On a correlated aggregate claims model with Poisson and Erlang risk processes

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Abstract

In this paper we consider a risk model with two dependent classes of insurance business. In this model the two claim number processes are correlated. Claim occurrences of both classes relate to Poisson and Erlang processes. We derive explicit expressions for the ultimate survival probabilities under the assumed model when the claim sizes are exponentially distributed. We also examine the asymptotic property of the ruin probability for this special risk process with general claim size distributions.

Introduction

In this paper we consider a risk model involving two dependent classes of insurance business. Let Xi be claim size random variables for the first class with common distribution function FX and Yi be those for the second class with common distribution function FY. Their means are denoted by μX and μY. Then the aggregate claim size process generated from the two classes of business is given by U(t)=∑i=1N1(t)Xi+∑i=1N2(t)Yi,where Ni(t) is the claim number process for class i (i=1,2). It is assumed that {Xi,i=1,2,…} and {Yi,i=1,2,…} are independent claim size random variables, and that they are independent of N1(t) and N2(t). The two claim number processes are correlated in the way that N1(t)=K1(t)+K̃(t)andN2(t)=K2(t)+K̃(t),with K1(t), K2(t), and K̃(t) being three independent renewal processes. As usual we define the surplus process S(t)=u+ct−U(t),where u is the amount of initial surplus and c is the rate of premium. The ultimate survival probability is φ(u)=Pr(S(t)≥0forallt≥0),and the ultimate ruin probability is ψ(u)=1−φ(u).

It is obvious that the correlation in (1.3) comes from the incorporation of the common component K̃(t) into the two claim number processes. This kind of correlated aggregate claims model has been studied extensively in recent years. For instance, Ambagaspitiya (1998) considered a general method of constructing a vector of p (p≥2) dependent claim numbers from a vector of independent random variables, and derived formulas to compute the aggregate claims distribution for the book of p dependent classes of business; and Cossette and Marceau (2000) used a discrete-time approach to study how such a dependence relation affects the finite-time ruin probabilities and the adjustment coefficient.

Erlang distribution is one of the most commonly used distributions in queueing theory which is closely related to risk theory; see for example, Asmussen, 1987, Asmussen, 1989 and Takács (1962). Dickson (1998) showed how methods that have been applied to derive results for the compound Poisson model can be adapted to derive results for a class of risk processes in which claims occur as an Erlang process. For this kind of Erlang risk process, Dickson and Hipp (1998) considered the ultimate survival probability as a compound geometric random variable and obtained both analytical and numerical solutions for ruin probabilities. Motivated by these papers, we explore the possibility of further applications of Erlang processes to actuarial problems. Specifically we study the ultimate survival (ruin) probability for model (1.3) with K1(t) and K2(t) being Poisson processes and K̃(t) being an Erlang process. The inclusion of an Erlang process in the compound Poisson model makes the problem more interesting. It also complicates the derivation of the probability of ruin.

In Section 2 we consider a transformed surplus process with two independent classes of business for which one claim number process is Poisson while the other is Erlang. The transformed surplus process can be used to obtain the ruin probability for S(t). Section 3 is devoted to studying the ultimate survival probability when the original claim size random variables are exponential. For general claim size distributions, an asymptotic result for the probability of ruin is presented in Section 4.

Section snippets

Model transformation

Let the parameters of the two Poisson processes, K1(t) and K2(t), be λ1 and λ2, respectively. Assume that K̃(t) is an Erlang(2) process with parameter λ̃. That is, the claim interarrival times for K̃(t) follow Erlang(2,λ̃) with density function f(t)=λ̃2texp{−λ̃t} for t>0. For investigating the probability of ruin for S(t), we make use of the following transformed surplus process: S′(t)=u+ct−∑i=1K12(t)Xi′−∑i=1K̃(t)Yi′,where {Xi′,i=1,2,…} and {Yi′,i=1,2,…} are independent claim size random

Ruin probability for exponential claims

Let V1,V2,… be the interarrival times for Xi′. Note that V1 is the occurrence time of X1′. These times are independent and exponentially distributed with mean (λ1+λ2)−1. For Yi′ the interarrival times form a sequence of independent and identically distributed random variables {L1,L2,…} following Erlang(2, λ̃) distribution. Equivalently we write L1=L11+L12,L2=L21+L22,…, where L11,L12,L21,L22,… are independent exponential random variables with mean λ̃−1. Since (λ1+λ2)μX and 0.5 λ̃μY′ are the

Asymptotic result for general claim sizes

For claim size distributions other than exponential, differential equations for φ1 are not available in general. In this section we are interested in studying the asymptotic behaviour (as u→∞) of the ruin probabilities when dealing with general claim size distributions.

Integrating (3.3) both sides from 0 to u yields φ(u)=φ(0)−λ̃c0uφ1(s)ds+λc0uφ(s)ds+λ12c0u0sφ(s−x)d(1−FX′(x))ds.By interchanging integral signs and performing integration by parts, we get 0u0sφ(s−x)d(1−FX′(x))ds=∫0u

Acknowledgements

This research was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. HKU 7202/99H).

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