  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 $\Delta$-power variations, introduced
herein, along with the H\"older-Zygmund norms. Consequently, we prove a new
central limit theorem for $\Delta$-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 Cialenco, Kim, Lototsky (2019) 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.
