Estimating the Spectral Measure of a Multivariate Stable Distribution via Spherical Harmonic Analysis

Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
Journal of Multivariate Analysis (Impact Factor: 0.93). 11/2003; 87(2):219-240. DOI: 10.1016/S0047-259X(03)00052-6
Source: RePEc


A new method is developed for estimating the spectral measure of a multivariate stable probability measure, by representing the measure as a sum of spherical harmonics.

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    • "Stable processes can be used to model real-world phenomena [32] [20], and in particular they are commonly used in Mathematical Finance; see for example [26] [11] [27] [28] [29] [8] and references therein. The infinitesimal generator of any symmetric stable Lévy process is of the form "
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    ABSTRACT: We establish sharp regularity estimates for solutions to $Lu=f$ in $\Omega\subset\mathbb R^n$, being $L$ the generator of any stable and symmetric L\'evy process. Such nonlocal operators $L$ depend on a finite measure on $S^{n-1}$, called the spectral measure. First, we study the interior regularity of solutions to $Lu=f$ in $B_1$. We prove that if $f$ is $C^\alpha$ then $u$ belong to $C^{\alpha+2s}$ whenever $\alpha+2s$ is not an integer. In case $f\in L^\infty$, we show that the solution $u$ is $C^{2s}$ when $s\neq1/2$, and $C^{2s-\epsilon}$ for all $\epsilon>0$ when $s=1/2$. Then, we study the boundary regularity of solutions to $Lu=f$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, in $C^{1,1}$ domains $\Omega$. We show that solutions $u$ satisfy $u/d^s\in C^{s-\epsilon}(\overline\Omega)$ for all $\epsilon>0$, where $d$ is the distance to $\partial\Omega$. Finally, we show that our results are sharp by constructing two counterexamples.
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    • "Concerning the approximations, Davydov and Nagaev (2002) have outlined possible directions although their results are still theoretical. For the estimation of the parameters, see Pivato and Seco (2003) and Davydov and Paulauskas (1999) and for the hypothesis testing see Mittnik et al. (1999). Although there are many other researches, accurate calculations of the multivariate stable densities are clearly of basic importance. "
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    ABSTRACT: We consider the numerical evaluation of one-dimensional projections of general multivariate stable densities introduced by Abdul-Hamid and Nolan [H. Abdul-Hamid, J.P. Nolan, Multivariate stable densities as functions of one dimensional projections, J. Multivariate Anal. 67 (1998) 80-89]. In their approach higher order derivatives of one-dimensional densities are used, which seems to be cumbersome in practice. Furthermore there are some difficulties for even dimensions. In order to overcome these difficulties we obtain the explicit finite-interval integral representation of one-dimensional projections for all dimensions. For this purpose we utilize the imaginary part of complex integration, whose real part corresponds to the derivative of the one-dimensional inversion formula. We also give summaries on relations between various parametrizations of stable multivariate density and its one-dimensional projection.
    Preview · Article · Mar 2009 · Journal of Multivariate Analysis
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