The distribution of total dividend payments in a Sparre Andersen model

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Abstract

We study the distribution of the total dividend payments in a Sparre Andersen model with phase-type inter-claim times in the presence of a constant dividend barrier. This paper shows that the distribution of the total dividend payments prior to the time of ruin is a mixture of a degenerate distribution at zero and a phase-type distribution. Further, the total dividends prior to ruin can be expressed as a compound geometric sum.

Introduction

Consider a continuous-time Sparre Andersen model in which the surplus process is given by U(t)=u+cti=1N(t)Xi,t0, where u0 is the initial surplus, c0 is the premium income rate, while the random sum represents aggregate claims. The Xi’s are i.i.d. random variables with common distribution function (d.f.) P(x)=P(Xx) (P(0)=0) and density function p(x)=P(x). Denote the Laplace transform (LT) of p by pˆ(s)=0esxp(x)dx. The renewal process {N(t)}t0 denotes the number of claims up to time t and is defined as follows.

For k=1,2,, let Vk denote the time when the kth claim occurs. Let W1=V1 and Wi=ViVi1 for i=2,3,. We assume that W1,W2, are independent and the inter-claim times W2,W3, have a common distribution function A(x)=P(Wx) and density function a(x)=A(x). Denote by aˆ(s)=0esxa(x)dx the LT of a. Then N(t)=max{nN+:W1+W2++Wnt}.

Further assume that {Wi}i1 and {Xi}i1 are independent and that cE(W)>E(X), providing a positive safety loading factor.

In this paper, we assume that the distribution of the inter-claim times A is phase-type with representation (α,Q), where Q=(qi,j)i,j=1m is an m×m matrix with qi,i<0, qi,j0 for ij, and j=1mqi,j0 for any i=1,2,,m,α=(α1,α2,,αm) with i=1mαi=1. Then A(t)=1αetQe,t0,a(t)=αetQq,t0,aˆ(s)=α(sIQ)1q, where e is a row vector of length m with each element being 1, I is the m×m identity matrix, and q=Qe.

It follows that W corresponds to the time to absorption in a terminating continuous-time Markov chain {J(t)}t0 with m+1 states, one of which is absorbing. The state space of {J(t)}t0 is {1,2,3,,m,0}=E{0} and the initial distribution is (α1,α2,,αm,0). The generator of {J(t)}t0 is (Qq00). A detailed introduction to phase-type distributions and their properties can be found in Neuts (1981) and Asmussen (1992), and the references therein.

Note that W1 may not follow the same distribution as the inter-claim times. In the case of an ordinary renewal risk process, W1 follows the distribution A, that is, a claim has taken place at time 0 and u is the surplus after the claim is paid. For iE, if J(0)=i, then W1 follows a distribution with density function eietQq, where ei is a 1×m row vector with the ith element being 1 and all other elements being 0. In the rest of the paper, we employ P to denote the probability measure when the surplus process is an ordinary renewal risk process and employ Pi to denote the probability measure given J(0)=i, i.e., the density function of W1 is eietQq. E and Ei represent the expectation operators under P and Pi, respectively.

Let T denote the time of ruin, T=inf{t0;U(t)<0} (T= if ruin does not occur). Define Ψ(u)=P(T<|U(0)=u),u0, as the infinite-horizon ruin probability in the corresponding ordinary renewal risk model. Further, define Ψi(u)=Pi(T<|U(0)=u),iE,u0, to be the ruin probability given that the initial state is i. Then Ψ(u)=αΨ(u), where Ψ(u)=(Ψ1(u),Ψ1(u),,Ψm(u)). In Section 2, each Ψi(u) will be further decomposed into m components according to the state at the time of recovery after ruin.

The Sparre Andersen model with Erlang inter-claim times has been studied by Li and Garrido (2004) and Gerber and Shiu (2005), and the references therein. The Sparre Andersen model with phase-type inter-claim times has been studied by Avram and Usábel (2004), Schmidli (2005), Albrecher and Boxma (2005), Ren (2007) and Li, 2008a, Li, 2008b. Li and Garrido (2005) study the discounted penalty (Gerber–Shiu) function in a general renewal process with Kn-family distribution as the inter-claim distribution, which includes the phase-type distribution as a special case, and Albrecher and Boxma (2005) analyze the Gerber–Shiu function through the Laplace–Stieltjes transforms for a semi-Markovian risk model which also includes the Sparre Andersen model with phase-type inter-claim times as a special case.

The barrier strategy was initially proposed by De Finetti (1957) for a binomial model. More general barrier strategies have been studied in a number of papers and books. See Lin et al. (2003), Dickson and Waters (2004), Li and Lu (2007) and Lu and Li (2009), and the references therein for details. The main focus of the mentioned papers is on optimal dividend payouts and problems associated with time of ruin, under various barrier strategies and other economic conditions. Cheung (2007) studies the moments of the discounted dividend payments prior to ruin in a Sparre Andersen model with phase-type claim inter-arrival times in the presence of a constant dividend barrier. The purposes of this note is to extend some results in Dickson and Waters (2004) and Li and Dickson (2006) and to find the distribution of total dividend payments prior to ruin.

Section snippets

The decomposition of ruin probabilities

In this paper, we allow the surplus process to continue if ruin occurs and define T=inf{t:t>T,U(t)=0} to be the time of the first upcrossing of the surplus process through level zero after ruin. This time is also called the time of recovery in the literature, see Gerber and Shiu (1998). Since we assume that the loading factor is positive, it is certain that the surplus process will attain this level. Define Ψi,j(u)=Pi(T<,J(T)=j|U(0)=u),i,jE,u0, to be the probability of ruin with the state

Main results

Now we consider the surplus process (1) modified by the payment of dividends. When the surplus exceeds a constant barrier b(u), dividends are paid continuously so the surplus stays at the level b until a new claim occurs. Let {Ub(t)}t0 be the surplus process with initial surplus Ub(0)=u under the above barrier strategy, i.e., Ub(t)=u+cti=1N(t)XiD(t), where D(t) is the aggregate dividends paid by time t.

Define T̄=inf{t0:Ub(t)<0} to be the time of ruin. Then D(T̄) is the total dividend

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