A nonparametric estimation problem for linear SPDEs

Randolf Altmeyer/ Markus Reiß, HU Berlin WIAS, Erhard-Schmidt Hörsaal, Mohrenstraße 39, 10117 Berlin10:00 - 12:30

It is well-known that parameters in the drift part of a stochastic ordinary differential equation, observed continuously on a time interval [0, T ], are generally only identifiable, if either T → ∞, the driving noise becomes small or if a sequence of independent samples is observed. On the other hand, in the case of a linear stochastic partial differential equation

dX(t,x) = θAX(t,x)dt + dW(t,x), x ∈ Ω ⊂ Rd, (1)

for a nonpositive self-adjoint operator A and an unknown parameter θ > 0, [1] showed that consistent estimation of θ is also possible in finite time T < ∞, if ⟨X(t,·),ek⟩ is observed continuously on [0,T] for k = 1,...,N as N → ∞, where the test functions ek are the eigenfunctions of A.
Our goal is to study this estimation problem for general test functions ek. Using an MLE-inspired estimator, we extend the results of [1] and give a precise understanding of how the estimation error depends on the interplay between A and the test functions ek. In particular, we show that more localized test functions improve the estimation considerably. It turns out that one local measurement ⟨X(t,·),uh⟩ is already sufficient for identifying θ, as long as h → 0, where uh(x) = h−d/2u(x/h) for a smooth kernel u. Central limit theorems are provided, as well. We further show that the same techniques extend to the more difficult nonparametric estimation problem, when θ is space-dependent. Indeed, we can show that θ(x0) at x0 ∈ Ω is identifiable using only local information. The rate of convergence, however, is affected by the bias, which is non-local and difficult to analyse, even when T → ∞. Possible solutions are discussed, along with questions of efficiency.

References
[1] M. Huebner and B.L. Rozovskii. On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE’s. Probability theory and related fields, 103, 1995, 143-163.