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On the Laplace transform of the Lognormal distribution

By Søren Asmussen, Jens Ledet Jensen and Leonardo Rojas-Nandayapa
Thiele Research Reports
No. 06, November 2013
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 analyze a closed-form approximation $\widetilde{\mathcal{L}}(\theta)$ of the Laplace transform $\mathcal{L}(\theta)$ which is obtained via a modified version of Laplace's method. This approximation, given in terms of the Lambert $W(\cdot)$ function, is tractable enough for applications. We prove that $\widetilde{\mathcal{L}}(\theta)$ is asymptotically equivalent to $\mathcal{L}(\theta)$ as $\theta\to \infty$. We apply this result to construct a reliable Monte Carlo estimator of $\mathcal{L}(\theta)$ and prove it to be logarithmically efficient in the rare event sense as $\theta\to \infty$.

Keywords: Lognormal distribution, Laplace transform, Characteristic function, Moment generating function, Laplace's method, Saddlepoint method, Lambert W function, rare event simulation, Monte Carlo method, Efficiency.

MSC: 60E05, 60E10, 90-04

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