Statistical inference for stochastic parabolic equations: a spectral approach. (English) Zbl 1157.62057

Lototsky, S. V.

Publ. Mat., Barc. 53, No. 1, 3-45 (2009).

Summary: A parameter estimation problem is considered for a stochastic parabolic equation driven by additive Gaussian noise that is white in time and space. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed. A necessary and sufficient condition for consistency and asymptotic normality of the estimator is derived in terms of the eigenvalues of the operators in the equation, and a detailed proof is provided. Other estimation problems are briefly surveyed.

MSC:

62M05	Markov processes: estimation; hidden Markov models
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
62F12	Asymptotic properties of parametric estimators
60G15	Gaussian processes
60G30	Continuity and singularity of induced measures

Keywords:

cylindrical Brownian motion; Ornstein-Uhlenbeck process; singular statistical models