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Asymptotic properties of Monte Carlo estimators of diffusion processes

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  • Detemple, Jerome
  • Garcia, Rene
  • Rindisbacher, Marcel

Abstract

This paper studies the limit distributions of Monte Carlo estimators of diffusion processes. Two types of estimators are examined. The first one is based on the Euler scheme applied to the original processes; the second applies the Euler scheme to a variance-stabilizing transformation of the processes. We show that the transformation increases the speed of convergence of the Euler scheme. The limit distribution of this estimator is derived in explicit form and is found to be non-centered. We also study estimators of conditional expectations of diffusions with known initial state. Expected approximation errors are characterized and used to construct second-order bias corrected estimators. Such bias correction eliminates the size distortion of asymptotic confidence intervals and allows to examine the relative efficiency of estimators. Finally, we derive the limit distributions of Monte Carlo estimators of conditional expectations with unknown initial state. The variance-stabilizing transformation is again found to increase the speed of convergence. For comparison we also study the Milshtein scheme. We derive new convergence results for this scheme and show that it does not improve on the convergence properties of the Euler scheme with transformation. Our results are illustrated in the context of a dynamic portfolio choice problem and of simulated-based estimation of diffusion processes. Dans cet article, nous étudions les distributions limites d'estimateurs de Monte Carlo de processus de diffusion. Nous examinons deux types d'estimateurs. Le premier est fondé sur un schéma d'Euler appliqué aux processus originaux, tandis que le second applique le schéma d'Euler à une transformation des processus qui stabilise la variance. Nous montrons que la transformation augmente la vitesse de convergence du schéma d'Euler. La distribution limite de cet estimateur, dérivée sous forme explicite, se révèle non centrée. Nous étudions également des estimateurs d'espéranc

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Econometrics.

Volume (Year): 134 (2006)
Issue (Month): 1 (September)
Pages: 1-68

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Handle: RePEc:eee:econom:v:134:y:2006:i:1:p:1-68

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Web page: http://www.elsevier.com/locate/jeconom

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Cited by:
  1. Carrasco, Marine & Chernov, Mikhaël & Florens, Jean-Pierre & Ghysels, Eric, 2000. "Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions," IDEI Working Papers 116, Institut d'Économie Industrielle (IDEI), Toulouse, revised 2002.
  2. Mikhail Chernov & A. Ronald Gallant & Eric Ghysels & George Tauchen, 2002. "Alternative Models for Stock Price Dynamics," CIRANO Working Papers 2002s-58, CIRANO.
  3. Castaneda, Pablo, 2006. "Long Term Risk Assessment in a Defined Contribution Pension System," MPRA Paper 3347, University Library of Munich, Germany, revised 30 Apr 2007.
  4. Kristensen, Dennis & Shin, Yongseok, 2012. "Estimation of dynamic models with nonparametric simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 167(1), pages 76-94.
  5. Alfonsi, Aurélien, 2013. "Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 602-607.

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