The Annals of Probability

Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes

Alexander Lindner and Ken-iti Sato

Full-text: Open access

Abstract

Properties of the law μ of the integral 0cNtdYt are studied, where c>1 and {(Nt, Yt), t≥0} is a bivariate Lévy process such that {Nt} and {Yt} are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalized Ornstein–Uhlenbeck process. The law μ is parametrized by c, q and r, where p=1−qr, q, and r are the normalized Lévy measure of {(Nt, Yt)} at the points (1, 0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p>0 and q>0, μc, q, r is infinitely divisible if and only if rpq. The infinite divisibility of the symmetrization of μ is also characterized. The law μ is either continuous-singular or absolutely continuous, unless r=1. It is shown that if c is in the set of Pisot–Vijayaraghavan numbers, which includes all integers bigger than 1, then μ is continuous-singular under the condition q>0. On the other hand, for Lebesgue almost every c>1, there are positive constants C1 and C2 such that μ is absolutely continuous whenever qC1pC2r. For any c>1 there is a positive constant C3 such that μ is continuous-singular whenever q>0 and max {q, r}≤C3p. Here, if {Nt} and {Yt} are independent, then r=0 and q=b/(a+b).

Article information

Source
Ann. Probab., Volume 37, Number 1 (2009), 250-274.

Dates
First available in Project Euclid: 17 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1234881690

Digital Object Identifier
doi:10.1214/08-AOP402

Mathematical Reviews number (MathSciNet)
MR2489165

Zentralblatt MATH identifier
1173.60007

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions 60G10: Stationary processes 60G30: Continuity and singularity of induced measures 60G51: Processes with independent increments; Lévy processes

Keywords
Decomposable distribution generalized Ornstein–Uhlenbeck process infinite divisibility Lévy process Peres–Solomyak (P.S.) number Pisot–Vijayaraghavan (P.V.) number symmetrization of distribution

Citation

Lindner, Alexander; Sato, Ken-iti. Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes. Ann. Probab. 37 (2009), no. 1, 250--274. doi:10.1214/08-AOP402. https://projecteuclid.org/euclid.aop/1234881690


Export citation

References

  • [1] Bertoin, J., Lindner, A. and Maller, R. (2008). On continuity properties of the law of integrals of Lévy processes. Séminaire de Probabilités XLI. Lecture Notes in Math. 1934 137–159. Springer, Berlin.
  • [2] Bunge, J. (1997). Nested classes of C-decomposable laws. Ann. Probab. 25 215–229.
  • [3] Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion. Bibl. Rev. Mat. Iberoamericana 73–130. Rev. Mat. Iberoamericana, Madrid.
  • [4] Carmona, P., Petit, F. and Yor, M. (2001). Exponential functionals of Lévy processes. In Lévy Processes (O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, eds.) 41–55. Birkhäuser, Boston, MA.
  • [5] Erdös, P. (1939). On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61 974–976.
  • [6] Erickson, K. B. and Maller, R. A. (2005). Generalised Ornstein–Uhlenbeck processes and the convergence of Lévy integrals. Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 70–94. Springer, Berlin.
  • [7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. 2, 2nd ed. Wiley, New York.
  • [8] Gjessing, H. K. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stochastic Process. Appl. 71 123–144.
  • [9] Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables, rev. ed. Addison-Wesley, Reading, MA.
  • [10] Gradshteyn, I. S. and Ryzhik, I. M. (1965). Table of Integrals, Series, and Products, 4th ed. Academic Press, New York.
  • [11] de Haan, L. and Karandikar, R. L. (1989). Embedding a stochastic difference equation into a continuous-time process. Stochastic Process. Appl. 32 225–235.
  • [12] Katti, S. K. (1967). Infinite divisibility of integer-valued random variables. Ann. Math. Statist. 38 1306–1308.
  • [13] Klüppelberg, C., Lindner, A. and Maller, R. (2006). Continuous time volatility modeling: COGARCH versus Ornstein–Uhlenbeck models. In From Stochastic Calculus to Mathematical Finance (Yu. Kabanov, R. Liptser and J. Stoyanov, eds.) 393–419. Springer, Berlin.
  • [14] Kondo, H., Maejima, M. and Sato, K. (2006). Some properties of exponential integrals of Lévy processes and examples. Electron. Comm. Probab. 11 291–303 (electronic).
  • [15] Lang, S. (1965). Algebra, 3rd ed. Addison-Wesley, Reading, MA.
  • [16] Lindner, A. and Maller, R. (2005). Lévy integrals and the stationarity of generalised Ornstein–Uhlenbeck processes. Stochastic Process. Appl. 115 1701–1722.
  • [17] Niedbalska-Rajba, T. (1981). On decomposability semigroups on the real line. Colloq. Math. 44 347–358 (1982).
  • [18] Paulsen, J. (1998). Ruin theory with compounding assets—a survey. Insurance Math. Econom. 22 3–16.
  • [19] Peres, Y., Schlag, W. and Solomyak, B. (2000). Sixty years of Bernoulli convolutions. In Fractal Geometry and Stochastics II (Greifswald/Koserow, 1998) (C. Bandt, S. Graf and M. Zähle, eds.). Progr. Probab. 46 39–65. Birkhäuser, Basel.
  • [20] Peres, Y. and Solomyak, B. (1998). Self-similar measures and intersections of Cantor sets. Trans. Amer. Math. Soc. 350 4065–4087.
  • [21] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [22] Steutel, F. W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. Monographs and Textbooks in Pure and Applied Mathematics 259. Dekker, New York.
  • [23] Watanabe, T. (2000). Absolute continuity of some semi-selfdecomposable distributions and self-similar measures. Probab. Theory Related Fields 117 387–405.
  • [24] Watanabe, T. (2001). Temporal change in distributional properties of Lévy processes. In Lévy Processes (O. E. Barndorff-Nielsen, T. Mikosch and S.I. Resnick, eds.) 89–107. Birkhäuser, Boston, MA.
  • [25] Wolfe, S. J. (1983). Continuity properties of decomposable probability measures on Euclidean spaces. J. Multivariate Anal. 13 534–538.
  • [26] Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.