Adaptive estimation for affine stochastic delay differential equations

Abstract

For stationary solutions of the affine stochastic delay differential equation dX(t) = (γ0X(t) + γrX(t - r) + ∫-r0 X (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 Bp,1s and LP-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/log T)-s/(2s+3) under certain restrictions on p and p. This rate is shown to be optimal in a minimax sense. © 2005 ISI/BS.