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On the Laplace Transform of the Lognormal Distribution

Author

Listed:
  • Søren Asmussen

    (Aarhus University)

  • Jens Ledet Jensen

    (Aarhus University)

  • Leonardo Rojas-Nandayapa

    (University of Queensland)

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 ℒ ~ ( 𝜃 ) $\widetilde {\mathcal {L}}(\theta )$ of the Laplace transform ℒ ( 𝜃 ) $\mathcal {L}(\theta )$ 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 → ∞.

Suggested Citation

  • Søren Asmussen & Jens Ledet Jensen & Leonardo Rojas-Nandayapa, 2016. "On the Laplace Transform of the Lognormal Distribution," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 441-458, June.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:2:d:10.1007_s11009-014-9430-7
    DOI: 10.1007/s11009-014-9430-7
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    Citations

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    Cited by:

    1. Manuel D. Ortigueira, 2022. "A New Series Representation and the Laplace Transform for the Lognormal Distribution," Mathematics, MDPI, vol. 10(19), pages 1-13, September.
    2. Azar, Macarena & Carrasco, Rodrigo A. & Mondschein, Susana, 2022. "Dealing with uncertain surgery times in operating room scheduling," European Journal of Operational Research, Elsevier, vol. 299(1), pages 377-394.
    3. Dang, Chao & Xu, Jun, 2020. "Unified reliability assessment for problems with low- to high-dimensional random inputs using the Laplace transform and a mixture distribution," Reliability Engineering and System Safety, Elsevier, vol. 204(C).
    4. Furman, Edward & Hackmann, Daniel & Kuznetsov, Alexey, 2020. "On log-normal convolutions: An analytical–numerical method with applications to economic capital determination," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 120-134.
    5. Lorenzo Cappello & Stephen G. Walker, 2018. "A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 777-797, June.
    6. Zied Chaieb & Djibril Gueye, 2022. "Pricing zero-coupon CAT bonds using the enlargement of ltration theory: a general framework," Papers 2208.02609, arXiv.org.
    7. Claire Mouminoux & Christophe Dutang & Stéphane Loisel & Hansjoerg Albrecher, 2022. "On a Markovian Game Model for Competitive Insurance Pricing," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1061-1091, June.
    8. McFadden, Daniel, 2022. "Instability in mixed logit demand models," Journal of choice modelling, Elsevier, vol. 43(C).
    9. Zied Chaieb & Djibril Gueye, 2022. "Pricing zero-coupon CAT bonds using the enlargement of ltration theory: a general framework ," Post-Print hal-03745077, HAL.
    10. Laurence Carassus & Massinissa Ferhoune, 2021. "Efficient approximations for utility-based pricing," Papers 2105.08804, arXiv.org, revised Feb 2024.
    11. Christopher Dobronyi & Christian Gouri'eroux, 2020. "Consumer Theory with Non-Parametric Taste Uncertainty and Individual Heterogeneity," Papers 2010.13937, arXiv.org, revised Jan 2021.

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