On Laplace transforms, generalized gamma convolutions, and their applications in risk aggregation

This dissertation begins with two introductory chapters to provide some relevant background information: an introduction on the Laplace transform and an introduction on Generalized Gamma Convolutions (GGCs). The heart of this dissertation is the final three chapters comprised of three contributions to the literature.

In Chapter 3, we study the analytical properties of the Laplace transform of the log-normal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the log-normal distribution 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; we show that the integral expression derived by Leipnik in \cite{Leipnik1991} is incorrect. We present two approximations for the Laplace transform of the log-normal distribution, both valid in $\backslash C\setminus (-\infty ,0]$. In the last section, we discuss how one may use our results to compute the density of a sum of independent log-normal random variables.

In Chapter 4, we explore the topic of risk aggregation with moment matching \approximations. We put forward a refined moment matching approximation (MMA) method for approximating the distributions of the sums of insurance risks. Our method approximates the distributions of interest to any desired precision, works equally well for light and heavy-tailed distributions, and is reasonably fast irrespective of the number of the involved summands.

In Chapter 5, we study the convergence of the Gaver-Stehfest algorithm. The Gaver-Stehfest algorithm is widely used for numerical inversion of Laplace transform. In this chapter we provide the first rigorous study of the rate of convergence of the Gaver-Stehfest algorithm. We prove that the Gaver-Stehfest approximations of order $n$ converge exponentially fast if the target function is analytic in a neighbourhood of a point and they converge at a rate $o(n-k)$ if the target function is $(2k+3)$-times differentiable at a point.