On the Laplace Transform of the Lognormal Distribution

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 \(\widetilde {\mathcal {L}}(\theta )\) 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.

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Correspondence to Leonardo Rojas-Nandayapa.

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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

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Keywords

  • 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