A note on wavelet density deconvolution for weakly dependent data

Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuous time processes and discrete time models will be discussed.The key insight for the analysis is a transformation of the volatility density estimation problem to a deconvolution model for which standard methods exist. Three type of nonparametric density estimators are reviewed: the Fourier-type deconvolution kernel density estimator, a wavelet deconvolution density estimator and a penalized projection estimator. The performance of these estimators will be compared.

show abstract

“…We now have the following result, see Van Zanten and Zareba [32], for the wavelet density estimatorĝ n of g defined by (16).…”

Section: An Application To the Amsterdam Aex Indexmentioning

confidence: 98%

Nonparametric Methods for Volatility Density Estimation

Spreij

Zanten

2011

Advanced Mathematical Methods for Finance

Self Cite

View full text Add to dashboard Cite

show abstract

“…, from ( ) ∈Z . Some related works are Masry [44], Kulik [45], Comte et al [46], and Van Zanten and Zareba [47].…”

Section: Density Deconvolutionmentioning

confidence: 99%

A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator underα-Mixing Dependence

Chesneau¹

2014

Journal of Probability and Statistics

View full text Add to dashboard Cite

We consider the estimation of an unknown function for weakly dependent data ( -mixing) in a general setting. Our contribution is theoretical: we prove that a hard thresholding wavelet estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context. Another application concerns the regression model with random design.

show abstract

“…Van Zanten and Zareba [24] consider wavelet estimators of the density of the accumulated squared volatility over intervals of length ∆ with ∆ fixed for the model without drift and with the same observation scheme. Under similar conditions, they found this rate for the supremum of the mean integrated squared error, the supremum taken over densities in some Sobolev ball.…”

Section: Remark 37mentioning

confidence: 99%

“…Statement (17) follows by combining standard arguments of kernel density estimation applied to expression (24) in Lemma 5.1 with Lemma 5.2. We will now show that the bound in Lemma 5.2 is essentially a negative power of n, whereas h 2 is of logarithmic order.…”

Section: Lemma 51mentioning

confidence: 99%

“…Using ideas from deconvolution theory, we will propose a procedure for the estimation of this density at a number of fixed time instants. Related work on estimating a stationary univariate density of the volatility process has been done by Van Es et al [22], Comte and Genon-Catalot [3], Van Zanten and Zareba [24], whereas a deconvolution approach has also been adopted to estimate a regression function for a discrete time stochastic volatility model by Franke et al [8], Comte [1] and Comte et al [2]. In [4] it is assumed that the volatility process solves a stochastic differential equation and nonparametric estimators for the drift and diffusion coefficients of that equation are studied.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Estimation of a multivariate stochastic volatility density by kernel deconvolution

Spreij

2011

Journal of Multivariate Analysis

View full text Add to dashboard Cite

a b s t r a c tWe consider a continuous time stochastic volatility model. The model contains a stationary volatility process. We aim to estimate the multivariate density of the finite-dimensional distributions of this process. We assume that we observe the process at discrete equidistant instants of time. The distance between two consecutive sampling times is assumed to tend to zero.A multivariate Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the multivariate volatility density. An expansion of the bias and a bound on the variance are derived.

show abstract

A note on wavelet density deconvolution for weakly dependent data

Cited by 7 publications

References 21 publications

Nonparametric Methods for Volatility Density Estimation

Nonparametric Methods for Volatility Density Estimation

A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator underα-Mixing Dependence

Estimation of a multivariate stochastic volatility density by kernel deconvolution

Contact Info

Product

Resources

About