On estimate of the unknown parameter in parabolic systems with rapid random oscillations
The observed data field \(U(t,x)\) is supposed to be a solution of the Cauchy problem for the parabolic equation NEWLINE\[NEWLINE\partial U/\partial t=L_{t,x} U+A(t,x,\vartheta_0)+B(t,x)\eta(t/\varepsilon),NEWLINE\]NEWLINE where \(L\) is a uniformly elliptic operator, \(\eta\) is a stationary \(\phi\)-mixing random process with \(\Pr\{\sup|\eta(t)|>H\}\leq C_1\exp(-C_2 H)\), and \(\vartheta_0\) is an unknown parameter to be estimated. The author constructs a quasi-maximum likelihood estimate \(\hat\vartheta_\varepsilon\) for \(\vartheta_0\) and obtains a large deviations type inequality for the probability NEWLINE\[NEWLINEPr\{\varepsilon^{-1/2}|\hat\vartheta_\varepsilon-\vartheta_0|>H\}NEWLINE\]NEWLINE (for large \(H\) and small \(\varepsilon\)).
scientific article; zbMATH DE number 1646311