Summary
We investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic SPDEs of the form
whereA 0 andA 1 are partial differential operators andW is a cylindrical Brownian motion. We introduce a spectral method for computing MLEs based on finite dimensional approximations to solutions of such systems, and establish criteria for consistency, asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity. We derive the asymptotic properties of the MLE from a condition on the order of the operators. In particular, the MLE is consistent if and only if ord(A 1)≧1/2(ord(A 0+θA 1)−d).
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References
[A] Aihara, S.I.: Regularized maximum likelihood estimate for an infinite dimensional parameter in stochastic parabolic systems. SIAM J. Cont. Optim.30, 745–764 (1992)
[B-B] Bagchi, A., Borkar, V.: Parameter identification in infinite dimensional linear systems. Stoch.12, 201–213 (1984)
[DZ] Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge: Cambridge University Press 1992
[G-Sk] Gikhman, I.I., Skorokhod, A.V.: Stochastic processes I. Berlin: Springer 1979
[H] Huebner, M.: Parameter estimation for stochastic differential equations. Thesis, University of Southern California 1993
[H-K-R] Huebner, M., Khasminskii, R., Rozovskii, B.: Two examples of parameter estimation. In: Cambanis, Ghosh, Karandikar, Sen (eds.) Stochastic processes. Berlin: Springer 1992
[I-K] Ibragimov, I.A., Khasminskii, R.Z.: Statistical estimation (asymptotic theory). Berlin: Springer 1981
[J-Sh] Jacod, J., Shiryayev, A.N.: Limit theorems for stochastic processes. Berlin: Springer 1987
[Ku1] Kutoyants, Yu.A.: Parameter estimation for stochastic processes. Heldermann 1984
[Ku2] Kutoyants, Yu.A.: Identification of dynamical systems with small noise. Berlin: Springer 1994
[Ko] Kozlov, S.M.: Some problems in stochastic partial differential equations. Proc. of Petrovski's Seminar4, 148–172 (1978) (in Russian)
[Lo] 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, 243–263 (1984)
[L-Sh] Liptser, R.S., Shiryayev A.N.: Statistics of random processes. Berlin: Springer 1992
[M-R] Mikulevicius, R., Rozovskii, B.L.: Absolute continuity of measures generated by solutions of SPDE's. preprint
[Sh] Shimakura, N.: Partial differential operators of elliptic type. AMS Transl.99, (1992)
[Shir] Shiryayev, A.N.: Probability, New York: Springer 1984
[Shu] Shubin, M.A.: Pseudodifferential operators and spectral theory. Berlin: Springer 1987
[Ro] Rozovskii, B.L.: Stochastic evolutions systems. Dordrecht: Kluwer Academic Publ. 1990
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This work was partially supported by ONR Grant # N00014-91-J-1526 and NSF Grant # DMS-9002997
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Huebner, M., Rozovskii, B.L. On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE's. Probab. Th. Rel. Fields 103, 143–163 (1995). https://doi.org/10.1007/BF01204212
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DOI: https://doi.org/10.1007/BF01204212