Electronic Journal of Probability

Fluctuation theory for Lévy processes with completely monotone jumps

Mateusz Kwaśnicki

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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

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

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

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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]

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

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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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  • [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.