On the Laplace Transform of the Lognormal Distribution

  • Søren Asmussen
  • Jens Ledet Jensen
  • Leonardo Rojas-Nandayapa
Article

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.

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 

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References

  1. Abramowitz M, Stegun I (1964) Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Applied mathematics series. Dover PublicationsGoogle Scholar
  2. Aitchison I, Brown JAC (1957) The lognormal distribution with special reference to its uses in economics. Cambridge University Press, CambridgeMATHGoogle Scholar
  3. Asmussen S, Glynn PW (2007) Stochastic simulation: algorithms and analysis. Springer-Verlag, New YorkMATHGoogle Scholar
  4. Asmussen S, Jensen JL, Rojas-Nandayapa L (2014) Exponential family techniques for the lognormal left tail. Research ReportGoogle Scholar
  5. Barakat R (1976) Sums of independent lognormally distributed random variables. J Opt Soc Am 66: 211–216MathSciNetCrossRefGoogle Scholar
  6. Barouch E, Kaufman GM (1976) On sums of lognormal random variables. Working paper. A. P. Sloan School of Management, MITGoogle Scholar
  7. Beaulieu NC, Xie Q (2004) An optimal lognormal approximation to lognormal sum distributions. IEEE Trans Veh Technol 53(2):479–489CrossRefGoogle Scholar
  8. Butler RW (2007) Saddlepoint approximations with applications. Cambridge University PressGoogle Scholar
  9. Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5:329–359MathSciNetCrossRefMATHGoogle Scholar
  10. Crow EL, Shimizu K (1988) Lognormal distributions: theory and applications. Marcel Dekker Inc., New YorkMATHGoogle Scholar
  11. de Bruijn NG (1970) Asymptotic methods in analysis. Courier Dover PublicationsGoogle Scholar
  12. Dufresne D (2008) Sums of lognormals. In: Actuarial research conference proceedingsGoogle Scholar
  13. Foss S, Korshunov D, Zachary S (2011) An introduction to heavy-tailed and subexponential distributions. SpringerGoogle Scholar
  14. Gubner JA (2006) A new formula for lognormal characteristic functions. IEEE Trans Veh Technol 55(5):1668–1671CrossRefGoogle Scholar
  15. Heyde C (1963) On a property of the lognormal distribution. J Roy Stat Soc Ser B 29:392–393MathSciNetMATHGoogle Scholar
  16. Holgate P (1989) The lognormal characteristic function. Commun Stat-Theor Methods 18:4539–4548MathSciNetCrossRefMATHGoogle Scholar
  17. Jensen JL (1994) Saddlepoint approximations. Oxford Science Publications, OxfordMATHGoogle Scholar
  18. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley, New YorkGoogle Scholar
  19. Leipnik RB (1991) On lognormal random variables: I. the characteristic function. J Aust Math Soc Ser B 32:327–347MathSciNetCrossRefMATHGoogle Scholar
  20. Limpert E, Stahel WA, Abbt M (2001) Log-normal distributions across the sciences: keys and clues. BioScience 51:341–352CrossRefGoogle Scholar
  21. Rojas-Nandayapa L (2008) Risk probabilities: asymptotics and simulation. Ph.D. thesis, Aarhus UniversityGoogle Scholar
  22. Rossberg AG (2008) Laplace transforms or probability distributions and their inversions are easy on logarithmic scales. J Appl Probab 45:531–541MathSciNetCrossRefMATHGoogle Scholar
  23. Small CG (2013) Expansions and asymptotics for statistics. Chapman & Hall/CRCGoogle Scholar
  24. Tellambura C, Seranarte D (2010) Accurate computation of the MGF of the lognormal distribution and its application to sum of lognormals. IEEE Trans Commun 58:1568–1577CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

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

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