Skip to main content
Log in

Moment estimators for the parameters of Ornstein-Uhlenbeck processes driven by compound Poisson processes

  • Published:
Discrete Event Dynamic Systems Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Billingsley P (1995) Probability and measure. Wiley series in probability and mathematical statistics. Wiley, New York

    MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Cassandras CG, Lygeros J (2006) Stochastic hybrid systems, vol 24. CRC Press, Boca Raton

    Book  Google Scholar 

  • Cassandras CG, Wardi Y, Panayiotou CG, Yao C (2010) Perturbation analysis and optimization of stochastic hybrid systems. Eur J Control 16(6):642

    Article  MathSciNet  MATH  Google Scholar 

  • Chaussé P (2010) Computing generalized method of moments and generalized empirical likelihood with R. J Stat Softw 34(11):1–35

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Griffin JE, Steel MF (2006) Inference with non-gaussian Ornstein–Uhlenbeck processes for stochastic volatility. J Econ 134(2):605–644

    Article  MathSciNet  MATH  Google Scholar 

  • Hansen LP (1982) Large sample properties of generalized method of moments estimators. Econometrica 50(4):1029–1054

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Mai H (2014) Efficient maximum likelihood estimation for Lévy-driven Ornstein–Uhlenbeck processes. Bernoulli 20(2):919–957

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Ross SM (2010) Introduction to probability models. Academic Press, Cambridge

    MATH  Google Scholar 

  • 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

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang SB, Zhang XS, Sun SG (2006) Parametric estimation of discretely sampled gamma-ou processes. Sci China Ser A Math 49(9):1231–1257

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jianqiang Hu.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10626-019-00276-y

Keywords

Navigation