On the Laplace Transform of the Lognormal Distribution

Søren Asmussen (), Jens Ledet Jensen () and Leonardo Rojas-Nandayapa ()
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Søren Asmussen: Aarhus University
Jens Ledet Jensen: Aarhus University
Leonardo Rojas-Nandayapa: University of Queensland

Methodology and Computing in Applied Probability, 2016, vol. 18, issue 2, 441-458

Abstract: 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 → ∞.

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; 60E05; 60E10; 90-04 (search for similar items in EconPapers)
Date: 2016
References: View complete reference list from CitEc
Citations: View citations in EconPapers (12)

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DOI: 10.1007/s11009-014-9430-7

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