Analysis of a Multivariate Claim Process

Abstract

The first part of this paper introduces a class of discrete multivariate phase-type (MPH) distributions. Recursive formulas are found for joint probabilities. Explicit expressions are obtained for means, variances and co-variances. The discrete MPH-distributions are used in the second part of the paper to study multivariate insurance claim processes in risk analysis, where claims may arrive in batches, the arrivals of different types of batches may be correlated and the amounts of different types of claims in a batch may be dependent. Under certain conditions, it is shown that the total amounts of claims accumulated in some random time horizon are discrete MPH random vectors. Matrix-representations of the discrete MPH-distributions are constructed explicitly. Efficient computational methods are developed for computing risk measures of the total claims of different types of claim batches and individual types of claims (e.g., joint distribution, mean, variance, correlation and value at risk.)

CMPH: a multivariate phase-type aggregate loss distribution

“…where U h,k (i), i ≥ , are independent copies of the random variable U h,k , which is the size of type k claim in the combination h. [16,17]. Certainly, the CMPH distribution bears resemblance with, but di ers from, the multivariate phase-type distribution studied by Assaf, Langberg, Savits and Shaked [1], and Kulkarni [20].…”

Calculating luminosity distance versus redshift in FLRW cosmology via homotopy perturbation method

“…Since the HPM has now become standard and for brevity, the reader is referred to [9]- [13] for the basic ideas of HPM. In this section, we shall apply the HPM to solve equation (11).…”

“…At the same time, Dr. Ji-Huan He [9] proposed an analytical method for solving differential and integral equations, HPM, which is a combination of standard homotopy and the perturbation. The HPM has a significant advantage in that it provides an analytical approximate solution to a wide range of nonlinear problems in applied sciences.…”

Numerical solution of fractionally damped beam by homotopy perturbation method

“…Proof of convergence of the above series may be found in [10,11]. The rapid convergence means that only a few terms are required to get the approximate solutions.…”

“…Recently, the homotopy perturbation method has been found to be a powerful tool for analysing this type of system involving fractional derivatives. The Homotopy Perturbation Method (HPM) was first developed by Ji-Huan He in 1999 [9][10][11][12][13] and many authors applied this method to solve various linear and non-linear functional equations of scientific and engineering problems. The solution is considered as the sum of infinite series, which converges rapidly to accurate solutions.…”