Nonparametric inference for parabolic stochastic partial differential equations

Abstract

Consider a parabolic stochatic partial differential equation of the type du(t,x) = Au(t,x) dt+θ(t)dW (t,x) 0 ≤ t ≤ T, x ∈ G where A is a partial differential operator, θ(t) is a positive valued function with θ(t) ∈ C^m([0,∞)] for some m ≥ 1 and W(t,x) is a cylindrical Brownian motion in L₂(G), G being a bounded domain in R^d with the boundary ∂G as a C^∞ manifold of dimension (d-1) and locally G is totally on one side of ∂G. We obtain an estimator for the function θ(t) based on the Fourier coefficients u_i(t),1 ≤ i ≤ N of the random field u(t,x) observed at discrete times and study its asymptotic properties.