Some ruin problems for the MAP risk model

https://doi.org/10.1016/j.insmatheco.2015.08.001Get rights and content

Highlights

  • We present a variety of new results for the MAP risk model.

  • We use classical arguments and Laplace transform inversion to obtain two different expressions for the density of the time of ruin.

  • We find the probability function of the number of claims until ruin.

  • We find a recursive formula from which moments of the time of ruin can be calculated.

Abstract

We consider ruin problems for a risk model with a Markovian arrival process (MAP). In particular, we study (1) the density of the time of ruin under two different assumptions on the premium income, by using two approaches; (2) the probability function of the number of claims until the time of ruin; (3) the moments of the time of ruin by developing a recursive approach.

Introduction

In this paper we consider a general class of risk models with claims arriving according to a Markovian arrival process (MAP) (see  Neuts, 1979). We denote by U(t) the insurer’s surplus at time t, which is given by U(t)=u+0tcJ(s)dsS(t), where u is the initial surplus and S(t)=i=1N(t)Xi is the aggregate claim amount at time t, with N(t) being the number of claims by time t and Xi being the amount of the ith claim. We assume that the evolution of the insurer’s surplus is driven by an underlying irreducible continuous-time Markov chain {J(t);t0} on the state space E={1,,m}, such that the premiums are collected linearly at rate ci>0 while J(t)=i,iE. The intensity matrix for {J(t);t0} is D=(dij)m×m=D0+D1 with D1=0, where 1=(1,1,,1),D0=(d0,ij)m×m represents the intensity of state changes without claim arrivals and D1=(d1,ij)m×m represents the intensity of state changes with an accompanying claim, with d1,ij0 for all i and j. Furthermore, claims arriving with a transition from state i to state j are assumed to be independent of all other claims, to have distribution function Fi,j with density fi,j, mean μi,j and Laplace transform f̃i,j(s)=0esxfi,j(x)dx. For this model, the positive loading condition is i=1mπi(cij=1md1,ijμi,j)>0, where π=(π1,π2,,πm) is the stationary distribution of {J(t);t0} with πD=0. This condition is assumed to hold throughout.

Now define Tu=inf{t0:U(t)<0} to be the time of ruin, with Tu= if U(t)0 for all t0. Letting I denote the indicator function, we define for δ0,u0 and i,jEϕi,j(u)=E[eδTuI(Tu<,J(Tu)=j)|J(0)=i] to be the Laplace transform of the time of ruin with ruin caused by a claim in environment state j, given initial surplus u and initial environment state iE. We remark that it follows from Feng and Shimizu (2014) that ϕi,j(u) can be expressed in terms of the δ-scale function defined in Ivanovs and Palmowski (2012). Further, ϕi(u)=j=1mϕi,j(u) is the Laplace transform of the time of ruin, given initial surplus u and initial environment state iE. In particular, when δ=0,ϕi,j(u) is denoted ψi,j(u) which is defined as ψi,j(u)=Pr(Tu<,J(Tu)=j|J(0)=i),i,jE. So ψi,j(u) is the ultimate ruin probability with ruin caused by a claim in environment state j, given that the initial surplus is u and the initial environment state is i. Hence ψi(u)=j=1mψi,j(u) is the probability of ultimate ruin given that the initial surplus is u and the initial environment state is i, and χi(u)=1ψi(u) is the non-ruin probability given that the initial surplus is u and the initial environment state is i.

An introduction to the Markovian arrival process can be found in Neuts (1979) and Breuer (2002). Studies on ruin-related quantities and dividend problems for the perturbed and non-perturbed MAP risk model can be found in Badescu et al., 2005, Badescu et al., 2007, Badescu (2008), Cheung and Landriault (2009), Cheung and Feng (2013), Li and Ren (2013), Feng and Shimizu (2014) and references therein. Ren (2008) studies the Laplace transform of the aggregate claims over (0,t] for the MAP risk model.

