Open Access
2010 Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise
Axel Munk, Johannes Schmidt-Hieber
Electron. J. Statist. 4: 781-821 (2010). DOI: 10.1214/10-EJS568


We consider the models Yi,n=0i/nσ(s)dWs+τ(i/n)εi,n, and i,n=σ(i/n)Wi/n+τ(i/n)εi,n, i=1,,n, where (Wt)t[0,1] denotes a standard Brownian motion and εi,n are centered i.i.d. random variables with E (εi,n2)=1 and finite fourth moment. Furthermore, σ and τ are unknown deterministic functions and (Wt)t[0,1] and (ε1,n,,εn,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 εi,n introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of εi,n. The performance of the estimates is illustrated by a small numerical study.


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Axel Munk. Johannes Schmidt-Hieber. "Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise." Electron. J. Statist. 4 781 - 821, 2010.


Published: 2010
First available in Project Euclid: 8 September 2010

zbMATH: 1329.62366
MathSciNet: MR2684388
Digital Object Identifier: 10.1214/10-EJS568

Primary: 62M09 , 62M10
Secondary: 62G08 , 62G20

Keywords: Brownian motion , microstructure noise , Minimax rate , Sobolev embedding , variance estimation

Rights: Copyright © 2010 The Institute of Mathematical Statistics and the Bernoulli Society

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