  We study the parameter estimation for parabolic, linear, second-order,
stochastic partial differential equations (SPDEs) observing a mild solution on
a discrete grid in time and space. A high-frequency regime is considered where
the mesh of the grid in the time variable goes to zero. Focusing on volatility
estimation, we provide an explicit and easy to implement method of moments
estimator based on squared increments. The estimator is consistent and admits a
central limit theorem. This is established moreover for the joint estimation of
the integrated volatility and parameters in the differential operator in a
semi-parametric framework. Starting from a representation of the solution of
the SPDE with Dirichlet boundary conditions as an infinite factor model and
exploiting mixing-type properties of time series, the theory considerably
differs from the statistics for semi-martingales literature. The performance of
the method is illustrated in a simulation study.
