Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach

Abstract

Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are proved by using the notion of ∆-power variations, introduced herein, along with the Hölder-Zygmund norms. Consequently, we prove a new central limit theorem for ∆-power variations of the iterated integrals of a fractional Brownian motion (fBm). These abstract results, besides being of independent interest, in the second part of the paper are applied to estimation of the drift and volatility coefficients of semilinear stochastic partial differential equations in dimension one, driven by an additive Gaussian noise white in time and possibly colored in space. In particular, we solve the earlier conjecture from [CKL20] about existence of a nontrivial bias in the estimators derived by naive approximations of derivatives by finite differences. We give an explicit formula for the bias and derive the convergence rates of the corresponding estimators. Theoretical results are illustrated by numerical examples.

Nonparametric estimation for linear SPDEs from local measurements

“…While the estimation of a scalar parameter in front of the highest order operator A i is well studied in the literature [24,28,12,13,20], there is little known about estimating the lower order coefficients or the full multivariate parameter ϑ. Relying on discrete space-time observations X(t k , x j ) in case of (1.3) and in dimension d = 1, [9,23,44] have analysed power variations and contrast estimators. For two parameters in front of operators A 1 and A 2 , [37] computed the maximum likelihood estimator from M spectral measurements ( X(t), e j ) 0≤t≤T , j = 1, .…”