Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise

Abstract

We consider the models Y_i,n=∫₀^i/nσ(s)dW_s+τ(i/n)ε_i,n, and Ỹ_i,n=σ(i/n)W_i/n+τ(i/n)ε_i,n, i=1,…,n, where (W_t)_t∈[0,1] denotes a standard Brownian motion and ε_i,n are centered i.i.d. random variables with E (ε_i,n²)=1 and finite fourth moment. Furthermore, σ and τ are unknown deterministic functions and (W_t)_t∈[0,1] and (ε_1,n,…,ε_n,n) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for σ² and τ² and investigate their rate of convergence of the MISE  in dependence of their smoothness. To this end specific basis functions and their corresponding Sobolev ellipsoids are introduced and we show that our estimators are optimal in minimax sense. Our work is motivated by microstructure noise models. A major finding is that the microstructure noise ε_i,n introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of ε_i,n. The performance of the estimates is illustrated by a small numerical study.