On the Laplace Transform of the Lognormal Distribution

Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyse a closed-form approximation ℒ~(𝜃)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\widetilde {\mathcal {L}}(\theta )$\end{document} of the Laplace transform ℒ(𝜃)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}(\theta )$\end{document} which is obtained via a modified version of Laplace’s method. This approximation, given in terms of the Lambert W(⋅) function, is tractable enough for applications. We prove that ~(𝜃) is asymptotically equivalent to ℒ(𝜃) as 𝜃 → ∞. We apply this result to construct a reliable Monte Carlo estimator of ℒ(𝜃) and prove it to be logarithmically efficient in the rare event sense as 𝜃 → ∞.