Consistency and Asymptotic Normality of an Approximate Maximum Likelihood Estimator for Discretely Observed Diffusion Processes

Asger Roer Pedersen
Vol. 1, No. 3 (Sep., 1995), pp. 257-279
DOI: 10.2307/3318480
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Page Count: 23
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Consistency and Asymptotic Normality of an Approximate Maximum Likelihood Estimator for Discretely Observed Diffusion Processes
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Most often the likelihood function based on discrete observations of a diffusion process is unknown, and estimators alternative to the well-behaved maximum likelihood estimator must be found. Traditionally, such estimators are defined with origin in the theory for continuous observation of the diffusion process, and are as a consequence strongly biased unless the discrete observation time-points are close. In contrast to these estimators, an estimator based on an approximation to the (unknown) likelihood function was proposed in Pedersen (1994). We prove consistency and asymptotic normality of this estimator with no assumptions on the distance between the discrete observation time-points.

Notes and References

This item contains 16 references.

  • Barndorff-Nielsen, O.E. and Sørensen, M. (1994) A review of some aspects of asymptotic likelihood theory for stochastic processes. Internat. Statist. Rev., 62, 133-165.
  • Billingsley, P. (1961) Statistical Inference for Markov Processes. Chicago: University of Chicago Press.
  • Dacunha-Castelle, D. and Duflo, M. (1983) Probabilités et Statistiques. Paris: Masson.
  • Dacunha-Castelle, D. and Florens-Zmirou, D. (1986) Estimation of the coefficients of a diffusion from discrete observations. Stochastics, 19, 263-284.
  • Florens-Zmirou, D. (1989) Approximate discrete-time schemes for statistics of diffusion processes. Statistics, 20, 547-557.
  • Friedman, A. (1975) Stochastic Differential Equations and Applications, Volume 1. New York: Academic Press.
  • Genon-Catalot, V. (1990) Maximum contrast estimation for diffusion processes from discrete observations. Statistics, 21, 99-116.
  • Hutton, J.E. and Nelson, P.I. (1986) Quasi-likelihood estimation for semi-martingales. Stochastic Process. Appl., 22, 245-257.
  • Jacod, J. and Shiryayev, A.N. (1987) Limit Theorems for Stochastic Processes. Berlin: Springer-Verlag.
  • Jensen, J.L. (1986) Nogle Asymptotiske Resultater. Aarhus University (in Danish).
  • Kloeden, P.E. and Platen, E. (1992) Numerical Solution of Stochastic Diferential Equations. New York: Springer-Verlag.
  • 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.
  • Revuz, D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Berlin: Springer-Verlag.
  • Rogers, L.C.G. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales, Volume 2: Ito Calculus. Chichester: Wiley.
  • Stroock, D.W. and Varadhan, S.R.S. (1979) Multidimensional Diffusion Processes. Berlin: Springer-Verlag.
  • Sweeting, T.J. (1980) Uniform asymptotic normality of the maximum likelihood estimator. Ann. Statist., 8, 1375-1381.