On the Laplace Transform of the Lognormal Distribution

  • Søren Asmussen
  • Jens Ledet Jensen
  • Leonardo Rojas-NandayapaEmail author


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 \(\widetilde {\mathcal {L}}(\theta )\) of the Laplace transform \(\mathcal {L}(\theta )\) 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.


Characteristic function Efficiency Importance sampling Lambert W function Laplace transform Laplace’s method Lognormal distribution Moment generating function Monte Carlo method Rare event simulation 

Mathematics Subject Classifications (2010)

60E05 60E10 90-04 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Søren Asmussen
    • 1
  • Jens Ledet Jensen
    • 1
  • Leonardo Rojas-Nandayapa
    • 2
    Email author
  1. 1.Department of MathematicsAarhus UniversityAarhusDenmark
  2. 2.School of Mathematics and PhysicsUniversity of QueenslandBrisbaneAustralia

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