On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE's

Abstract

We investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic SPDEs of the form $$du(t,x) = (A_0 + \theta A_1 )u(t,x)dt + dW(t,x),$$ whereA 0 andA 1 are partial differential operators andW is a cylindrical Brownian motion. We introduce a spectral method for computing MLEs based on finite dimensional approximations to solutions of such systems, and establish criteria for consistency, asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity. We derive the asymptotic properties of the MLE from a condition on the order of the operators. In particular, the MLE is consistent if and only if ord(A 1)≧1/2(ord(A 0+θA 1)−d).