The Laplace transform of the lognormal distribution

By Justin Miles. 2018

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In this paper, we explore the analytic continuation of the Laplace transform of the lognormal distribution. Two integral expressions of the analytic continuation are provided, one of which takes the form of a Mellin-Barnes integral. As a corollary, we obtain an integral expression for the characteristic function and we claim that the integral expression derived by Leipnik in [11] is incorrect. Two approximations for the Laplace transform of the lognormal distribution are presented; both of which are valid for arguments in $\mathbb{C}\setminus(-\infty,0]$. The first is a convergent series that is asymptotic to the Laplace transform of the lognormal distribution as the magnitude of the argument decreases to zero. Furthermore, the error of the appoximation is uniformly bounded on $\mathbb{C}\setminus(-\infty,0]$ by a constant that can be made arbitrarily small. The second approximation is a sum which is asymptotic to the Laplace transform of the lognormal distribution as a parameter ($\sigma$) increases to infinity. These approximations are used to compute the Laplace transform for several real arguments. In the last section, we discuss how one may use the analytic continuation to compute the density of a sum of independent lognormals, via the inverse Laplace transform. By deforming the contour of the Bromwich integral to a Hankel contour, one may obtain a real integral with an integrand which decays exponentially. The result is an integral which is easily evaluated numerically.

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