The Markovian regime-switching risk model with a threshold dividend strategy

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Abstract

In this paper, we study a regime-switching risk model with a threshold dividend strategy, in which the rate for the Poisson claim arrivals and the distribution of the claim amounts are driven by an underlying (external) Markov jump process. The purpose of this paper is to study the unified Gerber–Shiu discounted penalty function and the moments of the total dividend payments until ruin. We adopt an approach which is akin to the one used in [Lin, X.S., Pavlova, K.P., 2006. The compound Poisson risk model with a threshold dividend strategy. Insu.: Math. and Econ. 38, 57–80] to extend the results for the classical risk model with a threshold dividend strategy to our model. The matrix form of systems of integro-differential equations is presented and the analytical solutions to these systems are derived. Finally, numerical illustrations with exponential claim amounts are also given.

Introduction

In recent years, the risk models with dividend strategies are of particular interest to some researchers. The threshold strategy is the one under which the dividends are paid at a rate that is less than the premium rate when the surplus exceeds a constant level (threshold) and no dividends are paid otherwise. Some recent references in this area include Albrecher and Boxma (2005), Asmussen (2000), Gerber and Shiu (2006), Lin and Pavlova (2006), Zhu and Yang (2008), and references therein. Lin and Pavlova (2006) studied the Gerber–Shiu function and related problems for the classical compound Poisson model with such a dividend strategy. In this paper, we consider a Markovian regime-switching risk model with a fixed threshold dividend strategy, which is a natural extension of the classical risk model. In the model, we assume that both the frequency of the claim arrivals and the distribution of the claim amounts are influenced by an external environment process. This type of process is also known as the Markov-modulated risk process and was studied in Asmussen (1989) and Reinhard (1984) two decades ago. The primary motivation for this generalization is to enhance the flexibility of the model parameter settings for the classical risk process. The examples usually given are weather conditions and epidemic outbreaks, even though seasonality would play a role and can probably not be modeled by a Markovian regime-switching model. Zhu and Yang (2008) referred to states of the environment process as economic circumstances or political regime switchings. It is therefore theoretically appealing to include in the classical risk process assumptions in the variation in both claim frequencies and claim severities as a result of external environmental factors. The modeling framework that is advocated in this paper achieves this.

Zhu and Yang (2008) studied a more general Markovian regime-switching risk model in which the premium, the claim intensity, the claim amount, the dividend payment rate and the dividend threshold level were influenced by an external Markovian environment process. Some ruin related functions were investigated and closed-form solutions to systems of integro-differential equations were obtained when the underlying Markovian environment process has only two states and the claim amounts are exponentially distributed. However, in this paper we assume that only the Poisson claim arrival rates and the claim amount distributions vary in time depending on the states of the environment process. Under these particular settings, we adopt an approach which is akin to the one used in Lin and Pavlova (2006) to study the Gerber–Shiu discounted penalty function and the moments of the discounted dividend payments.

Lin and Sendova (2008) further considered a multi-threshold compound Poisson risk model. A piecewise integro-differential equation was derived for the Gerber–Shiu discounted penalty function, and an elegant recursive approach for obtaining general solutions to the equation was presented. By a similar approach, the results in this paper for the Markovian regime-switching risk model with a fixed threshold level can be mathematically extended to the one under a multi-threshold dividend strategy. Piecewise integro-differential equations in matrix form would be derived and corresponding solutions would be obtained recursively in terms of analytical matrix expressions. The risk model with multi-layer (multi-threshold) dividend strategy was also studied in Albrecher and Hartinger (2007), Zhou (2007), and Yang and Zhang (2008), while the risk model discussed in Lin and Pavlova (2006) can be considered as a two-layer model. See also Badescu et al. (2007) for an analysis of a threshold dividend strategy for a risk model with the Markovian arrival process, which includes the Markovian regime-switching risk model investigated in this paper as a special case. However a different approach is used.

Now denote by {J(t);t0} the external environment process, and suppose that it is a homogeneous, irreducible and recurrent Markov process with a finite state space E={1,2,,m} and intensity matrix Λ=(αi,j)i,j=1m, where αi,iαi for iE. Let N(t) be the number of claims occuring in (0,t]. If J(s)=i for all s in a small interval (t,t+h], then the number of claims occuring in that interval, N(t+h)N(t), is assumed to follow a Poisson distribution with parameter λi(>0), and the corresponding claim amounts have distribution Fi with density function fi and finite mean μi (iE). Moreover, we assume that premiums are received continuously at a positive constant rate c1. The corresponding surplus process {U(t);t0} is then given by U(t)=u+c1tn=1N(t)Xn,t0, where u0 is the initial surplus and Xn is the amount of the n-th claim.

In this paper we consider the surplus process (1.1) modified by the payment of dividends. Let d (0dc1) be the dividend rate. When the surplus exceeds the constant barrier b(u), dividends are paid continuously at rate d so that the net premium rate after dividend payments is c1d=c2. Let {Ub(t);t0} be the surplus process with initial surplus Ub(0)=u under the threshold dividend strategy above; then it is defined by Ub(t)=u+0tc[Ub(s)]dsn=1N(t)Xn,t0, where c(y) is c1 for 0y<b, and is c2 for y>b. Further, we assume that i=1mπi(c2λiμi)>0 so that the loading is positive and the ruin is not certain, where π=(π1,,πm) is the stationary distribution of {J(t);t0}.

