Elliptical Tempered Stable Distribution and Fractional Calculus

A definition for elliptical tempered stable distribution, based on the characteristic function, have been explained which involve a unique spectral measure. This definition provides a framework for creating a connection between infinite divisible distribution, and particularly elliptical tempered stable distribution, with fractional calculus. Finally, some analytical approximations for the probability density function of tempered infinite divisible distribution, which elliptical tempered stable distributions are a subclass of them, are considered.Comment: 16 pages, working pape

Fourier Approximation for Integral Equations on the Real Line

Based on Guass quadradure method a class of integral equations having unknown periodic solution on the real line is investigated, by using Fourier series expansion for the solution of the integral equation and applying a process for changing the interval to the finite interval −1; 1 , the Chebychev weights become appropriate and examples indicate the high accuracy and very good approximation to the solution of the integral

Fourier Approximation for Integral Equations on the Real Line

Based on Guass quadradure method a class of integral equations having unknown periodic solution on the real line is investigated, by using Fourier series expansion for the solution of the integral equation and applying a process for changing the interval to the finite interval (−1; 1), the Chebychev weights become appropriate and examples indicate the high accuracy and very good approximation to the solution of the integral

Quantile-based inference for tempered stable distributions

A simple, fast, and accurate method for the estimation of numerous distributions that belong to the tempered stable class is introduced. The method is based on the Method of Simulated Quantiles and it consists of matching empirical and theoretical functions of quantiles that are informative about the parameters of interest. In the Monte Carlo study we show that MSQ is significantly faster than Maximum Likelihood and the estimates are almost as precise as under MLE. A Value-at-Risk and Expected Shortfall study for 13 years of daily data and for an array of market indexes world-wide shows that the tempered stable estimation with MSQ estimates provides reasonable risk assessment