Recovery of density functions in Lévy driven models by \begin{document}$ SK $\end{document} -splines

The main aim of this article is to construct an efficient method of recovery of density functions arising in Levy driven models. Density function of any Levy process in \begin{document}$ \mathbb{R}^{n} $\end{document} can be expressed as a Fourier transform of the respective characteristic function defined by the Khinchine-Levy formula. To approximate Fourier transforms of functions induced by the Khinchine-Levy formula we introduce \begin{document}$ sk $\end{document} -splines on \begin{document}$ \mathbb{R}^{n} $\end{document} with knots and points of interpolation on \begin{document}$ \mathbf{A}\mathbb{Z}^{n} $\end{document} , where \begin{document}$ \mathbf{A} $\end{document} is an arbitrary nonsingular \begin{document}$ n\times n $\end{document} matrix. Such sets of points are analogs for \begin{document}$ \mathbb{R}^{n} $\end{document} of Number Theoretic Korobov and Sparse Grids on the torus \begin{document}$ \mathbb{T}^{n} $\end{document} and proved to be useful in problems of high dimensionality. Representations of cardinal \begin{document}$ sk $\end{document} -splines are given in terms of Fourier transforms of special functions which allows us to construct explicit forms of approximants for density functions without calculation of Fourier transforms. High efficiency of the proposed algorithm is underlined by simple numerical examples.