A Note on a Sum of Lognormals

TL;DR

This paper explores numerical methods for computing the moment generating function and derivatives of a sum of lognormal variables, focusing on techniques like Gauss-Hermite quadrature, asymptotic preprocessing, and inversion methods.

Contribution

It introduces a segmentation approach based on the derivative structure of the mgf/chf to improve the computation of sums of lognormals.

Findings

Gauss-Hermite quadrature effectively computes the characteristic function.

Asymptotic preprocessing improves numerical stability.

Segmentation based on derivative activity enhances inversion accuracy.

Abstract

This note considers the applicability of Gauss-Hermite quadrature and direct numerical quadrature for computation of moment generating function (mgf) and the derivatives. A preprocessing using the asymptotic technique is employed while computing the characteristic function (chf) using Gauss Hermite quadrature while this is optional for mgf. The mgf of the low and high amplitude regions of a single lognormal variable and the derivatives is examined and attention is drawn to the effect of variance. The problem of inversion of the mgf/chf of a sum of lognormals to obtain the CDF/pdf is considered with special reference to methods related to Post Widder technique, Gaussian quadrature and the Fourier series method. The method based on the complex exponential integral which makes use of the derivative of the cumulant is an alternative. Segmentation of the mgf/chf on the basis of the derivative structure which indicates activity rate is shown to be useful.