Abstract
We consider the dual of the generalized Erlang risk model with a barrier dividend strategy. We derive integro-differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin and the moment-generating function of the discounted dividend payments until ruin, respectively. The results are illustrated by several examples.
1. Introduction
Many interesting results have been obtained on a model that is dual to the classical insurance risk model in recent years. See, for example, Albrecher et al. [1], Ng [2], Avanzi et al. [3], and Avanzi and Gerber [4]. In the classical dual model (see Grandell [5]), the surplus at time is where and are constants, is the initial surplus and is the rate of expenses, is the aggregate positive gains process, is a sequence of independent and identically distributed nonnegative random variables with a common probability distribution function , and is a Poisson process with rate . Moreover, it is assumed that and are independent. In (1.1), the expected increment of the surplus per unit time is It is assumed that .
In the model (1.1), the premium rate is negative, causing the surplus to decrease. Claims, on the other hand, cause the surplus to jump up. Thus the premium rate should be viewed as an expense rate and claims should be viewed as profits or gains. While not very popular in insurance mathematics, this model has appeared in various literature (see Cramér [6], Seal [7], Tákacs [8], and the references cited therein). In Avanzi et al. [3], the authors studied the expected total discounted dividends until ruin for the dual model under the barrier strategy by means of integro-differential equations.
Recently, the research to models with two-sided jumps has been attracting a lot of attention in applied probability. For example, Perry et al. [9] studied the one- and two-sided first exit problems for a compound Poisson process with negative and positive jumps and linear deterministic decrease between jumps and assumed that the jumps have hyperexponential distributions. Kou and Wang [10] used a double exponential jump diffusion process to model the asset return. Asmussen et al. [11] considered the stock price models as an exponential Lévy process with phase-type jumps in two directions. For some related work see, among others, Jacobsen [12], Dong and Wang [13], Dong and Wang [14], Cai et al. [15], Zhang et al. [16], Chi and Lin [17], Cai and Kou [18], and the references therein.
Motivated by some related work mentioned above, we consider a more general risk process. We will assume that the number of gains up to time is an ordinary renewal process: where the random variables are independent and identically generalized Erlang()-distributed, that is, the ’s are distributed as the sum of independent and exponentially distributed random variables: where may have different exponential parameters . We also assumed that the jumps are two-sided. The upward jumps can be interpreted as the random gains of the company, while the downward jumps are interpreted as the random loss of the company. The common density of the jumps is given by where and are two arbitrary probability density functions on and are two constants such that . Denoted are the probability distribution functions of and , respectively, by and .
We then consider the modification of the surplus process by a barrier strategy with a barrier . Whenever the surplus exceeds the barrier, the excess is paid out immediately as a dividend. But when is below , no dividends are paid. The modified surplus at time is given by where denote the aggregate dividends paid between time and time , that is, Let be the time of ruin for the modified surplus , and let be the sum of the discounted dividend payments, where is the force of interest for valuation.
In this paper, we consider the expectation and the moment-generating function of the sum of the discounted dividends until ruin. In Section 2, we derive an integro-differential equation with boundary conditions for the expectation of the discounted dividends until ruin. In Section 3, we obtain an integro-differential equation with boundary conditions for the moment-generating function of the discounted dividend payments until ruin.
2. Expectation of the Discounted Dividends
Denote by the expectation of the discounted dividends until ruin if the barrier strategy with parameter is applied: Note that
Let denote the differentiation operator with respect to . And we define .
Theorem 2.1. The function satisfies the following integro-differential equation: with boundary conditions
Proof. We let denote the expectation of the discounted dividends if the risk process is in state . Eventually, we are interested in . Conditioning on the occurrence of a (sub-) claim within an infinitesimal time interval, we obtain for and ,
Note that we have
Substituting these formulas into (2.6), after some careful calculations, we have for
For , we have
which leads to
It follows from (2.8) that
and subsequently
which together with (2.10) yields (2.3).
Since the ruin is immediate if , we have the boundary condition (2.4).
For , we obtain analogously for that
which by Taylor expansion leads to
Comparing these equations with the corresponding ones in (2.8), the continuity of at then implies that
Similarly, one can verify that (2.15) also holds for . For , (2.15) is equivalent to (2.5) with . It follows from (2.11) that
Applying the operator to both sides of the above equation, we get
which is the boundary condition (2.5).
Remark 2.2. When the gains waiting time have an exponential distribution with for , , , , and , we can get the integro-differential equation of : This is the result (2.3) in Avanzi et al. [3].
Example 2.3. For , , let , , where and are two positive constants. Then From (2.3), we have Applying the operator to both sides of (2.20), we get The characteristic equation of (2.21) is That is, The expression on the left-hand side is a linear function of , while the expression on the right-hand side is a rational function with poles at . By a graphical arguments, it can be verified that the characteristic equation above has exactly three real roots satisfying Hence, we set where , , and are constants need to be determined. It follows from (2.4) and (2.5) that we have Substituting (2.25) into (2.20), and since this equation must be satisfied for , we have which can be rewritten as where Solving system (2.26) and (2.28) gives , where
Example 2.4. For , and , , , we have
Applying the operator to both sides of (2.31), we get
from which we get the characteristic equation
By a graphical arguments, it can be verified that the characteristic equation above has exactly three real roots satisfying
Hence we get
where , , and are constants. It follows from (2.4) and (2.5) that
Substituting (2.35) into (2.31), and because this equation must be satisfied for all , the sum of the coefficients of must be zero. Therefore,
It follows from (2.36)–(2.38) that
where
3. Moment-Generating Function of the Discounted Dividends
We denote moment-generating function of by Let denote the differentiation operator with respect to and correspondingly the differentiation operator with respect to .
Theorem 3.1. The moment-generating function satisfies the following integro-differential equation: with boundary conditions
Proof. As in Albrecher et al. [19], let denote the moment-generating function of if the risk process is in state . Eventually, we are interested in . Conditioning on the occurrence of a (sub-)claim within an infinitesimal time interval, we obtain for and ,
It follows from (3.4) that
For , we have
It follows from (3.5) that we have
which together with (3.6) yields (3.2).
For , we obtain analogously for
which leads to
Comparing these equations with the corresponding equations in (3.5), the continuity of at implies
Similarly, we can show that (3.10) holds true for . For , (3.10) is equivalent to (3.3) for . Now it just remains to express equations (3.10) for in terms of , which is done by virtue of (3.9).
For , we denote the th moment of by
Theorem 3.2 :. The th moment satisfies the following integro-differential equation with boundary conditions
Proof. Since the result follows if we equate the coefficients of in (3.2) and (3.3).
Remark 3.3. We remark that when , , and , we reobtained the result of Theorem 2.1; when , , and , (3.12) reduces to (2.3) of Cheung and Drekic [20]; when , , (3.2) and (3.3) reduce to (2) and (3) of Albrecher et al. [19], and (3.12) and (3.13) reduce to (9) and (10) of Albrecher et al. [19].
Acknowledgments
The authors would like to thank the anonymous referees for their constructive and insightful suggestions and comments on the previous version of this paper. This paper is supported by the National Natural Science Foundation of China (no. 11171179).