The MAP risk model is very broad and it contains the classical risk model with m=1, the phase-type renewal risk model (Ren, 2007), the semi-Markovian risk model studied by Albrecher and Boxma (2005), as well as the Markov-modulated risk model proposed by Asmussen (1989) and further studied by a number of authors including Asmussen and Rolski (1994), Ng and Yang (2006), Li and Lu (2008), and references therein. In particular, the MAP risk model simplifies to the Markov-modulated risk model if Fi,j(x)=Fi(x)I(i=j), (so that fi,j(x)=fi(x)I(i=j)), D0+D1=A and D1=Λ=diag(λ1,λ2,,λm) with A=(αi,j)m×m being the intensity matrix for the Markov process {J(t);t0}, with αi,iαi.

This paper is set out as follows. In the next section we give some notation and basic results. We then consider the distribution of the time of ruin by using two approaches in Sections  3 The distribution of the time of ruin under a constant premium rate, 4 The density of the time of ruin with varying premium rates. The probability function of the number of claims by the time of ruin is given in Section  5, and moments of the time of ruin are considered in Section  6.

Section snippets

Notation and basic relationships

In this section we introduce some notation and state some basic results. We start with claim numbers, and define qi,j(n)(t)=Pr(N(t)=n,J(t)=j|J(0)=i),i,jE,nN to be the probability that n claims occur before time t, with the initial state being i and the state at time t being j. Then qi(n)(t)=Pr(N(t)=n|J(0)=i)=j=1mqi,j(n)(t) is the probability that n claims occur up to time t with the initial state being i.

We next consider aggregate claims distributions. Define Gi,j(x,t)=Pr(S(t)x,J(t)=j|J(0)=i

The distribution of the time of ruin under a constant premium rate

Define χi,j(u,t)=Pr(Tu>t,J(t)=j|J(0)=i),i,jE, to be the probability that ruin does not occur by time t and that the environment state at time t is j, given that the initial environment state is i. Then χi(u,t)=j=1mχi,j(u,t)=Pr(Tu>t|J(0)=i) is the non-ruin probability by time t given that the initial environment state is i. Let wi,j(u,t)=χi,j(u,t)/t denote the defective density of the time of ruin with ruin caused by a claim in environment state j given that the initial environment state is

The density of the time of ruin with varying premium rates

In this section we derive an alternative expression for the density of the time of ruin using transform inversion, and we let the premium rate vary according to the environment state. We recall that ϕi(u)=E[eδTuI(Tu<)|J(0)=i] is the Laplace transform of the time of ruin from initial surplus u given that the initial environment state is i.

Conditioning on the events that can occur over an infinitesimal interval, we can show that cidduϕi(u)=δϕi(u)k=1md0,ikϕk(u)k=1m0ud1,ikfi,k(x)ϕk(ux)dxk=

The distribution of the number of claims by the time of ruin

We now consider the distribution of the number of claims until ruin. This problem has been studied by a number of authors including Egídio dos Reis (2002) and Dickson (2012) for the classical risk model, and Stanford et al. (2000) who consider certain renewal risk models. The distribution of the number of claims until ruin can be used as a proxy to the distribution of Tu, although caution should be exercised, as discussed in Stanford et al. (2000). Landriault et al. (2011) show that in certain

The moments of the time of ruin

We now consider moments of the time of ruin. Define ψi,j(n)(u)=E[TunI(Tu<,J(Tu)=j)|J(0)=i],nN, to be the nth moment of the time of ruin with ruin caused by a claim in environment state j, given that the initial surplus is u and the initial environment state is i. Further, ψi(n)(u)=j=1mψi,j(n)(u) is the nth moment of the time of ruin, given that the initial surplus is u and the initial environment state is i. We note that ψi,j(0)(u)=ψi,j(u)=1limtχi,j(u,t). Further, let ϕ(u)=(ϕi,j(u))m×m

Acknowledgements

This paper has benefited from comments from referees, and from colleagues at the University of Melbourne.

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