Lp estimation of the diffusion coefficient

Abstract

We study the functional estimation of the space-dependent diffusion coefficient in a one-dimensional framework. The sample path is observed at discrete times. We study global $L_p$-loss errors $(1\le p<+\mathrm{\infty })$ over Besov spaces $B_{\mathrm{sp}\mathrm{\infty }}$. We show that, under suitable conditions, the minimax rate of convergence is the usual $n^{-s/(1+2s)}$. Linking our model to nonparametric regression, we provide an estimating procedure based on a linear wavelet method which is optimal in the minimax sense.