Title: Volatility estimation for stochastic PDEs using high-frequency observations,
Abstract: 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 estimation of the
integrated volatility in a semi-parametric framework and for the joint
estimation of the volatility and an unknown parameter in the differential
operator. Starting from a representation of the solution 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.