Numerical Inversion of Laplace Transforms of Probability Distributions

The purpose of this document is to summarize main points of the paper, “Numerical Inversion of Laplace Transforms of Probability Distributions”, and provide R code for the Euler method that is described in the paper. The code is used to invert the Laplace transform of some popular functions. Context Laplace transform is a useful mathematical tool that one must be familiar with, while doing applied work. It is widely used in Queueing models where probability distributions are characterized in terms of transforms. Inverting a Laplace transform to get to the probability distribution is an essential task in Queueing theory. For textbook examples and simple Markovian models, one might be fortunate to find convenient forms for LT inversion. However for most of the real life situations, a practitioner needs to know a way to numerically invert LT. The paper titled,“Numerical Inversion of Laplace Transforms of Probability Distributions”, written by Joseph Abate and Ward Whitt gives two methods. This document focuses on one of the methods, Euler Objective The objective is to calculate values of a real valued function f(t) of a positive real variable t for various t from the Laplace transform f̂(s) = ∫ ∞ 0 e−stf(t)dt, where s is a complex variable with nonnegative real part. The following are the assumptions made about the function f(t): ˆ |f | ≤ 1 for all t in the error analysis ˆ f is sufficiently smooth. ˆ The algorithm requires that one is able to evaluate the real part of the Laplace transform f̂(s) at any desired complex s ˆ The algorithm is intended for computing f(t) at single values of t.