Theoretical Statistics and Mathematics Unit, ISI Delhi

Estimation for some stochastic partial differential equations

by B. L. S. Prakasa Rao

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
$u_\epsilon (x, t)$ satisfying the SPDE over the region $[0, 1] \times [0, T ]$ 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 $u_\epsilon (x, t)$ observed at discrete times $t$ and at
discrete positions $x$ or from the Fourier coefficients $u_{i\epsilon (t)$ observed at discrete time instants.
We construct consistent and asymptotically normal estimators of the parameter based on the
Fourier coefficients $u_{i\epsilon (t)$ observed at discrete times $t_j = j \bigtriangleup, 0 \le j \le n$ where $\bigtriangleup > 0$ and $n$
tends to infinity.

isid/ms/2003/06 [fulltext]

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