Adaptive estimation for affine stochastic delay differential equations

Abstract

For stationary solutions of the affine stochastic delay differential equation d X(t)=( γ 0X(t)+γ rX(t-r)+∫ - r 0X(t+u)g(u)du )dt+σdW(t), we consider the problem of nonparametric inference for the weight function g and for γ0,γr from the continuous observation of one trajectory up to time T>0. For weight functions in the scale of Besov spaces Bsp,1 and Lρ-type loss functions, convergence rates are established for long-time asymptotics. The estimation problem is equivalent to an ill-posed inverse problem with error in the data and unknown operator. We propose a wavelet thresholding estimator that achieves the rate (T/logT)-s/(2s+3) under certain restrictions on p and ρ. This rate is shown to be optimal in a minimax sense.