On the Class of Erlang Mixtures with Risk Theoretic Applications

Abstract

A wide variety of distributions are shown to be of mixed-Erlang type. Useful computational formulas result for many quantities of interest in a risk-theoretic context when the claim size distribution is an Erlang mixture. In particular, the aggregate claims distribution and related quantities such as stop-loss moments are discussed, as well as ruin-theoretic quantities including infinitetime ruin probabilities and the distribution of the deficit at ruin. A very useful application of the results is the computation of finite-time ruin probabilities, with numerical examples given. Finally, extensions of the results to more general gamma mixtures are briefly examined.

An insurance risk model with stochastic volatility

“…(5.44) where q i ≥ 0 and ∞ i=1 q i = 1. We follow the results in Willmot and Woo (2007) to give expressions of Φ 0 (u) and Φ 1 (u). …”

“…For notational convenience, for t ∈ (0, 1], negative binomial type probability h m (i, t) is defined in Willmot and Woo (2007) as …”

“…When the claim size distribution is hyper-exponential, we extract all roots of the Lundberg fundamental equation and provide the explicit form of the EDP function approximation. When the claim distribution is mixed-Erlang, to avoid the computation of a matrix exponential, we employ the results in Willmot and Woo (2007) and express the approximated terms as mixed-Erlang type. Finally, in Section 6 we provide some concluding remarks.…”

Error bounds in approximations of random sums using gamma-type operators

“…16,130)). A recent work in which random sums of Erlang mixtures are considered in the context of risk theory is Willmot and Woo (2006).…”

Hypoexponential Distribution with Different Parameters

“…In particular the sum of exponential random has important applications in the modeling in many domains such as communications and computer science [3,4], Markov process [5,6], insurance [7,8] and reliability and performance evaluation [4,5,9,10]. Nadarajah [11], presented a review of some results on the sum of random variables.…”