Methodology and Computing in Applied Probability

, Volume 13, Issue 3, pp 619–656 | Cite as

Transition Law-based Simulation of Generalized Inverse Gaussian Ornstein–Uhlenbeck Processes

  • Shibin Zhang


In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of three independent random variables—one follows a distribution whose density is a deconvolution of the densities of two generalized inverse Gaussian distributions, and the two others all have compound Poisson distributions. Based on the representation of the stochastic integral, a simulation procedure for obtaining discretely observed values of Ornstein–Uhlenbeck processes with given generalized inverse Gaussian distribution is provided. For some subclasses of the generalized inverse Gaussian Ornstein–Uhlenbeck process, the innovations can be sampled exactly. The performance of the simulation method is evidenced by some empirical results.


Self-decomposability Generalized inverse Gaussian Ornstein–Uhlenbeck Random sample generation Estimation 

AMS 2000 Subject Classifications

60E07 62E15 65C10 62M05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andrews TG (1984) Special functions for engineers and applied mathematicians. Macmillan Publishing Company, New YorkGoogle Scholar
  2. Asmussen S, Glynn PW (2007) Stochastic simulation: algorithms and analysis. SpringerGoogle Scholar
  3. Atkinson AC (1982) The simulation of generalized inverse Gaussian and hyperbolic random variables. SIAM J Sci Statist Comput 3:502–515MATHCrossRefMathSciNetGoogle Scholar
  4. Barndorff-Nielsen OE (1998) Processes of normal inverse Gaussian type. Finance Stoch 2:41–68MATHCrossRefMathSciNetGoogle Scholar
  5. Barndorff-Nielsen OE, Halgreen C (1977) Infinite divisibility of the hyperbolic and generalized inverse gaussian distributions. Z Wahrscheinlichkeitstheor Verw Geb 38:309–311CrossRefMathSciNetGoogle Scholar
  6. Barndorff-Nielsen OE, Shephard N (2001a) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc B Stat Methodol 63:167–241MATHCrossRefMathSciNetGoogle Scholar
  7. Barndorff-Nielsen OE, Shephard N (2001b) Normal modified stable processes. Theory Probab Math Stat 65:1–19MATHMathSciNetGoogle Scholar
  8. Barndorff-Nielsen OE, Shephard N (2001c) Modelling by Lévy processes for financial econometrics. In: Barndorff-Nielsen OE, Mikosch T, Resnick S (eds) Lévy processes-theory and applications. Birkhäuser, Boston, pp 283–318Google Scholar
  9. Barndorff-Nielsen OE, Shephard N (2003) Integrated OU processes and non-Gaussian OU-based stochastic volatility models. Scand J Statist 30:277–295MATHCrossRefMathSciNetGoogle Scholar
  10. Barndorff-Nielsen OE, Leonenko NN (2005) Spectral properties of superpositions of Ornstein–Uhlenbeck type processes. Methodol Comput Appl Probab 7:335–352MATHCrossRefMathSciNetGoogle Scholar
  11. Barndorff-Nielsen OE, Jensen JL, Sørensen M (1998) Some stationary processes in discrete and continuous time. Adv Appl Probab 30:989-1007MATHCrossRefGoogle Scholar
  12. Blaesild P (1978) The shape of the generalized inverse gaussian and hyperbolic distributions. Research Report 37, Department of Theoretical Statistics, Aarhus University, Aarhus, DenmarkGoogle Scholar
  13. Bondesson L (1982) On simulation from infinitely divisible distributions. Adv Appl Probab 14:855–869MATHCrossRefMathSciNetGoogle Scholar
  14. Cariboni C, Schoutens W (2009) Jumps in intensity models: investigating the performance of Ornstein–Uhlenbeck processes in credit risk modeling. Metrika 69:173–198CrossRefMathSciNetGoogle Scholar
  15. Dagpunar JS (1989) An easily implemented generalized inverse Gaussian generator. Commun Stat, Simul 18:703–710CrossRefMathSciNetGoogle Scholar
  16. Devroye L (1986) Non-uniform random variate generation. Springer, New YorkMATHGoogle Scholar
  17. Eberlein E, von Hammerstein EA (2004) Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In: Seminar on stochastic analysis, random fields and applications IV (Progress Prob 58). Birkhäuser, Basel, pp 221–264Google Scholar
  18. Feller W (1971) An introduction to probability theory and its applications, vol II, 2nd edn. Wiley, New YorkMATHGoogle Scholar
  19. Ferguson TS, Klass MJ (1972) A representation of independent increment processes without Gaussian components. Ann Math Stat 43:1634–1643MATHCrossRefMathSciNetGoogle Scholar
  20. Fotopoulos SB, Jandhyala VK (2004) Bessel inequalities with applications to conditional log returns under GIG scale mixtures of normal vectors. Stat Probab Lett 66:117–125MATHCrossRefMathSciNetGoogle Scholar
  21. Gander MPS, Stephens DA (2007a) Stochastic volatility modelling in continous time with general marginal distributions: inference, prediction and model selection. J Stat Plan Inference 137:3068–3081MATHCrossRefMathSciNetGoogle Scholar
