Abstract
Consider a parabolic stochastic partial differential equation perturbed by small noise observed on a time interval [0,T]. We construct the maximum likelihood estimators of the coefficients of the operators involved in these equations based on partial observations in the form of diffusion processes and show the asymptotic efficiency for loss functions with polynomial majorant as the variance goes to zero.
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References
Barndorff-Nielsen, O. B. and Sørensen, M.: A review of some aspects of asymptotic likelihood theory for stochastic processes. International Statistical Review 62 (1994), 133–165.
Frankignoul, C. and Reynolds, R.: Testing a dynamical model for mid-latitude SST anomalies. J. Phys. Oceanogr. 13 (1983), 1131–1143.
Genon-Catalot, V. and Jacod, J.: On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. H. Poincare 29 (1993), 119–151.
Gikhman, I. I. and Skorokhod, A. V.: Stochastic Differential Equations. Springer, Berlin, 1972.
Holden, H., Øksendal, B., Ubøe, J. and Zhang, T.: Stochastic Partial Differential Equations, Birkhäuser, Boston, 1996.
Huebner, M.: A characterization of asymptotic behaviour of maximum likelihood estimators for stochastic evolution equations. Methods Math. Statistics 6 (1997), 395–415.
Huebner, M., Khasminskii, R. and Rozovskii, B.: Two examples of parameter estimation. in Cambanis, Ghosh, Karandikar, Sen (eds), Stochastic Processes, Springer-Verlag, New York, 1980.
Huebner, M. and Rozovskii, B.: On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDEs. Prob. Theory Rel. Fields 103 (1995), 143–163.
Ibragimov, I. and Khasminskii, R.: Statistical Estimation. Springer, New York, 1982.
Jacod, J.: La variation quadratique de brownian en pr´ esence d'erreurs d'arrondi. Preprint, 1994.
Jacod, J. and Shiryayev, A.: Limit theorems for stochastic processes. Springer, New York, 1987.
Kessler, M. and Sørensen, M.: Estimating equations based on eigenfunctions for a discretely observed diffusion process. Research Report, University of Aarhus, Denmark, 1995.
Kutoyants Yu, A.: Parameter estimation for stochastic processes. Heldermann Verlag, Berlin, 1984.
Kutoyants Yu, A.: Identifaction of dynamical systems with small noise. Kluwer Academic Publishers, 1994.
Loges, W.: Girsanov's theorem in Hilbert space and an application to the statistics of Hilbert space valued stochastic differential equations. Stoch. Proc. Appl. 17 (1984), 243–263.
Mikulevicius, R. and Rozovskii, B.: Uniqueness and absolute continuity of weak solutions for parabolic SPDEs. Acta Applicanda Math. 35 (1994), pp. 179–192.
Piterbarg, L. and Rozovskii, B.: Maximum likelihood estimators in the equations of physical oceanography. in R. Adler, P. Muller, B. Rozovskii, (eds) Stochastic Modeling in Physical Oceanography,Birkhäuser, Boston, (1996), pp. 397–421.
Piterbarg, L. and Rozovskii, B.: On asymptotic problems of parameter estimation in stochastic PDEs: discrete time sampling Math. Meth. Statistics 6, (1997), 200–223.
Rozovskii, B. L.: Stochastic Evolution Systems. Kluwer Academic Publ, 1991.
Safarov, Yu. and Vassiliev, D.: The asymptotic distribution of eigenvalues of partial differential operators. AMS Transl. 155 (1997).
Skorokhod, A. V.: Studies in the Theory of Random Processes. Addison-Wesley, 1965.
Walsh, J.: An introduction to stochastic partial differential equations. Lecture Notes in Mathematics 1180 (1984), 265–439.
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Huebner, M. Asymptotic Properties of the Maximum Likelihood Estimator for Stochastic PDEs Disturbed by Small Noise. Statistical Inference for Stochastic Processes 2, 57–68 (1999). https://doi.org/10.1023/A:1009990504925
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DOI: https://doi.org/10.1023/A:1009990504925