Asymptotic properties of Monte Carlo estimators of diffusion processes
AbstractThis 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 Ã©tudions les distributions limites d'estimateurs de Monte Carlo de processus de diffusion. Nous examinons deux types d'estimateurs. Le premier est fondÃ© sur un schÃ©ma d'Euler appliquÃ© aux processus originaux, tandis que le second applique le schÃ©ma d'Euler Ã une transformation des processus qui stabilise la variance. Nous montrons que la transformation augmente la vitesse de convergence du schÃ©ma d'Euler. La distribution limite de cet estimateur, dÃ©rivÃ©e sous forme explicite, se rÃ©vÃ¨le non centrÃ©e. Nous Ã©tudions Ã©galement des estimateurs d'espÃ©ranc
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Bibliographic InfoArticle provided by Elsevier in its journal Journal of Econometrics.
Volume (Year): 134 (2006)
Issue (Month): 1 (September)
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Web page: http://www.elsevier.com/locate/jeconom
Other versions of this item:
- Jérôme B. Detemple & René Garcia & Marcel Rindisbacher, 2003. "Asymptotic Properties of Monte Carlo Estimators of Diffusion Processes," CIRANO Working Papers 2003s-11, CIRANO.
- Marcel Rindisbacher & Jérôme Detemple & René Garcia, 2004. "Asymptotic Properties of Monte Carlo Estimators of Diffusion Processes," Econometric Society 2004 North American Winter Meetings 483, Econometric Society.
- C0 - Mathematical and Quantitative Methods - - General
- C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
- G0 - Financial Economics - - General
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995.
"Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes,"
Econometric Society, vol. 63(4), pages 767-804, July.
- Lars Peter Hansen & Jose Alexandre Scheinkman, 1993. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," NBER Technical Working Papers 0141, National Bureau of Economic Research, Inc.
- repec:cup:etheor:v:12:y:1996:i:4:p:657-81 is not listed on IDEAS
- Tauchen, George E. & Gallant, A. Ronald, 1995.
"Which Moments to Match,"
95-20, Duke University, Department of Economics.
- Chamberlain, Gary, 1987. "Asymptotic efficiency in estimation with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 34(3), pages 305-334, March.
- Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
- Gourieroux, C. & Monfort, A. & Renault, E., 1992.
92.279, Toulouse - GREMAQ.
- Jérôme Detemple & René Garcia & Marcel Rindisbacher, 2005. "Representation formulas for Malliavin derivatives of diffusion processes," Finance and Stochastics, Springer, vol. 9(3), pages 349-367, 07.
- BROZE, Laurence & SCAILLET, Olivier & ZAKOIAN, Jean-Michel, 1995.
"Quasi Indirect Inference for Diffusion Processes,"
CORE Discussion Papers
1995005, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Chacko, George & Viceira, Luis M., 2003. "Spectral GMM estimation of continuous-time processes," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 259-292.
- Jérôme B. Detemple & René Garcia & Marcel Rindisbacher, 2000.
"A Monte-Carlo Method for Optimal Portfolios,"
CIRANO Working Papers
- Duffie, Darrell & Singleton, Kenneth J, 1993. "Simulated Moments Estimation of Markov Models of Asset Prices," Econometrica, Econometric Society, vol. 61(4), pages 929-52, July.
- Merton, Robert C., 1971.
"Optimum consumption and portfolio rules in a continuous-time model,"
Journal of Economic Theory,
Elsevier, vol. 3(4), pages 373-413, December.
- R. C. Merton, 1970. "Optimum Consumption and Portfolio Rules in a Continuous-time Model," Working papers 58, Massachusetts Institute of Technology (MIT), Department of Economics.
- Chamberlain, Gary, 1992. "Efficiency Bounds for Semiparametric Regression," Econometrica, Econometric Society, vol. 60(3), pages 567-96, May.
- Hansen, Lars Peter, 1985. "A method for calculating bounds on the asymptotic covariance matrices of generalized method of moments estimators," Journal of Econometrics, Elsevier, vol. 30(1-2), pages 203-238.
- Cox, John C. & Huang, Chi-fu, 1989. "Optimal consumption and portfolio policies when asset prices follow a diffusion process," Journal of Economic Theory, Elsevier, vol. 49(1), pages 33-83, October.
