathematics 1980, Springer, 2009.
  • [9] K. Bogdan, T. Grzywny, M. Ryznar, Density and tails of unimodal convolution semigroups. J. Funct. Anal. 266(6) (2014): 3543–3571.
  • [10] K. Bogdan, T. Grzywny, M. Ryznar, Barriers, exit time and survival probability for unimodal Lévy processes. Probab. Theory Related Fields 162(1–2) (2015): 155–198.
  • [11] K. Bogdan, T. Grzywny, M. Ryznar, Dirichlet heat kernel for unimodal Lévy processes. Stoch. Proc. Appl. 124(11) (2014): 3612–3650.
  • [12] J. Burridge, A. Kuznetsov, A. E. Kyprianou, M. Kwaśnicki, New families of subordinators with explicit transition probability semigroup. Stoch. Proc. Appl. 124(10) (2014): 3480–3495.
  • [13] Z.-Q. Chen, P. Kim, R. Song, Dirichlet Heat Kernel Estimates for Subordinate Brownian Motions with Gaussian Components. J. Reine Angewandte Math. 711 (2016): 111–138.
  • [14] G. Coqueret, On the supremum of the spectrally negative stable process with drift. Stat. Probab. Lett. 107 (2015): 333–340.
  • [15] W. Cygan, T. Grzywny, B. Trojan, Asymptotic behavior of densities of unimodal convolution semigroups. Trans. Amer. Math. Soc. 369(8) (2017): 5623–5644.
  • [16] D. A. Darling, The maximum of sums of stable random variables. Trans. Amer. Math. Soc. 83 (1956) 164–169.
  • [17] R. A. Doney, On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Probab. 15(4) (1987) 1352–1362.
  • [18] R. A. Doney, Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897, Springer, Berlin, 2007.
  • [19] R. A. Doney, V. Rivero, Asymptotic behaviour of first passage time distributions for Lévy processes. Probab. Theory Related Fields 157(1) (2013): 1–45.
  • [20] R. A. Doney, M. S. Savov, The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab. 38(1) (2010): 316–326.
  • [21] S. Fotopoulos, V. Jandhyala, J. Wang, On the joint distribution of the supremum functional and its last occurrence for subordinated linear Brownian motion. Stat. Probab. Lett. 106 (2015): 149–156.
  • [22] B. E. Fristedt, Sample functions of stochastic processes with stationary, independent increments. In: Advances in Probability and Related Topics, vol. 3, Dekker, New York, 1974, 241–396.
  • [23] P. Graczyk, T. Jakubowski, On Wiener–Hopf factors of stable processes. Ann. Inst. Henri Poincaré (B) 47(1) (2010): 9–19.
  • [24] P. Graczyk, T. Jakubowski, On exit time of stable processes. Stoch. Proc. Appl. 122(1) (2012): 31–41.
  • [25] T. Grzywny, Potential theory of one-dimensional geometric stable processes. Colloq. Math. 129(1) (2012): 7–40.
  • [26] T. Grzywny, On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes. Potential Anal. 41(1) (2014): 1–29.
  • [27] T. Grzywny, M. Ryznar, Hitting Times of Points and Intervals for Symmetric Lévy Processes. Potential Anal. 46(4) (2017): 739–777.
  • [28] T. Grzywny, K. Szczypkowski, Estimates of heat kernels of non-symmetric Lévy processes. Preprint, 2017, arXiv:1710.07793.
  • [29] T. Grzywny, K. Szczypkowski, Heat kernels of non-symmetric Lévy-type operators. Preprint, 2018, arXiv:1804.01313.
  • [30] D. Hackmann, A. Kuznetsov, Approximating Lévy processes with completely monotone jumps. Ann. Appl. Probab. 26(1) (2016): 328–359.
  • [31] C. C. Heyde, On the maximum of sums of random variables and the supremum functional for stable processes. J. Appl. Probab. 6 (1969): 419–429.
