Adaptive estimation for affine stochastic delay differential equations

Abstract

For stationary solutions of the affine stochastic delay differential equation $$dX\left(t\right)=\left({\gamma }_{0}X\right(t)+{\gamma }_{\mathrm{rX}}(t-r)+{\int }_{-r}^{0}X(t+u\left)g\right(u)du)dt+\sigma dW\left(t\right),$$ we consider the problem of nonparametric inference for the weight function g and for γ₀,γ_r from the continuous observation of one trajectory up to time T>0. For weight functions in the scale of Besov spaces B^s_p,1 and L^ρ-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.