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# Asymptotic Properties of Monte Carlo Estimators of Diffusion Processes

## Author

Listed:
• Jérôme B. Detemple
• René Garcia
• Marcel Rindisbacher

## 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'espérances conditionnelles de diffusions à partir d'un état initial connu. Nous caractérisons les erreurs d'approximation attendues et utilisons les expressions obtenues pour construire des estimateurs corrigés du biais de deuxième ordre. La correction de ce biais élimine la distorsion de niveau des intervalles de confiance asymptotiques et nous permet d'évaluer l'efficacité relative des estimateurs. Enfin, nous dérivons les distributions limites des estimateurs de Monte Carlo d'espérances conditionnelles de diffusions avec état initial inconnu. Nous trouvons de nouveau que la transformation stabilisatrice de la variance augmente la vitesse de convergence. À titre comparatif, nous étudions également le schéma de Milshtein. Nous dérivons de nouveaux résultats de convergence pour ce schéma et montrons qu'il n'améliore pas les propriétés de convergence du schéma d'Euler avec transformation. Nos résultats sont illustrés dans le contexte d'un problème de choix de portefeuille dynamique et d'estimation de processus de diffusion par méthodes simulées.

## Suggested Citation

• Jérôme B. Detemple & René Garcia & Marcel Rindisbacher, 2003. "Asymptotic Properties of Monte Carlo Estimators of Diffusion Processes," CIRANO Working Papers 2003s-11, CIRANO.
• Handle: RePEc:cir:cirwor:2003s-11
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File URL: http://www.cirano.qc.ca/files/publications/2003s-11.pdf
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## References listed on IDEAS

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## Citations

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Cited by:

1. Kristensen, Dennis & Shin, Yongseok, 2012. "Estimation of dynamic models with nonparametric simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 167(1), pages 76-94.
2. Detemple, Jerome & Rindisbacher, Marcel, 2007. "Monte Carlo methods for derivatives of options with discontinuous payoffs," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3393-3417, April.
3. Carrasco, Marine & Chernov, Mikhail & Florens, Jean-Pierre & Ghysels, Eric, 2007. "Efficient estimation of general dynamic models with a continuum of moment conditions," Journal of Econometrics, Elsevier, vol. 140(2), pages 529-573, October.
4. 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.
5. Chernov, Mikhail & Ronald Gallant, A. & Ghysels, Eric & Tauchen, George, 2003. "Alternative models for stock price dynamics," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 225-257.
6. Jérôme Detemple & René Garcia & Marcel Rindisbacher, 2005. "Asymptotic Properties of Monte Carlo Estimators of Derivatives," Management Science, INFORMS, vol. 51(11), pages 1657-1675, November.
7. 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.
8. repec:spr:joptap:v:161:y:2014:i:1:d:10.1007_s10957-012-0208-1 is not listed on IDEAS
9. 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.

## More about this item

### Keywords

Monte Carlo errors; Monte Carlo estimators; Estimation of Diffusion Processes; Doss transformation; Discretization schemes; Erreurs de Monte Carlo; estimateurs de Monte Carlo; estimation de processus de diffusion; transformation de Doss; schémas de discrétisation;

### JEL classification:

• C0 - Mathematical and Quantitative Methods - - General
• C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
• G0 - Financial Economics - - General

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