Approximating the Laplace transform of the sum of dependent lognormals

TL;DR

This paper introduces an approximation method for the Laplace transform of sums of dependent lognormal variables, providing a closed-form component and analyzing the error factor, with numerical validation and applications.

Contribution

It develops a novel approximation for the Laplace transform of dependent lognormals using Taylor expansion and asymptotic analysis, including algorithms for error estimation.

Findings

The approximation $ ilde{ m L}( heta)$ is accurate for large $ heta$.

The error factor $I( heta)$ approaches 1 as $ heta$ increases.

Numerical methods effectively evaluate the approximation and invert the Laplace transform.

Abstract

Let $(X_1, \dots, X_n)$ be multivariate normal, with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$, and $S_n=\mathrm{e}^{X_1}+\cdots+\mathrm{e}^{X_n}$. The Laplace transform ${\cal L}(\theta)=\mathbb{E}\mathrm{e}^{-\theta S_n} \propto \int \exp\{-h_\theta(\boldsymbol{x})\} \,\mathrm{d} \boldsymbol{x}$ is represented as $\tilde{\cal L}(\theta)I(\theta)$, where $\tilde{\cal L}(\theta)$ is given in closed-form and $I(\theta)$ is the error factor ($\approx 1$). We obtain $\tilde{\cal L}(\theta)$ by replacing $h_\theta(\boldsymbol{x})$ with a second order Taylor expansion around its minimiser $\boldsymbol{x}^*$. An algorithm for calculating the asymptotic expansion of $\boldsymbol{x}^*$ is presented, and it is shown that $I(\theta)\to 1$ as $\theta\to\infty$. A variety of numerical methods for evaluating $I(\theta)$ are discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace transform inversion for the density of $S_n$) are also given.