Asmussen, S., Jensen, J.L. & Rojas-Nandayapa, L. On the Laplace Transform of the Lognormal Distribution. Methodol Comput Appl Probab 18, 441–458 (2016). https://doi.org/10.1007/s11009-014-9430-7

View Publication

Abstract

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  of the Laplace transform  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 → ∞.

Authors: Søren Asmussen, Jens Ledet Jensen, Leonardo Rojas-Nandayapa
Publication date: 2016/6
Journal: Methodology and Computing in Applied Probability
Volume: 18
Issue: 2
Pages: 441-458
Publisher: Springer US