Nonparametric calibration for stochastic reaction–diffusion equations based on discrete observations

Nonparametric estimation for semilinear SPDEs, namely stochastic reaction–diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid time and with infill asymptotics both coordinates. Firstly, we derive nonparametric estimator reaction function underlying equation. The estimate chosen from finite-dimensional based least squares criterion. Oracle inequalities provide conditions to achieve usual rate convergence. Adaptivity provided via model selection. Secondly, show that asymptotic properties realized quadratic variation estimators diffusivity volatility carry over linear SPDEs. In particular, obtain rate-optimal joint two parameters. result relies our precise analysis Hölder regularity process its nonlinear component, which may be own interest. Both steps calibration can carried out simultaneously without prior knowledge