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) over tilde(theta) of the Laplace transform 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 (L) over tilde(theta) is asymptotically equivalent to L(theta) as theta -> infinity. We apply this result to construct a reliable Monte Carlo estimator of L(theta) and prove it to be logarithmically efficient in the rare event sense as theta -> infinity.