Skip to main content
SpringerLink
Log in

Local linear estimator for stochastic differential equations driven by α-stable Lévy motions

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We study the local linear estimator for the drift coefficient of stochastic differential equations driven by α-stable Lévy motions observed at discrete instants. Under regular conditions, we derive the weak consistency and central limit theorem of the estimator. Compared with Nadaraya-Watson estimator, the local linear estimator has a bias reduction whether the kernel function is symmetric or not under different schemes. A simulation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator, especially on the boundary.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aït-Sahalia Y, Jacod J. Testing for jumps in a discretely observed process. Ann Statist, 2009, 37: 184–222

    Article  MATH  MathSciNet  Google Scholar 

  2. Aït-Sahalia Y, Jacod J. Estimating the degree of activity of jumps in high frequency data. Ann Statist, 2009, 37: 2202–2224

    Article  MATH  MathSciNet  Google Scholar 

  3. Allen D M. The relationship between variable and data augmentation and a method of prediction. Technometrics, 1974, 16: 125–127

    Article  MATH  MathSciNet  Google Scholar 

  4. Andersen T, Benzoni L, Lund J. An empirical investigation of continuous time equity return models. J Finance, 2002, 57: 1239–1284

    Article  Google Scholar 

  5. Bandi F, Nguyen T. On the functional estimation of jump diffusion models. J Econometrics, 2003, 116: 293–328

    Article  MATH  MathSciNet  Google Scholar 

  6. Bandi F, Phillips P. Fully nonparametric estimation of scalar diffusion models. Econometrica, 2003, 71: 241–283

    Article  MATH  MathSciNet  Google Scholar 

  7. Bandi F, Phillips P. A simple approach to the parametric estimation of potentially nonstationary diffusions. J Econometrics, 2007, 137: 354–395

    Article  MathSciNet  Google Scholar 

  8. Barndorff-Nielsen O E, Mikosch T, Resnick S. Lévy Processes, Theory and Applications. Boston: Birkhäuser, 2001

    Book  MATH  Google Scholar 

  9. Baskshi G, Cao Z, Chen Z. Empirical performance of alternative option pricing models. J Finance, 1997, 52: 2003–2049

    Article  Google Scholar 

  10. Bosq D. Nonparametric Statistics for Stochastic Processes. Lecture Notes in Statistics, vol. 110. New York: Springer-Verlag, 1996

    Book  Google Scholar 

  11. Bradley R. Basic properties of strong mixing conditions: A survey and some open questions. Probab Surv, 2005, 2: 107–144

    Article  MATH  MathSciNet  Google Scholar 

  12. Cleveland W. Robust locally weighted regression and smoothing scatter plots. J Amer Statis Assoc, 1979, 74: 829–836

    Article  MATH  MathSciNet  Google Scholar 

  13. Duffie D, Pan J, Singleton K. Transform analysis and asset pricing for affine jump-diffusions. Econometrica, 2000, 68: 1343–1376

    Article  MATH  MathSciNet  Google Scholar 

  14. Fan J. Design-adaptive nonparametric regression. J Amer Statis Assoc, 1992, 87: 998–1004

    Article  MATH  Google Scholar 

  15. Fan J, Gijbels I. Variable bandwidth and local linear regression smoothers. Ann Statist, 1992, 20: 2008–2036

    Article  MATH  MathSciNet  Google Scholar 

  16. Fan J, Gijbels I. Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation. J R Stat Soc Ser B, 1995, 57: 371–394

    MATH  MathSciNet  Google Scholar 

  17. Fan J, Gijbels I. Local Polynomial Modeling and its Applications. New York: Chapman and Hall, 1996

    Google Scholar 

  18. Fan J, Zhang C. A re-examination of diffusion estimators with applications to financial model validation. J Amer Statist Assoc, 2003, 98: 118–134

    Article  MATH  MathSciNet  Google Scholar 

  19. Gasser T, Müller H G. Kernel estimation of regression functions. In: Smoothing Techniques for Curve Estimation. Lecture Notes in Math, vol. 757. New York: Springer, 1979, 23–68

    Chapter  Google Scholar 

  20. Hall P, Presnell B. Intentionally biased bootstrap methods. J R Stat Soc Ser B, 1999, 61: 143–158

    Article  MATH  MathSciNet  Google Scholar 

  21. Hanif M. Local linear estimation of recurrent jump-diffusion models. Comm Statist Theory Methods, 2012, 41: 4142–4163

    Article  MATH  MathSciNet  Google Scholar 

  22. Hu Y, Long H. Least squares estimator for Ornstein-Uhlenbeck processes driven by α-stable motions. Stochastic Process Appl, 1999, 119: 2465–2480

    Article  MathSciNet  Google Scholar 

  23. Jacod J, Kurtz T G, Méléard S, et al. The approximate Euler method for Lévy driven stochastic differential equations. Ann Inst H Poincaré Probab Statist, 2005, 41: 523–558

    Article  MATH  Google Scholar 

  24. Jacod J, Li Y, Mykland P A, et al. Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Process Appl, 2009, 119: 2249–2276

