Abstract
We develop new estimators for the parameters of Ornstein-Uhlenbeck processes driven by compound Poisson processes, which can be considered as a class of stochastic hybrid systems. Our estimators are derived based on the method of moments. We also establish the central limit theorem for the proposed estimators. Numerical experiments are provided to show that our method performs better when compared with the existing methods, especially in cases when the jumps of the compound Poisson process are relatively rare.
Similar content being viewed by others
References
Barboza LA, Viens FG (2017) Parameter estimation of gaussian stationary processes using the generalized method of moments. Electronic Journal of Statistics 11 (1):401–439
Barndorff-Nielsen OE, Shephard N (2001) Non-gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B (Stat Methodol) 63(2):167–241
Billingsley P (1995) Probability and measure. Wiley series in probability and mathematical statistics. Wiley, New York
Brockwell PJ (2009) Lévy–driven continuous–time arma processes. In: Handbook of financial time series. Springer, pp 457–480
Brockwell PJ, Davis RA, Yang Y (2007) Estimation for nonnegative Lévy-driven Ornstein-Uhlenbeck processes. J Appl Probab 44(4):977–989
Cassandras CG, Lygeros J (2006) Stochastic hybrid systems, vol 24. CRC Press, Boca Raton
Cassandras CG, Wardi Y, Panayiotou CG, Yao C (2010) Perturbation analysis and optimization of stochastic hybrid systems. Eur J Control 16(6):642
Chaussé P (2010) Computing generalized method of moments and generalized empirical likelihood with R. J Stat Softw 34(11):1–35
Gander MP, Stephens DA (2007) Stochastic volatility modelling in continuous time with general marginal distributions: inference, prediction and model selection. Journal of Statistical Planning and Inference 137(10):3068–3081
Griffin JE, Steel MF (2006) Inference with non-gaussian Ornstein–Uhlenbeck processes for stochastic volatility. J Econ 134(2):605–644
Hansen LP (1982) Large sample properties of generalized method of moments estimators. Econometrica 50(4):1029–1054
Hu J, Lygeros J, Sastry S (2000) Towards a theory of stochastic hybrid systems. In: International workshop on hybrid systems: computation and control. Springer, pp 160–173
Jongbloed G, Van Der Meulen FH, Van Der Vaart AW (2005) Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli 11(5):759–791
Mai H (2014) Efficient maximum likelihood estimation for Lévy-driven Ornstein–Uhlenbeck processes. Bernoulli 20(2):919–957
Peng Y, Fu MC, Hu JQ (2016) Gradient-based simulated maximum likelihood estimation for stochastic volatility models using characteristic functions. Quantitative Finance 16(9):1393–1411
Peng YJ, Fu MC, Hu JQ (2014) Gradient-based simulated maximum likelihood estimation for Lévy-driven Ornstein–Uhlenbeck stochastic volatility models. Quantitative Finance 14(8):1399–1414
Roberts GO, Papaspiliopoulos O, Dellaportas P (2004) Bayesian inference for non-gaussian Ornstein–Uhlenbeck stochastic volatility processes. J R Stat Soc Ser B (Stat Methodol) 66(2):369–393
Ross SM (2010) Introduction to probability models. Academic Press, Cambridge
Zhang SB, Zhang XS (2010) Moment estimation of parameters for discretely sampled ou-compound poisson processes. Chinese Journal of Applied Probability 26 (4):384–398
Spiliopoulos K (2009) Method of moments estimation of Ornstein-Uhlenbeck processes driven by general Lévy process. In: Annales de l’ISUP, Institut de statistique de l’Université de Paris, vol 53, pp 3–18
Valdivieso L, Schoutens W, Tuerlinckx F (2009) Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type. Stat Infer Stoch Process 12(1):1–19
Zhang SB, Zhang XS, Sun SG (2006) Parametric estimation of discretely sampled gamma-ou processes. Sci China Ser A Math 49(9):1231–1257
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grants 71571048 and 71720107003, by Fudan University under a ShuangYiLiu grant. We thank the associate editor and three anonymous reviewers for their comments and suggestions.
Rights and permissions
About this article
Cite this article
Wu, Y., Hu, J. & Zhang, X. Moment estimators for the parameters of Ornstein-Uhlenbeck processes driven by compound Poisson processes. Discrete Event Dyn Syst 29, 57–77 (2019). https://doi.org/10.1007/s10626-019-00276-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10626-019-00276-y