Faster than FFT: Conformal accelerations method

Abstract

Fourier-Laplace transform technique allows one to represent several classes of important probability distributions and solutions of basic boundary problems for wide classes of fractional differential equations as integrals of functions enjoying two key properties: analytic continuation to a cone or the union of a cone and tube domain, and regular decay at infinity. Integral representations for Wiener-Hopf factors, fractional moments and special functions enjoy these properties as well. In the paper, we present the general methodology which allows one to evaluate the integrals enjoying these properties very fast and accurately. Among applications, we derive new efficient realizations of the Fourier, Laplace and Z-transforms, representations for probability distributions in Levy models, stable ones including, and algorithms for pricing contingent claims, Monte-Carlo simulations and evaluation of special functions.