Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-12T05:28:23.049Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation for Nonnegative Lévy-Driven Ornstein-Uhlenbeck Processes

Part of: Inference from stochastic processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Peter J. Brockwell ,
Richard A. Davis  and
Yu Yang
Peter J. Brockwell*
Affiliation:
Colorado State University
Richard A. Davis*
Affiliation:
Columbia University
Yu Yang*
Affiliation:
Colorado State University
*
∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523-1877, USA.
∗∗∗Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA.
∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523-1877, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.

Keywords

Continuous-time autoregressionOrnstein-Uhlenbeck processLevy processstochastic differential equationsampled process

MSC classification

Secondary: 62M10: Time series, auto-correlation, regression, etc. 60H10: Stochastic ordinary differential equations 62M09: Non-Markovian processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 977 - 989
Copyright
Copyright © Applied Probability Trust 2007 

References

Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. Ser. B 63, 167–241.Google Scholar
Brockwell, P. J. and Brown, B. M. (1978). Expansions for the positive stable laws. Z. Wahrscheinlichkeitst. 45, 213–224.Google Scholar
Brockwell, P. J. and Marquardt, T. (2005). Fractionally integrated continuous-time ARMA processes. Statistica Sinica 15, 477–494.Google Scholar
Davis, R. and McCormick, W. (1989). Estimation for first-order autoregressive processes with positive and bounded innovations. Stoch. Process. Appl. 31, 273–250.Google Scholar
Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace Transformation. Springer, New York.CrossRefGoogle Scholar
Jongbloed, G., van der Meulen, F. H. and van der Vaart, A. W. (2005). Nonparametric inference for Lévy-driven Ornstein–Uhlenbeck processes. Bernoulli 11, 759–791.Google Scholar
Phillips, A. W. (1959). The estimation of parameters in systems of stochastic differential equations. Biometrika 46, 67–76.Google Scholar
Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, New York.Google Scholar
Shtatland, E.S. (1965). On local properties of processes with independent increments. Theory Prob. Appl. 10, 317–322.CrossRefGoogle Scholar