  • [32] F. Hubalek, A. Kuznetsov, A convergent series representation for the density of the supremum of a stable process. Elect. Comm. Probab. 16 (2011): 84–95.
  • [33] T. Juszczyszyn, M. Kwaśnicki, Hitting times of points for symmetric Lévy processes with completely monotone jumps. Electron. J. Probab. 20(48) (2015): 1–24.
  • [34] P. Kim, A. Mimica, Green function estimates for subordinate Brownian motions: stable and beyond. Trans. Amer. Math. Soc. 366(8) (2014): 4383–4422.
  • [35] P. Kim, A. Mimica, Estimates of Dirichlet heat kernels for subordinate Brownian motions. Electron. J. Probab. 23(64) (2018): 1–45.
  • [36] P. Kim, R. Song, Z. Vondraček, Potential theory of subordinate Brownian motions revisited. In: T. Zhang, X. Zhou (Eds.), Stochastic Analysis and Applications to Finance–Essays in Honour of Jia-an Yan, World Scientific, 2012, 243–290.
  • [37] P. Kim, R. Song, Z. Vondraček, Potential theory of subordinate Brownian motions with Gaussian components. Stoch. Proc. Appl. 123 (2013): 764–795.
  • [38] P. Kim, R. Song, Z. Vondraček, Global uniform boundary Harnack principle with explicit decay rate and its application. Stoch. Proc. Appl. 124 (2014): 235–267
  • [39] P. Kim, R. Song, Z. Vondraček, Boundary Harnack principle and Martin boundary at infinity for subordinate Brownian motions. Potential Anal. 41(2) (2014): 407–441.
  • [40] P. Kim, R. Song, Z. Vondraček, Heat kernels of non-symmetric jump processes: beyond the stable case. Potential Anal. 49(1) (2018): 37–90.
  • [41] V. Knopova, A. Kulik, Intrinsic compound kernel estimates for the transition probability density of a Lévy type processes and their applications. Probab. Math. Stat. 37(1) (2017): 53–100.
  • [42] T. Kulczycki, M. Kwaśnicki, J. Małecki, A. Stós, Spectral properties of the Cauchy process on half-line and interval. Proc. London Math. Soc. 101(2) (2010): 589–622.
  • [43] A. Kuznetsov, Analytic Proof of Pecherskiĭ–Rogozin Identity and Wiener–Hopf Factorization. Theory Probab. Appl. 55(3) (2010): 432–443.
  • [44] A. Kuznetsov, Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Prob. 20(5) (2010): 1801–1830.
  • [45] A. Kuznetsov, On extrema of stable processes. Ann. Probab. 39(3) (2011): 1027–1060.
  • [46] A. Kuznetsov, On the density of the supremum of a stable process. Stoch. Proc. Appl. 123(3) (2013): 986–1003.
  • [47] A. Kuznetsov, A. Kyprianou, J. C. Pardo, Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22(3) (2012): 1101–1135.
  • [48] A. Kuznetsov, M. Kwaśnicki, Spectral analysis of stable processes on the positive half-line. Electron. J. Probab. 23(10) (2018): 1–29.
  • [49] A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext, Springer-Verlag, Berlin, 2006.
  • [50] A. E. Kyprianou, Deep factorisation of the stable process. Electron. J. Probab. 21(23) (2016): 1–28.
  • [51] A. E. Kyprianou, V. Rivero, B. Şengül, Deep factorisation of the stable process II: Potentials and applications. Ann. Inst. H. Poincaré Probab. Statist. 54(1) (2018): 343–362.
  • [52] A. E. Kyprianou, V. Rivero, W. Satitkanitkul Deep factorisation of the stable process III: Radial excursion theory and the point of closest reach. Preprint, 2017, arXiv:1706.09924.
  • [53] A. E. Kyprianou, J. C. Pardo, A. R. Watson, The extended hypergeometric class of Lévy processes. J. Appl. Probab. 51(A) (2014): 391–408.
