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 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 → ∞ .