Estimation for some Stochastic Partial Differential Equations Based on Discrete Observations

Abstract

Stochastic partial differential equations (SPDE) are used for stochastic modelling, for instance, in the study of neuronal behaviour in neurophysiology and in building stochastic models for turbulence. Huebner, Khasminskii and Rozovskii (1993) started the investigation of the maximum likelihood estimation of the parameters involved in two types of SPDE's and extended their results for a class of parabolic SPDE's in Huebner and Rozovskii (1995). Prakasa Rao (1998,2000) obtained Bernstein - von Mises type theorems for a class of parabolic SPDE's and investigated the properties of Bayes estimators of parameters involved in such SPDE's. In all the papers cited earlier, it was assumed that a continuous observation of a random field u_ε(x, t) satisfying the SPDE over the region [0, 1] × [0, T] is available. It is obvious that this assumption is not tenable in practice and the problem of interest is to develop methods of estimation of parameters from the random field u_ε(x, t) observed at discrete times t and at discrete positions x or from the Fourier coefficients u_iε(t) observed at discrete time instants. We construct consistent and asymptotically normal estimators of the parameter based on the Fourier coefficients u_iε(t) observed at discrete times $t_j=j\Delta ,\hspace{0.17em}\hspace{0.17em}0\le j\le n\hspace{0.17em}\hspace{0.17em}\text{where}\hspace{0.17em}\Delta >0\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}n$ tends to infinity.

AMS (2000) Subject classification: Primary 62 M 40; Secondary 60 H 15.