On the Laplace Transform of the Lognormal Distribution

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 approximat...

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Bibliographic Details
Published in:Methodology and computing in applied probability Vol. 18; no. 2; pp. 441 - 458
Main Authors: Asmussen, Søren, Jensen, Jens Ledet, Rojas-Nandayapa, Leonardo
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2016
Springer Nature B.V
Subjects:
Approximation
Asymptotic properties
Business and Management
Distribution
Economics
Electrical Engineering
Exact solutions
Laplace transforms
Life Sciences
Mathematical analysis
Mathematical models
Mathematics and Statistics
Monte Carlo simulation
Probability
Statistics
Studies
Texts
United States > US
Monte Carlo method
Lambert W function
90-04
60E10
Rare event simulation
Laplace’s method
Lognormal distribution
Characteristic function
Efficiency
Laplace transform
Importance sampling
Moment generating function
60E05
ISSN:1387-5841, 1573-7713
Online Access:Get full text
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Summary: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 ) of the Laplace transform ℒ ( 𝜃 ) 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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ISSN:1387-5841
1573-7713
DOI:10.1007/s11009-014-9430-7