Estimation problems for coefficients of stochastic partial differential equations. III (Q2752949)

[For part II of this paper see the preceding entry, Zbl 0983.62052.]NEWLINENEWLINENEWLINEIn this part III the authors study the problem of estimation of a functional parameter, i.e., the coefficient \(\theta = a_0(t,x)\) of the SPDE NEWLINE\[NEWLINEdu_{\varepsilon}(t) = \sum_{|k|\leq 2p} a_k D_x^k u_{\varepsilon} dt + f dt + \varepsilon dw(t) = 0,NEWLINE\]NEWLINE with specified initial and boundary conditions. Asymptotically minimax estimates are provided for \(\theta\) as well as asymptotically efficient estimates for \(\Phi(\theta)\) (where \(\Phi\) is a certain functional) when \(\theta\) does not depend on \(t\). The construction of consistent estimates for \(\theta(x)\) employs approximation theory [see \textit{S.M. Nikol'skiĭ, Approximation of functions of several variables and imbedding theorems. (Russian) Moscow: Nauka (1969; Zbl 0185.37901); English translation Berlin etc.: Springer-Verlag (1975; Zbl 0307.46024)].}