Define τb=inf{t0:Ub(t)<0} to be the time of ruin and let w(x,y), for x,y0, be a non-negative penalty function. For notational convenience, let Pi()=P(|J(0)=i). Let δ0 be the force of interest for valuation. For iE, define Ei[eδτbw(Ub(τb),|Ub(τb)|)I(τb<)|Ub(0)=u]={ϕi1(u;b),0u<bϕi2(u;b),b<u<, to be the expected discounted penalty function at ruin, given the initial surplus u and the initial environment iE, for the surplus Ub(τb) before ruin and the deficit |Ub(τb)| at ruin, where I() is the indicator function. This so-called Gerber–Shiu function was introduced originally in their influential paper by Gerber and Shiu (1998). In particular, when δ=0 and w(x,y)=1, (1.3) simplifies to Ψi(u;b), the conditional ruin probability Ψi(u;b)=Pi{τb<|Ub(0)=u}={Ψi1(u;b),0u<bΨi2(u;b),b<u<,iE.

The rest of the paper is organized as follows. In Section 2 we review the main results for the Gerber–Shiu discounted penalty function at ruin for a Markovian regime-switching risk model without dividends involved. Systems of integro-differential equations in matrix form for the discounted penalty functions under the threshold dividend strategy are presented in Section 3. Then, in Section 4, the analytical formulas for the discounted penalty functions when the initial surplus is below and above the dividend threshold b are derived, respectively. A constant vector which is crucial for completing the results in Section 4 is determined in Section 5. The moment of the dividend payments for the model is considered in Section 6. Matrix forms of the integro-differential equations are derived and their analytical solutions are presented in Theorem 2. Finally, numerical examples for a two-state model are illustrated in Section 7 for the ruin probability, and the expected total dividend payments until ruin when claim amounts are exponentially distributed.

Section snippets

Preliminaries

In this section, we first review some results for surplus process (1.1) where no dividends are involved, i.e., b=. As we will see in the next section, the discounted penalty functions ϕi1(u;b) and ϕi2(u;b) defined in (1.3) under the threshold strategy are associated with the discounted penalty function for the process without such a strategy. Define τ=inf{t0:U(t)<0} to be the time of ruin, and for δ0ϕi(u)=Ei[eδτw(U(τ),|U(τ)|)I(τ<)U(0)=u],u0,iE, to be the expected discounted penalty

Systems of integro-differential equations

In this section, we derive integro-differential equations for the discounted penalty functions defined by (1.3) when the initial surplus below or above the barrier b. By similar arguments used in Zhu and Yang (2008) (see Theorem 4.1), we have for 0u<b, (λi+δ)ϕi1(u;b)=c1ϕi1(u;b)+k=1mαi,kϕk1(u;b)+λi[0uϕi1(ux;b)dFi(x)+uw(u,xu)dFi(x)],iE, and for b<u<, (λi+δ)ϕi2(u;b)=c2ϕi2(u;b)+k=1mαi,kϕk2(u;b)+λi0ubϕi2(ux;b)dFi(x)+λi[ubuϕi1(ux;b)dFi(x)+uw(u,xu)dFi(x)],iE. Integro-differential

Analytical expressions for ϕ1(u;b) and ϕ2(u;b)

In this section, using the result in Lemma 1, we derive firstly an analytical expression for the discounted penalty function ϕ1(u;b), then obtain an equation which shows the relationship between discounted penalty functions ϕ(u;b) and ϕ1(u;b). The latter equation allows us to determine a constant vector and then derive the analytical expression for ϕ2(u;b).

Obviously, vector function ϕ1(u;b) (0u<b) in (3.3) satisfies a non-homogeneous integro-differential equation. By the similar arguments

Determine the constant vector κ(b)

For simplicity, we write ϕi(u;b)=ϕi(u) for i=1,2 in this section. We adopt an approach which is similar to the one used in Lin and Pavlova (2006) for the compound Poisson risk model under the threshold dividend strategy. By multiplying both sides of Eq. (3.2) by es(ub) and integrating with respect to u from b to , and after some manipulations, we can write Eq. (3.2) as [sIPc2Gˆc2(s)]Tsϕ2(b)=ϕ2(b)+0bTsGc2(bt)ϕ1(t)dt+Tsζ2(b). Let Ac2(s)=sIPc2Gˆc2(s). Similarly, we can show that the

Moments for the dividend payments

For modified surplus process (1.2) with Ub(0)=u and δ>0 define Du,b=0τbeδtdD(t)=d0τbeδt1(Ub(t)>b)dD(t) to be the total discounted dividends until time of ruin τb, where D(t) is the aggregate dividends paid by time t and d=c1c2. Define the moment-generating function of Du,b, given that the initial environment state is i, by Mi(u,y;b)=Ei[eyDu,b]={Mi1(u,y;b),0u<bMi2(u,y;b),b<u<,iE, where y is such that Mi(u,y;b) exists. Further for 0u< define Vi[n](u;b)=Ei[Du,bn]={Vi1[n](u;b),0u<bVi2[n](

Numerical illustrations for a two-state model

In this section, we illustrate some results numerically. Consider a two-state regime-switching risk model under a threshold dividend strategy, that is, {J(t);t0} is a two-state Markov process, which reflects the random environmental effects due to, probably, normal risk or abnormal risk conditions.

In the case where the claim amount distributions f1 and f2 are exponentially distributed with Laplace transformations fiˆ(s)=βi/(s+βi), βi>0 and i=1,2, Li and Lu (2007) obtained the explicit

Acknowledgments

The authors would like to thank an anonymous referee for the valuable comments and suggestions to improve the paper presentation. This research for Dr. Yi Lu was supported by the Natural Science and Engineering Research Council of Canada.

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