Spectral asymptotics of some functionals arising in statistical inference for SPDEs
Let \(M\) be a \(d\)-dimensional compact orientable \(C^\infty \) manifold, \(L\) an elliptic, positive definite, self-adjoint differential operator of order \(2m\) with real \(C^\infty\) coefficients on \(M\). Let \(A\), \(B\), \(N\) be differential operators (all with real \(C^\infty\) coefficients) on \(M\) of orders less than \(2m\), and let \(W\) be a standard cylindrical Wiener process in \(L^2(M)\). A random field \(u\) on \(M\) defined by a stochastic partial differential equation \[ (1)\qquad\roman du(t) + [\theta _1 (L+A) + \theta _2 B + N]u(t) = \roman dW(t),\quad 0<t\leq T,\quad u(0) = u_0, \] is studied, the task being to estimate one of the parameters \(\theta _1\), \(\theta _2\) assuming that the other is known. Closely related problems were solved by \textit{M. Huebner} and \textit{B. Rozovskii} [Probab. Theory Relat. Fields 103, No. 2, 143-163 (1995; Zbl 0831.60070)] and by \textit{L. Piterbarg} and \textit{B. Rozovskii} [Math. Methods Stat. 6, No. 2, 200-223 (1997; Zbl 0884.65140)] under the additional assumption that the differential operators in (1) commute. In this paper this hypothesis is relaxed. Quasi maximum likelihood estimates of the unknown parameter based on a finite number of spatial Fourier coefficients of the solution are proposed and their consistency, asymptotic normality and asymptotic efficiency is established under suitable non-degeneracy assumptions.
scientific article; zbMATH DE number 1554274
scientific article