Loss Data Analysis

6Laplace transforms and fractional moment problems

This chapter is a detour from the track were were on, but as it is an essential part of the methodology that we are to use to deal with the problem of loss data analysis, we decided to gather the material under one heading. Since the Laplace transform technique is rather important we shall try to weave a line between generality and the particular applications that we have in mind. To make it generic, let us denote by X a positive random variable, and throughout this chapter we suppose that it is continuous and has a density fX(x), with respect to the usual Lebesgue measure on [0, ∞) unless otherwise specified. Its Laplace transform is defined by

$\varphi \left(\alpha \right)=\text{E}\left[e^{-\alpha X}\right]={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\alpha x}{f}_X\left(x\right)dx.}$