  22. Gander MPS, Stephens DA (2007b) Simulation and inference for stochastic volatility models driven by Lévy processes. Biometrika 94(3):627–646MATHCrossRefMathSciNetGoogle Scholar
  23. Grosswald E (1976) The student-t distribution of any degree of freedom is infinitely divisible. Z Wahrscheinlichkeitstheor Verw Geb 36:103–109MATHCrossRefMathSciNetGoogle Scholar
  24. Halgreen C (1979) Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Zeit Wahrsch Verw Gebiete 47:13–17MATHCrossRefMathSciNetGoogle Scholar
  25. Hömann W, Leydold J, Deringer G (2004) Automatic nonuniform random variate generation. Springer, BerlinGoogle Scholar
  26. Ihaka R, Gentleman R (1996) R: a language for data analysis and graphics. J Comput Graph Stat 5:299–314CrossRefGoogle Scholar
  27. Jacod J (2004) The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann Probab 32(3A):1830–1872MATHCrossRefMathSciNetGoogle Scholar
  28. Jacod J, Kurtz TG, Méléard S, Protter P (2005) The approximate Euler method for Lévy driven stochastic differential equations. Ann Inst H Poincaré Probab Statist 41:523–558MATHCrossRefGoogle Scholar
  29. Janicki A, Weron A (1994) Can one see α-stable variables and processes? Stat Sci 9:109–126MATHCrossRefMathSciNetGoogle Scholar
  30. Jorgensen B (1982) Statistical properties of the generalized inverse Gaussian distribution. In: Lecture notes in statistics, vol 9. Springer, BerlinGoogle Scholar
  31. Kallenberg O (1997) Foundations of modern probability. SpringerGoogle Scholar
  32. Kessler M (2000) Simple and explicit estimating functions for a discretely observed diffusion process. Scand J Statist 27:65–82MATHCrossRefMathSciNetGoogle Scholar
  33. Khintchine A Ya (1937) Zur theorie der unbeschränkt teilbaren verteilungsgesetze mat. Sbornik 2:79–119MATHGoogle Scholar
  34. L’Ecuyer P (2004) Random number generation. In: Gentle J, Härdle W, Mori Y (eds) Handbook of computational statistics: concepts and methods. Spinger, HeidelbergGoogle Scholar
  35. Magnus W, Oberhettinger F (1948) Formeln und Sätze für die speziellen Funktionen der mathematischen Physik. Springer, BerlinMATHGoogle Scholar
  36. Marsaglia G, Tsang WW (2001) A simple method for generating gamma variables. ACM Trans Math Softw 24:341–350CrossRefMathSciNetGoogle Scholar
  37. Michael JR, Schucany WR, Haas RW (1976) Generating random variates using transformations with multiple roots. Am Stat 30:88–90MATHCrossRefGoogle Scholar
  38. Piessens R, deDoncker-Kapenga E, Uberhuber C, Kahaner D (1983) Quadpack: a subroutine package for automatic integration. Springer, BerlinMATHGoogle Scholar
  39. Rosinski J (2001) Contribution to the discussion of a paper by Barndorff-Nielsen and Shephard. J R Stat Soc B 63:230–231Google Scholar
  40. Rosinski J (2002) Series representations of Lévy processes from the perspective of point processes. In: Barndorff-Nielsen OE, Mikosch T, Resnick S (eds) Lévy processes—theory and applications. Birkhäuser, Boston, pp 401–415Google Scholar
  41. Rubenthaler S (2003) Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stoch Process Appl 103:311–349MATHCrossRefMathSciNetGoogle Scholar
  42. Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, CambridgeMATHGoogle Scholar
  43. Schoutens W (2003) Lévy processes in finance. Wiley, ChichesterCrossRefGoogle Scholar
  44. Taufer E, Leonenko N (2008) Simulation of Lévy-driven Ornstein–Uhlenbeck processes with given marginal distribution. Comput Stat Data Anal 53:2427–2437CrossRefMathSciNetGoogle Scholar
  45. Watson GN (1944) A treatise on the theory of Bessel functions. Cambridge University Press, CambridgeMATHGoogle Scholar
  46. Watson GN (1966) A treatise on the theory of Bessel functions, 2nd edn. Cambridge University Press, LondonMATHGoogle Scholar
  47. Wolfe J (1982) On a continuous analogue of the stochastic difference equation X n = ρX n − 1 + B n. Stoch Process their Appl 12:301–312MATHCrossRefMathSciNetGoogle Scholar
  48. Zhang S (2008) Simulation of non-Gaussian OU-based stochastic volatility models. In: Ai C, Wu D (eds) Proceedings of international symposium on financial engineering and risk management 2008, pp 234–238Google Scholar
  49. Zhang S, Zhang X (2008) Exact simulation of IG-OU processes. Methodol Comput Appl Probab 10(3):337–355MATHCrossRefMathSciNetGoogle Scholar
  50. Zhang S, Zhang X (2009) On the transition law of tempered stable Ornstein–Uhlenbeck processes. J Appl Probab 46(3):721–731MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Maritime UniversityShanghaiChina

Personalised recommendations