New Families of Integral Representations and Efficient Evaluation of Stable Distributions

Abstract

We suggest new families of integral representations of pdfs and cpds of stable distributions, which lead to efficient numerical methods. The first method is based on the approximation of the characteristic exponent by functions analytic in the complex plane with two cuts i(-&#8734,-&#955] and i[&#955, &#8734), which are slightly different from the characteristic exponents of processes of KoBoL class (Boyarchenko and Levendorskii, 1999). We justify Richardson's extrapolation, and calculate the approximating values for several small values of &#955 using the sinh-acceleration technique (Boyarchenko and Levendorskii, 2018). The second method starts with the change of variable &#958=eiw y in the integrals over R that represent pdfs and cpdfs. At the last step, in both methods, the simplified trapezoid rule is applied. For wide regions in the parameter space, the absolute error tolerance of the order of 10-15 can be satisfied in 0.005-0.1 msc, in Matlab implementation, even when the index of the distribution is small or close to 1. For calculation of quantiles in wide regions in the tails using the Newton or bisection method, it suffices to precalculate several hundred of values of the characteristic exponent at points of an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf, which require a fairly small number of elementary operations. The methods of the paper are applicable to other classes of integrals, highly oscillatory ones especially, and, typically, are faster than popular methods.