Abstract
In this article we propose a maximum likelihood methodology to estimate the parameters of a one-dimensional stationary process of Ornstein-Uhlenbeck type that is constructed via a self-decomposable distribution D. Our approach is based on the inversion of the characteristic function and the use of the classical or fractional discrete fast Fourier transform. The results are illustrated throughout an extensive simulation study. This includes the cases where D belongs to the gamma, tempered stable and normal inverse Gaussian family of distributions.
Similar content being viewed by others
References
Asmussen S, Rosiński J (2001) Approximations of small jumps of Lévy processes with a view towards simulation. J Appl Proba 38: 482–493
Bailey DH, Swarztrauber PN (1991) The fractional fourier transform and applications. SIAM Rev 33(3): 389–404
Barndorff-Nielsen OE (1988) Processes of normal inverse Gaussian type. Finance Stochastic 2: 41–68
Barndorff-Nielsen OE, Shepard N (2001) Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc B 63: 167–241
Barndorff-Nielsen OE, Shepard N (2002) Normal modified stable processes. Theory Probab Math Stat 2: 1–20
Cariboni J, Schoutens W (2006) Jumps in intensity models. Technical Report 1, University Centre for Statistics, K.U. Leuven, Belgium
Halgreen C (1979) Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z Wahrscheinlichkeit 47: 13–18
Jiang W, Pedersen J (2003) Parameter estimation for a discretely observed stochastic volatility model with jumps in the volatility. Chi Ann Math 24: 227–238
Jongbloed G, Van Der Meulen FH, Van Der Vaart AW (2005) Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli 11: 759–791
Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. Springer, New York
Kyprianou A (2006) Introductory lectures on fluctuations of Lévy processes with applications. Springer
Lukacs E (1970) Characteristic functions. Charles Griffin and Co, London
Michael JR, Schucany WR, Haas RW (1976) Generating random variates using transformations with multiple roots. Am Stat 30: 88–90
Prause K (1999) The generalized hyperbolic model: estimation, financial derivatives, and risk measures. PhD thesis, Freiburg
Protter P (1990) Stochastic integration and differential equations. Springer-Verlag, Heidelberg
Raible S (2000) Lévy processes in finance: theory, numerics, and empirical facts. PhD thesis, Freiburg
Rosiński J (2001) Contribution to the discussion of a paper by Barndorff-Nielsen and Shephard. J R Stat Soc 63: 230–231
Rosiński J (2002) Tempered stable processes. In: Bandorff-Nielsen OE (ed) Miniproceddings of 2nd MaPhySto Conference on Lévy processes: Theory and applications, pp 215–220
Rosiński J (2007) Tempering stable processes. Stoch Process Appl 117: 677–707
Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press
Schoutens W (2003) Lévy processes in finance: pricing financial derivatives. Wiley, Chichester
Storn R, Price K (1997) Differential evolution—a simple and efficient Heuristic for global optimization over continuous spaces. J Glob Optim 11: 341–359
Yamazato M (1978) Unimodality of infinitely divisible distribution functions of class L. Ann Prob 6: 523–531
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Valdivieso, L., Schoutens, W. & Tuerlinckx, F. Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type. Stat Inference Stoch Process 12, 1–19 (2009). https://doi.org/10.1007/s11203-008-9021-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11203-008-9021-8