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On Simulation of a Fractional Ornstein–Uhlenbeck Process of the Second Kind by the Circulant Embedding Method

  • José Igor Morlanes
  • Andriy Andreev
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)

Abstract

We demonstrate how to utilize the Circulant Embedding Method (CEM) for simulation of fractional Ornstein–Uhlenbeck process of the second kind (\({ fOU}_2\)). The algorithm contains two major steps. First, the relevant covariance matrix is embedded into a circulant one. Second, a sample from the \({ fOU}_2\) is obtained by means of fast Fourier transform applied on the circulant extended matrix. The main goal of this paper is to explain both steps in detail. As a result, we obtain an accurate and an efficient algorithm for generating \({ fOU}_2\) random vectors. We also indicate that the above described procedure can be extended to applications with non-Gaussian marginals.

Keywords

Fractional Brownian motion Fractional Ornstein–Uhlenbeck process Circulant embedding method Simulation 

Notes

Acknowledgements

We thank Michael Carlson for assistance with final English check.

References

  1. 1.
    Andreev, A., Morlanes, J.I.: Simulation-based studies of covariance structure for fractional Ornstein-Uhlenbeck process of the second kind, submitted manuscript (2018)Google Scholar
  2. 2.
    Asmussen, S.: Stochastic simulation with a view towards stochastic processes. University of Aarhus, Center for Mathematical Physics and Stochastics (MaPhySto) [MPS] (1998)Google Scholar
  3. 3.
    Azmoodeh, E.: Riemann-Stiltjes integrals with respect to fractional Brownian motion and applications. Ph.D. dissertation, Helsinki University of Technology Institute of Mathematics (2010)Google Scholar
  4. 4.
    Azmoodeh, E., Morlanes, J.I.: Drift parameter estimation for fractional Ornstein-Uhlenbeck process of the second kind. Statistics 49(1), 1–18 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Azmoodeh, E., Viitasaari, L.: Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind. Stat. Inference Stoch. Process. 18(3), 205–227 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chan, G., Wood, A.T.A.: An algorithm for simulating stationary Gaussian random fields. J. R. Stat. Soc.: Ser. C (Appl. Stat.) 46(1), 171–181 (1997)CrossRefGoogle Scholar
  7. 7.
    Chan, G., Wood, A.T.A.: Simulation of stationary Gaussian vector fields. Stat. Comput. 9(4), 265–268 (1999)CrossRefGoogle Scholar
  8. 8.
    Davies, R.B., Harte, D.S.: Tests for Hurst effect. Biometrika 74, 95–102 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dietrich, C.R., Newsam, G.N.: Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Stat. Comput. 18(4), 1088–1107 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gneiting, T., Sevcikova, H., Percival, D.B., Schalther, M., Jiang, Y.: Fast and exact simulation of large Gaussian lattice systems: exploring the limits. J. Comput. Graph. Stat. 15(3) (2006)Google Scholar
  11. 11.
    Grigoriu, M.: Simulation of stationary non-Gaussian translation processes. J. Eng. Mech. 124(2), 121–126 (1998)CrossRefGoogle Scholar
  12. 12.
    Hosking, J.R.M.: Modeling persistence in hydrological time series using fractional differencing. Water Resour. Res. 20(12), 1898–1908 (1984)CrossRefGoogle Scholar
  13. 13.
    Kaarakka, T., Salminen, P.: On fractional Ornstein-Uhlenbeck processes. Commun. Stoch. Anal. 5(1), 121–133 (2011)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mandelbrot, B., van Ness, J.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nualart, D.: The Malliavin Calculus and Related Topics, Probability and Its Applications. Springer, Berlin (2006)Google Scholar
  16. 16.
    Uhlenbeck, G.E., Ornstein, L.S.: On the theory of Brownian motion. Phys. Rev. 36(5), 823 (1930)CrossRefGoogle Scholar
  17. 17.
    Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5(2), 177–188 (1977)CrossRefGoogle Scholar
  18. 18.
    Wood, A.T.A., Chan, G.: Simulation of stationary Gaussian processes in [0,1]. J. Comput. Graph. Stat. 3(4), 409–432 (1994)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of StatisticsStockholm UniversityStockholmSweden

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