  • [54] M. Kwaśnicki, Spectral analysis of subordinate Brownian motions on the half-line. Studia Math. 206(3) (2011): 211–271.
  • [55] M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval. J. Funct. Anal. 262(5) (2012): 2379–2402.
  • [56] M. Kwaśnicki, Spectral theory for one-dimensional symmetric Lévy processes killed upon hitting the origin. Electron. J. Probab. 17 (2012), 83:1–29.
  • [57] M. Kwaśnicki, A new class of bell-shaped functions. Trans. Amer. Math. Soc., in press, arXiv:1710.11023.
  • [58] M. Kwaśnicki, Rogers functions and fluctuation theory. Unpublished, 2013, arXiv:1312.1866.
  • [59] M. Kwaśnicki, J. Małecki, M. Ryznar, First passage times for subordinate Brownian motions. Stoch. Proc. Appl 123 (2013): 1820–1850.
  • [60] M. Kwaśnicki, J. Małecki, M. Ryznar, Suprema of Lévy processes. Ann. Probab. 41(3B) (2013): 2047–2065.
  • [61] M. Kwaśnicki, J. Mucha, Extension technique for complete Bernstein functions of the Laplace operator. J. Evol. Equ. 18(3) (2018): 1341–1379.
  • [62] A.L. Lewis, E. Mordecki, Wiener–Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms. J. Appl. Probab. 45(1) (2008): 118–134.
  • [63] Z. Michna, Explicit formula for the supremum distribution of a spectrally negative stable process. Electron. Commun. Probab. 18 (2013), 10:1–6.
  • [64] H. Pantí, On Lévy processes conditioned to avoid zero. ALEA, Lat. Am. J. Probab. Math. Stat. 14 (2017): 657–690.
  • [65] P. Patie, M. Savov, Cauchy problem of the non-self-adjoint Gauss–Laguerre semigroups and uniform bounds for generalized Laguerre polynomials. J. Spectral Theory 7(3) (2017): 797–846.
  • [66] P. Patie, M. Savov, Spectral expansions of non-self-adjoint generalized Laguerre semigroups. Mem. Amer. Math. Soc., in press, arXiv:1506.01625.
  • [67] P. Patie, M. Savov, Y. Zhao, Intertwining, Excursion Theory and Krein Theory of Strings for Non-self-adjoint Markov Semigroups. Ann. Probab., in press, arXiv:1706.08995.
  • [68] P. Patie, Y. Zhao, Spectral decomposition of fractional operators and a reflected stable semigroup. J. Differ. Equations 262(3) (2017): 1690–1719.
  • [69] E.A. Pecherski, B.A. Rogozin, The joint distributions of random variables associated to fluctuations of a process with independent increments. Teor. Veroyatnost. Primenen.14(3) (1969): 431–444; English transl. in Theory Probab. Appl. 14(3) (1969): 410–423.
  • [70] L.C.G. Rogers, Wiener–Hopf factorization of diffusions and Lévy processes. Proc. London Math. Soc. 47(3) (1983): 177–191.
  • [71] B.A. Rogozin Distribution of certain functionals related to boundary problems for processes with independent increments. Teor. Veroyatnost. i Primenen. 11(4) (1966): 656–670.
  • [72] K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge, 1999.
  • [73] R. Schilling, R. Song, Z. Vondraček, Bernstein Functions: Theory and Applications. De Gruyter, Studies in Math. 37, Berlin, 2012.
  • [74] Y. Tamura, H. Tanaka, On a fluctuation identity for multidimensional Lévy processes. Tokyo J. Math. 25 (2002): 363–380.
  • [75] Y. Tamura, H. Tanaka, On a formula on the potential operators of absorbing Lévy processes in the half space. Stoch. Proc. Appl 118 (2008): 199–212.
  • [76] K. Yano, On harmonic function for the killed process upon hitting zero of asymmetric Lévy processes. J. Math-for-Industry 5 (2013): 17–24.