Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-17T23:23:52.314Z Has data issue: false hasContentIssue false
  • English
  • Français

NONPARAMETRIC ESTIMATION OF SECOND-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS

Published online by Cambridge University Press:  14 May 2007

João Nicolau
João Nicolau
Affiliation:
School of Economics and Management (ISEG) Universidade Técnica de Lisboa
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose nonparametric estimators of the infinitesimal coefficients associated with second-order stochastic differential equations. We show that under appropriate conditions, the proposed estimators are consistent. Also, we state conditions ensuring the asymptotic normality of these estimators. We conclude our paper with a Monte Carlo experiment in which we assess the response of the nonparametric estimators with respect to the step of discretization.I thank two anonymous referees who made valuable suggestions that led to considerable improvements in the paper. I am also grateful to Carlos Braumann and Tom Kundert for helpful comments. This research was supported by the Fundação para a Ciência e a Tecnologia (FCT) and by FEDER/POCI 2010.

Type
Research Article
Information
Econometric Theory , Volume 23 , Issue 5 , October 2007 , pp. 880 - 898
Copyright
© 2007 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aït-Sahalia, Y. (2002) Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70, 223–262.Google Scholar
Arnold, L. (1974) Stochastic Differential Equations: Theory and Application. Wiley.
Bandi, F. & P. Phillips (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241–283.Google Scholar
Bibby, B. & M. Sørensen (1997) A hyperbolic diffusion model for stock prices. Finance and Stochastics 1, 25–41.Google Scholar
Chen, X., L. Hansen, & M. Carrasco (1999) Nonlinearity and Temporal Dependence. Unpublished.
Dacunha-Castelle, D. & D. Florens-Zmirou (1986) Estimation of the coefficient of a diffusion from discretely sampled observations. Stochastics 19, 263–284.Google Scholar
Ditlevsen, S. & M. Sørensen (2004) Inference for observations of integrated diffusion processes. Scandinavian Journal of Statistics 31, 417–429.Google Scholar
Florens-Zmirou, D. (1989) Approximate discrete-time schemes for statistics of diffusion processes. Statistics 20, 547–557.Google Scholar
Florens-Zmirou, D. (1993) On estimating the diffusion coefficient from discrete observation. Journal of Applied Probability 30, 790–804.Google Scholar
Gloter, A. (2001) Parameter estimation for a discrete sampling of an integrated Ornstein-Uhlenbeck process. Statistics 35, 225–243.Google Scholar
Gloter, A. (2006) Parameter estimation for a discretely observed integrated diffusion process. Scandinavian Journal of Statistics 33, 83–104.Google Scholar
Gobet, E., M. Hoffmann, & M. Reiß (2004) Nonparametric estimation of scalar diffusions based on low frequency data. Annals of Statistics 32, 2223–2253.Google Scholar
Hansen, L. & J. Scheinkman (1995) Back to the future: Generating moment implications for continuous-time Markov processes. Econometrica 63, 767–804.Google Scholar
Jiang, G. & J. Knight (1997) A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest rate model. Econometric Theory 13, 615–645.Google Scholar
Kessler, M. (1997) Estimation of an ergodic diffusion from discrete observations. Scandinavian Journal of Statistics 24, 211–229.Google Scholar
Nicolau, J. (2003) Bias reduction in nonparametric diffusion coefficient estimation. Econometric Theory 19, 754–777.Google Scholar
Rao, B. (1983) Nonparametrical Functional Estimation. Academic Press.
Skorokhod, A. (1989) Asymptotic Methods in the Theory of Stochastic Differential Equations. Translation of Mathematical Monographs 78, American Mathematical Society.