- Darrell Duffie & Philip Protter, 1992. "From Discrete- to Continuous-Time Finance: Weak Convergence of the Financial Gain Process," Mathematical Finance, Wiley Blackwell, vol. 2(1), pages 1-15.
- Andrew W. Lo, 1986.
"Maximum Likelihood Estimation of Generalized Ito Processes with Discretely Sampled Data,"
NBER Technical Working Papers
0059, National Bureau of Economic Research, Inc.
- Lo, Andrew W., 1988. "Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data," Econometric Theory, Cambridge University Press, vol. 4(02), pages 231-247, August.
- Andrew W. Lo, . "Maximum Likelihood Estimation of Generalized Ito Processes with Discretely Sampled Data," Rodney L. White Center for Financial Research Working Papers 15-86, Wharton School Rodney L. White Center for Financial Research.
- Neil Shephard & Ola Elerian & Siddhartha Chib, 1998.
"Likelihood inference for discretely observed non-linear diffusions,"
Economics Series Working Papers
1998-W10, University of Oxford, Department of Economics.
- Elerain, Ola & Chib, Siddhartha & Shephard, Neil, 2001. "Likelihood Inference for Discretely Observed Nonlinear Diffusions," Econometrica, Econometric Society, vol. 69(4), pages 959-93, July.
- Elerian, O. & Chib, S. & Shephard, N., 1998. "Likelihood INference for Discretely Observed Non-linear Diffusions," Economics Papers 146, Economics Group, Nuffield College, University of Oxford.
- Ola Elerian & Siddhartha Chib & Neil Shephard, 2000. "Likelihood inference for discretely observed non-linear diffusions," OFRC Working Papers Series 2000mf02, Oxford Financial Research Centre.
- Yacine Aït-Sahalia, 1999. "Transition Densities for Interest Rate and Other Nonlinear Diffusions," Journal of Finance, American Finance Association, vol. 54(4), pages 1361-1395, 08.
- Nicholas Barberis, 2000. "Investing for the Long Run when Returns Are Predictable," Journal of Finance, American Finance Association, vol. 55(1), pages 225-264, 02.
- Gallant, A. Ronald & Tauchen, George, 2002. "Simulated Score Methods and Indirect Inference for Continuous-time Models," Working Papers 02-09, Duke University, Department of Economics.
- Newey, Whitney K, 1990. "Semiparametric Efficiency Bounds," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 5(2), pages 99-135, April-Jun.
- Singleton, Kenneth J., 2001. "Estimation of affine asset pricing models using the empirical characteristic function," Journal of Econometrics, Elsevier, vol. 102(1), pages 111-141, May.
- Eraker, Bjorn, 2001. "MCMC Analysis of Diffusion Models with Application to Finance," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(2), pages 177-91, April.
- Yacine Ait-Sahalia, 2002. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach," Econometrica, Econometric Society, vol. 70(1), pages 223-262, January.
- Durham, Garland B & Gallant, A Ronald, 2002. "Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 297-316, July.
- 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.
- Eric Ghysels & Jean-Pierre Florens & Mikhail Chernov & Marine Carrasco, 2003. "Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions," CIRANO Working Papers 2003s-02, CIRANO.
- Mikhail Chernov & A. Ronald Gallant & Eric Ghysels & George Tauchen, 2002.
"Alternative Models for Stock Price Dynamics,"
CIRANO Working Papers
- Castaneda, Pablo, 2006.
"Long Term Risk Assessment in a Defined Contribution Pension System,"
3347, University Library of Munich, Germany, revised 30 Apr 2007.
- Pablo Castañeda, 2007. "Long Term Risk Assessment in a Defined Contribution Pension System," Working Papers 20, Superintendencia de Pensiones, revised Oct 2007.
- Kristensen, Dennis & Shin, Yongseok, 2012.
"Estimation of dynamic models with nonparametric simulated maximum likelihood,"
Journal of Econometrics,
Elsevier, vol. 167(1), pages 76-94.
- Dennis Kristensen & Yongseok Shin, 2008. "Estimation of Dynamic Models with Nonparametric Simulated Maximum Likelihood," CREATES Research Papers 2008-58, School of Economics and Management, University of Aarhus.
- 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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