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

Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems higher order finite differences applied to stochastic processes with arbitrary finitely regular paths. These results are proved using the notion of \(\Delta \)-power variations, introduced herein, along Hölder-Zygmund norms. Consequently, prove a new theorem variations iterated integrals fractional Brownian motion. abstract results, besides being independent interest, in second part paper estimation drift and volatility coefficients semilinear partial differential equations dimension one, driven an additive Gaussian noise white time possibly colored space. In particular, solve earlier conjecture Cialenco et al. (Stat. Inference Stoch. Process. 23:83-103, 2020) about existence nontrivial bias estimators derived naive approximations derivatives differences. We give explicit formula convergence rates corresponding estimators. Theoretical illustrated numerical examples.