Nonparametric inference for a class of stochastic partial differential equations based on discrete observations

Abstract

Consider the stochastic partial differential equations of the type du,(t,x) = (Δu,(t,x)+u,(t,x))dt + ∊ Θ(t) dW_Q(t,x), Θ ≤ t ≤ T and du_∊,(t,x)= Δu_∊ (t,x)dt+ ∊ Θ(t) (I - Δ)^-1/2 dW(t,x), 0 ≤ t ≤ T where Δ = ∂²/∂x²,θ ∈ Θ and Θ is a class of positive valued functions such that Θ²(t)∈ L²(R). 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.