The Annals of Statistics

Estimators of diffusions with randomly spaced discrete observations: A general theory

Yacine Aït-Sahalia and Per A. Mykland

Full-text: Open access

Enhanced PDF (283 KB)

Abstract

We provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random. We introduce a new operator, the generalized infinitesimal generator, to obtain Taylor expansions of the asymptotic moments of the estimators. As a special case, our results apply to the situation where the data are discretely sampled at a fixed nonrandom time interval. We include as specific examples estimators based on maximum-likelihood and discrete approximations such as the Euler scheme.

Article information

Source
Ann. Statist., Volume 32, Number 5 (2004), 2186-2222.

Dates
First available in Project Euclid: 27 October 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1098883787

Digital Object Identifier
doi:10.1214/009053604000000427

Mathematical Reviews number (MathSciNet)
MR2102508

Zentralblatt MATH identifier
1062.62155

Subjects
Primary: 62F12: Asymptotic properties of estimators 62M05
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60

Keywords
Diffusions likelihood discrete and random sampling

Citation

Aït-Sahalia, Yacine; Mykland, Per A. Estimators of diffusions with randomly spaced discrete observations: A general theory. Ann. Statist. 32 (2004), no. 5, 2186--2222. doi:10.1214/009053604000000427. https://projecteuclid.org/euclid.aos/1098883787


Export citation

References

  • Aït-Sahalia, Y. (1996). Nonparametric pricing of interest rate derivative securities. Econometrica 64 527--560.
  • Aït-Sahalia, Y. (2002). Maximum-likelihood estimation of discretely-sampled diffusions: A closed-form approximation approach. Econometrica 70 223--262.
    Mathematical Reviews (MathSciNet): MR1926260
    Digital Object Identifier: doi:10.1111/1468-0262.00274
    Zentralblatt MATH: 1104.62323
  • Aït-Sahalia, Y. and Mykland, P. A. (2003). The effects of random and discrete sampling when estimating continuous-time diffusions. Econometrica 71 483--549.
    Mathematical Reviews (MathSciNet): MR1958137
    Digital Object Identifier: doi:10.1111/1468-0262.t01-1-00416
    Zentralblatt MATH: 1142.60381
  • Bibby, B. M., Jacobsen, M. and Sørensen, M. (2004). Estimating functions for discretely sampled diffusion-type models. In Handbook of Financial Econometrics (Y. Aït-Sahalia and L. P. Hansen, eds.). North-Holland, Amsterdam. To appear.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
    Zentralblatt MATH: 0219.60003
  • Florens-Zmirou, D. (1989). Approximate discrete-time schemes for statistics of diffusion processes. Statistics 20 547--557.
    Mathematical Reviews (MathSciNet): MR1047222
  • Godambe, V. P. (1960). An optimum property of regular maximum likelihood estimation. Ann. Math. Statist. 31 1208--1211.
    Mathematical Reviews (MathSciNet): MR123385
    Digital Object Identifier: doi:10.1214/aoms/1177705693
  • Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50 1029--1054.
    Mathematical Reviews (MathSciNet): MR666123
  • Hansen, L. P. and Scheinkman, J. A. (1995). Back to the future: Generating moment implications for continuous-time Markov processes. Econometrica 63 767--804.
    Mathematical Reviews (MathSciNet): MR1343081
  • Heyde, C. C. (1997). Quasi-Likelihood and Its Application. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1461808
    Zentralblatt MATH: 0879.62076
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1121940
    Zentralblatt MATH: 0734.60060
  • Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
    Mathematical Reviews (MathSciNet): MR611513
    Zentralblatt MATH: 0469.60001
  • Merton, R. C. (1992). Continuous-Time Finance. Basil Blackwell, New York.
    Zentralblatt MATH: 1019.91502
  • Protter, P. (1992). Stochastic Integration and Differential Equations: A New Approach. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1037262
    Zentralblatt MATH: 0694.60047

The Institute of Mathematical Statistics

The Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more