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Estimation for discretely observed diffusions using transform functions

Part of: Parametric inference Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Leah Kelly ,
Eckhard Platen  and
Michael Sørensen
Leah Kelly
Affiliation:
University of Technology Sydney, School of Finance and Economics and School of Mathematical Sciences, PO Box 123, Broadway, NSW 2007, Australia. Email address: leah.kelly@uts.edu.au
Eckhard Platen
Affiliation:
University of Technology Sydney, School of Finance and Economics and School of Mathematical Sciences, PO Box 123, Broadway, NSW 2007, Australia. Email address: eckhard.platen@uts.edu.au
Michael Sørensen
Affiliation:
University of Copenhagen, Department of Applied Mathematics and Statistics, Universitetspraken 5, DK-2100 Copenhagen Ø, Denmark. Email address: michael@math.ku.dk
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Abstract

This paper introduces a new estimation technique for discretely observed diffusion processes. Transform functions are applied to transform the data to obtain good and easily calculated estimators of both the drift and diffusion coefficients. Consistency and asymptotic normality of the resulting estimators is investigated. Power transforms are used to estimate the parameters of affine diffusions, for which explicit estimators are obtained.

Keywords

Discretely observed diffusiontransformation functionaffine diffusion processestimation

MSC classification

Primary: 62M05: Markov processes 62F12: Asymptotic properties of estimators
Secondary: 60G35: Signal detection and filtering 60G10: Stationary processes
Type
Part 2. Estimation methods
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 99 - 118
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70, 223–262.Google Scholar
[2] Barndorff-Nielsen, O. and S⊘rensen, M. (1994). A review of some aspects of asymptotic likelihood theory for stochastic processes. Internat. Statist. Rev. 62, 133–165.Google Scholar
[3] Bibby, B. M. (1994). Optimal combination of martingale estimating functions for discretely observed diffusions. Tech. Rep. 298, Department of Theoretical Statistics, University of Aarhus.Google Scholar
[4] Bibby, B. M. and S⊘rensen, M. (1995). Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1, 17–39.Google Scholar
[5] Bibby, B. M. and S⊘rensen, M. (1996). On estimation for discretely observed diffusions: a review. Theory Stoch. Process. 2, 49–56.Google Scholar
[6] Bibby, B. M., Jacobsen, M. and S⊘rensen, M. (2003). Estimating functions for discretely sampled diffusion-type models. To appear in Handbook of Financial Economics , eds Ait-Sahalia, Y. and Hansen, L. P., North Holland, Amsterdam.Google Scholar
[7] Brandt, M. and Santa-Clara, P. (2001). Simulated likelihood estimation of diffusions with application to exchange rate dynamics in incomplete markets. J. Financial Econom. 63, 161–210.Google Scholar
[8] Christensen, B. J., Poulsen, R. and S⊘rensen, M. (2001). Optimal inference in diffusion models of the short rate of interest. Tech. Rep. 102, Centre for Analytical Finance, University of Aarhus.Google Scholar
[9] Dacunha-Castelle, D. and Florens-Zmirou, D. (1986). Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19, 263–284.Google Scholar
[10] Dorogovcev, A. J. (1976). The consistency of an estimate of a parameter of a stochastic differential equation. Theory Prob. Math. Statist. 10, 73–82.Google Scholar
[11] Doukhan, P. (1994). Mixing (Lecture Notes Statist. 85). Springer, New York.Google Scholar
[12] Doukhan, P, Massart, P. and Rio, E. (1994). The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincaré Prob. Statist. 30, 63–82.Google Scholar
[13] Elerain, O., Chib, S. and Shepard, N. (2001). Likelihood inference for discretely observed non-linear diffusions. Econometrica 69, 959–993.Google Scholar
