zbMATH — the first resource for mathematics

On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE’s. (English) Zbl 0831.60070
Summary: We investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic SPDEs of the form \[ du(t, x)= (A_0+ \theta A_1) u(t, x) dt+ dW(t, x), \] where \(A_0\) and \(A_1\) are partial differential operators and \(W\) 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 \(\text{ord}(A_1)\geq {1\over 2}(\text{ord}(A_0+ \theta A_1)- d)\).

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
62F10 Point estimation
65C99 Probabilistic methods, stochastic differential equations
Full Text: DOI
[1] [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) · Zbl 0756.93077
[2] [B-B] Bagchi, A., Borkar, V.: Parameter identification in infinite dimensional linear systems. Stoch.12, 201–213 (1984) · Zbl 0541.93072
[3] [DZ] Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge: Cambridge University Press 1992 · Zbl 0761.60052
[4] [G-Sk] Gikhman, I.I., Skorokhod, A.V.: Stochastic processes I. Berlin: Springer 1979
[5] [H] Huebner, M.: Parameter estimation for stochastic differential equations. Thesis, University of Southern California 1993
[6] [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
[7] [I-K] Ibragimov, I.A., Khasminskii, R.Z.: Statistical estimation (asymptotic theory). Berlin: Springer 1981
[8] [J-Sh] Jacod, J., Shiryayev, A.N.: Limit theorems for stochastic processes. Berlin: Springer 1987
[9] [Ku1] Kutoyants, Yu.A.: Parameter estimation for stochastic processes. Heldermann 1984
[10] [Ku2] Kutoyants, Yu.A.: Identification of dynamical systems with small noise. Berlin: Springer 1994 · Zbl 0831.62058
[11] [Ko] Kozlov, S.M.: Some problems in stochastic partial differential equations. Proc. of Petrovski’s Seminar4, 148–172 (1978) (in Russian)
[12] [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) · Zbl 0553.93059
[13] [L-Sh] Liptser, R.S., Shiryayev A.N.: Statistics of random processes. Berlin: Springer 1992
[14] [M-R] Mikulevicius, R., Rozovskii, B.L.: Absolute continuity of measures generated by solutions of SPDE’s. preprint
[15] [Sh] Shimakura, N.: Partial differential operators of elliptic type. AMS Transl.99, (1992) · Zbl 0757.35015
[16] [Shir] Shiryayev, A.N.: Probability, New York: Springer 1984
[17] [Shu] Shubin, M.A.: Pseudodifferential operators and spectral theory. Berlin: Springer 1987 · Zbl 0616.47040
[18] [Ro] Rozovskii, B.L.: Stochastic evolutions systems. Dordrecht: Kluwer Academic Publ. 1990
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.