Non-parametric estimation of non-linear diffusion coefficient in parabolic SPDEs

In this article, we introduce a novel non-parametric predictor, based on conditional expectation, for the unknown diffusion coefficient function $\sigma$ in the stochastic partial differential equation $Lu = \sigma(u)\dot{W}$, where $L$ is a parabolic second order differential operator and $\dot{W}$ is a suitable Gaussian noise. We prove consistency and derive an upper bound for the error in the $L^p$ norm, in terms of discretization and smoothening parameters $h$ and $\varepsilon$. We illustrate the applicability of the approach and the role of the parameters with several interesting numerical examples.