[14] Eraker, B. (2001). MCMC analysis of diffusion models with application to finance. J. Business Econom. Statist. 19, 177–191.Google Scholar
[15] Florens-Zmirou, D. (1989). Approximate discrete-time schemes for statistics of diffusion processes. Stochastics 20, 547–557.Google Scholar
[16] Gallant, A. R. and Tauchen, G. (1996). Which moments to match? Econometric Theory 12, 657–681.Google Scholar
[17] Genon-Catalot, V., Jeantheau, T. and Laredo, C. (2000). Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6, 1051–1079.Google Scholar
[18] Godambe, V. P. and Heyde, C. C. (1987). Quasi-likelihood and optimal estimation. Internat. Statist. Rev. 55, 231–244.Google Scholar
[19] Gouriéroux, C., Monfort, A. and Renault, E. (1993). Indirect inference. J. Appl. Econometrics 8, 85–118.Google Scholar
[20] Hansen, L. P. (1982). Large sample properties of the generalized method of moments. Econometrica 50, 1029–1054.Google Scholar
[21] Heyde, C. C. (1997). Quasi-Likelihood and Its Application. Springer, New York.Google Scholar
[22] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes , 2nd edn. Springer, Berlin.Google Scholar
[23] Jensen, B. and Poulsen, R. (2002). Transition densities of diffusion processes: numerical comparison of approximation techniques. J. Derivatives 9, No. 4, 18–32.Google Scholar
[24] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math 113), 2nd edn. Springer, New York.Google Scholar
[25] Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist. 24, 211–229.Google Scholar
[26] Kessler, M. and S⊘rensen, M. (1999). Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli 5, 299–314.Google Scholar
[27] Kloeden, P. E. and Platen, E. (1999). Numerical Solution of Stochastic Differential Equations (Appl. Math. 23), 3rd edn. Springer, Berlin.Google Scholar
[28] Kloeden, P. E., Platen, E., Schurz, H. and S⊘rensen, M. (1996). On effects of discretization on estimators of drift parameters for diffusion processes. J. Appl. Prob. 33, 1061–1076.Google Scholar
[29] Kutoyants, Y. (1984). Parameter Estimation for Stochastic Processes. Helderman, Berlin.Google Scholar
[30] Liptser, R. and Shiryaev, A. (2001). Statistics of Random Processes: II. Applications (Appl. Math. 6), 2nd edn. Springer, Berlin.Google Scholar
[31] Milstein, G. N., Platen, E. and Schurz, H. (1998). Balanced implicit methods for stiff stochastic systems. SIAM J. Numerical Analysis 35, 1010–1019.Google Scholar
[32] Pedersen, A. R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist. 22, 55–71.Google Scholar
[33] Poulsen, R. (1999). Approximate maximum likelihood estimation of discretely observed diffusion processes. Tech. Rep. 29, Centre for Analytical Finance, University of Aarhus.Google Scholar
[34] Prakasa Rao, B. L. S. (1988). Statistical inference from sampled data for stochastic processes. Contemp. Math. 80, 249–284.Google Scholar
[35] Prakasa Rao, B. L. S. (1999). Statistical Inference for Diffusion Type Processes (Kendall's Library Statist. 8). Arnold, London.Google Scholar
[36] S⊘rensen, H. (2001). Discretely observed diffusions: approximation of the continuous-time score function. Scand. J. Statist. 28, 113–121.Google Scholar
[37] S⊘rensen, M. (1997). Estimating functions for discretely observed diffusions: a review. In Selected Proc. of the Symp. on Estimating Functions (IMS Lecture Notes Monogr. Ser. 32), eds Basawa, I. V., Godambe, V. P. and Taylor, R. L., Institute of Mathematical Statistics, Hayward, CA, pp. 305–325.Google Scholar
[38] S⊘rensen, M. (1999). On asymptotics of estimating functions. Brazilian J. Prob. Statist. 13, 111–136.Google Scholar
[39] S⊘rensen, M. (2000). Prediction based estimating functions. J. Econometrics 3, 123–147.Google Scholar