On the Laplace Transform of the Lognormal Distribution - Asmussen, Søren; Jensen, Jens Ledet; Rojas-Nandayapa, Leonardo (Methodology and computing in applied probability, 2016)
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 → ∞ .