    Article  MATH  MathSciNet  Google Scholar 

  25. Jacod J, Protter P. Discretization of Processes. Heidelberg-New York: Springer-Verlag, 2012

    Book  MATH  Google Scholar 

  26. Jacod J, Shiryaev A. Limit Theory for Stochastic Processes. Berlin: Springer, 2003

    Book  Google Scholar 

  27. Janicki A, Weron A. Simulation and Chaotic Behavior of α-Stable Stochastic Processes. New York: Marcel Dekker, 1994

    Google Scholar 

  28. Johannes M S. The economic and statistical role of jumps to interest rates. J Finance, 2004, 59: 227–260

    Article  Google Scholar 

  29. Kyprianou A, Schouten W, Wilmott P. Exotic Option Pricing and Advanced Lévy Models. Hoboken: Wiley, 2005

    MATH  Google Scholar 

  30. Long H, Qian L. Nadaraya-Watson estimator for stochastic processes driven by stable Lévy motions. Working paper, http://math.fau.edu/long/QianLongSPA08.pdf

  31. Masuda H. On multi-dimensional Ornstein-Uhlenbeck processes driven by a general Lévy process. Bernoulli, 2004, 10: 1–24

    Article  MathSciNet  Google Scholar 

  32. Masuda H. Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps. Stochastic Process Appl, 2007, 117: 35–56

    Article  MATH  MathSciNet  Google Scholar 

  33. Masuda H. Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes. Electron J Probab, 2010, 4: 525–565

    MATH  MathSciNet  Google Scholar 

  34. Moloche G. Local nonparametric estimation of scalar diffusions. Working paper, MIT, 2010

    Google Scholar 

  35. Nicolau J. Nonparametric estimation of second-order stochastic differential equations. Econometric Theory, 2007, 23: 880–898

    Article  MATH  MathSciNet  Google Scholar 

  36. Prakasa Rao B L S. Estimation of the drift for diffusion process. Statistics, 1985, 16: 263–275

    Article  MATH  MathSciNet  Google Scholar 

  37. Ruppert D, Wand M P. Multivariate locally weighted least squares regression. Ann Statist, 1994, 22: 1346–1370

    Article  MATH  MathSciNet  Google Scholar 

  38. Sato K I. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999

    MATH  Google Scholar 

  39. Schertzer D, Larchevêque M, Duan J, et al. Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises. J Math Phys, 2001, 42: 200–212

    Article  MATH  MathSciNet  Google Scholar 

  40. Schoutens W. Lévy Processes in Finance: Pricing Financial Derivatives. Hoboken: Wiley, 2003

    Book  Google Scholar 

  41. Shimizu Y, Yoshida N. Estimation of parameters for diffusion processes with jumps from discrete observation. Stat Inference Stoch Process, 2006, 9: 227–277

    Article  MATH  MathSciNet  Google Scholar 

  42. Silverman B W. Density Estimation for Statistics and Data Analysis. London: Chapman and Hall, 1986

    Book  MATH  Google Scholar 

  43. Stone C. Optimal global rates of convergence for nonparametric regression. Ann Statist, 1982, 10: 1040–1053

    Article  MATH  MathSciNet  Google Scholar 

  44. Stone M. Cross-validity choice and assesment of statistical predictions (with discussion). J R Stat Soc Ser B, 1974, 36: 111–147

    MATH  Google Scholar 

  45. Sun S, Zhang X. Empirical likelihood estimation of discretely sampled processes of OU type. Sci China Ser A, 2009, 52: 908–931

    Article  MATH  MathSciNet  Google Scholar 

  46. Wand M, Jone C. Kernel Smoothing. London: Chapman and Hall, 1995

    Book  MATH  Google Scholar 

  47. Wang H, Lin Z. Local linear estimation of second-order diffusion models. Comm Statist Theory Methods, 2011, 40: 394–407

    Article  MATH  MathSciNet  Google Scholar 

  48. West B J, Seshadri V. Linear systems with Lévy fluctuations. Physica A, 1982, 113: 203–216

    Article  MathSciNet  Google Scholar 

  49. Xu K. Empirical likelihood based inference for nonparametric recurrent diffusions. J Econometrics, 2009, 153: 65–82

    Article  MathSciNet  Google Scholar 

  50. Xu K. Re-weighted functional estimation of diffusion models. Econometric Theory, 2010, 26: 541–563

    Article  MATH  MathSciNet  Google Scholar 

  51. Zhang L, Mykland P A, Aït-Sahalia Y. A tale of two time scales: Determining integrated volatility with noisy high-frequency data. J Amer Statist Assoc, 2005, 100: 1394–1411

    Article  MATH  MathSciNet  Google Scholar 

  52. Zhao Z, Wu W B. Nonparametric inference of discretely sampled stable Lévy processes. J Econometrics, 2009, 153: 83–92

    Article  MathSciNet  Google Scholar 

  53. Zhou Q, Yu J. Asymptotic distributions of the least squares estimator for diffusion process. Working paper, Singapore Management University, 2010

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    ZhengYan Lin, YuPing Song & JiangSheng Yi

Authors
  1. ZhengYan Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YuPing Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. JiangSheng Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YuPing Song.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, Z., Song, Y. & Yi, J. Local linear estimator for stochastic differential equations driven by α-stable Lévy motions. Sci. China Math. 57, 609–626 (2014). https://doi.org/10.1007/s11425-013-4628-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4628-7

Keywords

MSC(2010)

Search

Navigation