Transform approach and its computer based numerical application in compound distributions

Purpose/Originality: To provide a basic concept of using transform approach to calculate the compound probability density function numerically. Key literature / theoretical perspective: Using the moment generating function of the compound distribution to get the Fourier transform of its probability density function (PDF), then applying IFFT to get the numerical data of PDF. Design/methodology/approach: We illustrate the relationships between Fourier transform, Laplace transform, characteristic function and moment generating function. Then we introduce discrete Fourier transform (DFT) and fast Fourier transform (FFT) technique, and explain periodic extension and sampling theory. The steps of computer-based transforming are listed. With the result of MATLAB computation, a numerical example of compound passion distribution is shown in the end. Findings: A numerical approach to calculate the probability density function of compound distribution. Research limitations/implications: The approaches used in this topic is based on that the moment generating function of the compound distribution is known. Practical and Social implications: It can be applied to estimate the distribution of aggregate loss in general insurance industry. It can also be a reference for calculating the distribution of aggregate sums in other fields.