The Gerber-Shiu Discounted Penalty Function of Sparre Andersen Risk Model with a Constant Dividend Barrier

where u ≥ 0 represents the initial capital, c is the insurer’s rate of premium income per unit time, and {N(t), t ≥ 0} is the claim number process representing the number of claims up to time t. {X i , i ≥ 1} is a sequence of i.i.d. random variables representing the individual claim amounts with distribution function F(x) and density function f(x) with mean μ. We assume that {N(t), t ≥ 0} and {X i , i ≥ 1} are independent. Let {T i , i ≥ 1} be sequence i.i.d. random variables, which represent the claim interarrival times, and T i has a density functionK(t),


The Risk Model
Consider a Sparre Andersen risk model, where  ≥ 0 represents the initial capital,  is the insurer's rate of premium income per unit time, and {(),  ≥ 0} is the claim number process representing the number of claims up to time .{  ,  ≥ 1} is a sequence of i.i.d.random variables representing the individual claim amounts with distribution function () and density function () with mean .We assume that {(),  ≥ 0} and {  ,  ≥ 1} are independent.Let {  ,  ≥ 1} be sequence i.i.d.random variables, which represent the claim interarrival times, and   has a density function (), where  ≥ 1 is a positive integer,  ≥ 0,  1 ,  2 ≥ 0, and  1 +  2 = 1.We further assume that [  ] > [  ] for all , which ensure that lim  → ∞ () = ∞ almost surely.Throughout the paper we use the convention that ∑ 0 =1   = 0.In recent years the Sparre Andersen model has been studied extensively.Ruin probabilities and many ruin related quantities such as the marginal and joint defective distributions of the time to ruin, the deficit at ruin, the surplus prior to ruin, and the claim size causing ruin have received considerable attention.Some related results can be found in Cai and Dickson [1], Sun and Yang [2], Gerber and Shiu [3], and Ko [4].Li and Garrido [5] consider a compound renewal (Sparre Andersen) risk process in the presence of a constant dividend barrier in which the claim waiting times are generalized Erlang(n) distributed.The Sparre Andersen model with phase-type interclaim times has been studied by Ren [6].Ng and Yang [7] study the ruin probability and the distribution of the severity of ruin in risk models with phasetype claims.Landriault and Willmot [8] study the Gerber-Shiu function in a Sparre Andersen model with general interclaim times.Yang and Zhang [9] study a Sparre Andersen model in which the interclaim times are generalized Erlang(n) distributed.They assume that the premium rate is a step function depending on the current surplus level.Landriault and Sendova [10] generalize the Sparre Andersen dual risk model with Erlang(n) interinnovation times by adding a budget-restriction strategy.Shi and Landriault [11] utilize the multivariate version of Lagrange expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely, the combination of  exponentials.Yang and Sendova [12] study the Sparre Andersen dual risk model in which the times 2 Mathematical Problems in Engineering between positive gains are independently and identically distributed and have a generalized Erlang(n) distribution.
The barrier strategy was initially proposed by De Finetti [13] for a binomial model.From then on, barrier strategies have been studied in a number of papers and books, including Lin et al. [14], Dickson and Waters [15], Li and Lu [16], Yu [17][18][19], Yao et al. [20], Zhu [21], Tan et al. [22], and references therein for details.The purpose of this paper is to extend some results in Li and Garrido [5] and Yang and Zhang [9].We study the Sparre Andersen risk model with a constant dividend barrier and the claim interarrival distribution is a mixture of an exponential distribution and an Erlang(n) distribution.
The contents of this paper are organized as follows.Section 2 introduces the risk model.In Section 3, we derive the higher-order integro-differential equation for the Gerber-Shiu discounted penalty function.Finally, in the special case we provide the numerical example in Section 4.

The Risk Model
Let   () be the surplus process with initial surplus   (0) =  under the barrier strategy.Thus, it can be expressed as where () = ∑ () =1   .Define   = inf{ :   () < 0} to be the first time that the surplus becomes negative.The stopping time   is referred to as the time of ruin.Let   () = Pr(  < ∞) be the ruin probability.
In this paper, we will study the time of ruin   and its related functions such as the surplus before ruin   (  −) and the deficit at ruin |  (  )|.By using a renewal equation approach, we will be able to get a number of analytic and probabilistic properties of these quantities.Our analysis will involve the Gerber-Shiu discounted penalty function that is defined below.
Let (, ), 0 ≤ ,  < ∞, be a nonnegative function.For  ≥ 0, define where 4) is useful for deriving results in connection with joint and marginal distributions of   ,   (  −) and |  (  )|.While  may be interpreted as a force of interest, function (4) may also be viewed in terms of a Laplace transform with  serving as the argument.In particular, if we let (, ) = 1, (4) is the Laplace transform of the time of ruin   .If we let  = 0 and (, ) = 1, then   () becomes the ruin probability ().If we let  = 0 and (, ) = ( ≤  1 )( ≤  2 ), (4) becomes the joint df of the surplus before ruin and the deficit at ruin.Furthermore, if  = 0 and (, ) =   1 , we obtain the th moment of the surplus before ruin.Likewise, if  = 0 and (, ) =    2 , we obtain the th moment of the deficit at ruin.

An Integro-Differential Equation
In this section, we show   () satisfies a higher-order integro-differential equation.
Theorem 2. The Gerber-Shiu discounted penalty function   () satisfies the higher-order integro-differential equation Proof.Let  be the time of the first claim and let  be the amount of the claim.There are two possibilities.First,  < (−)/ and the surplus has not yet reached the barrier.In this case, the surplus immediately before time  is  + .Second,  ≥ ( − )/ and the surplus immediately before time  is .And since the "probability" that the claim occurs at time  is () and the "probability" of the claim amount being  is (), we have, for 0 ≤  ≤ , Using the substitution  =  + , we have which implies that where (, ) is defined in Lemma 1. Differentiating the above equation  times and using condition (6) yield Multiplying ( 12) by   (− − ) −1−   −1 for  = 0, 1, 2, . . .,  − 1, then adding up these equations, and using (5), we obtain Differentiating ( 13) again, we have which, together with ( 13), implies Moreover, note that So, it follows from ( 16) that and thus the result follows from ( 15) and ( 17).
we obtain the integro-differential equation for Erlang (2) risk model with no dividend barrier, which has been considered in Dickson and Hipp [25].
Theorem 8.The Laplace transform of   () is where Proof.It is easy to see that Taking the Laplace transform on both sides of (8), and together with (20), (21), and ( 22), we have which implies (8).

Numerical Illustration for Ruin Probability
In this section, we give the numerical illustration for   () when the claim number process has Erlang ,  − )  () ,
Science Foundation of China (no.11301303), Natural Science Foundation of Shandong Province (nos.ZR2012AQ013 and ZR2010GL013), Humanities and Social Sciences Project of the Ministry of Education of China (no.14YJA630088, no.13YJC630150, no.10YJC630092, and no.09YJC910004), and the Doctorate Scientific Research Foundation of Shandong Jiaotong University.