Dividend Problems with a Barrier Strategy in the Dual Risk Model until Bankruptcy
The paper studies the dual risk model with a barrier strategy under the concept of bankruptcy, in which one has a positive probability to continue business despite temporary negative surplus. Integrodifferential equations for the expectation of the discounted dividend payments and the probability of bankruptcy are derived. Moreover, when the gain size distribution is exponential, explicit solutions for the expected dividend payments and the bankruptcy probability are obtained for constant bankruptcy rate function. It also provided some numerical examples to illustrate the applications of the explicit solutions.
In the continuous time dual risk model, the company's surplus process with initial surplus is given by where is the constant rate of expenses per unit time, is a Poisson process with intensity , and the gain size is a sequence of independent and identically distributed positive continuous random variables with finite mean, independent of . Assume all have the same distribution as a generic random variable , which has probability density function and cumulative distribution function . Under model (1), the expected increase in surplus per unit time is , and it is assumed that .
Since De Finetti  proposed dividend strategies for an insurance risk model, the risk model in the presence of dividend payments has become a more and more popular topic in risk theory. In recent years, quite a few interesting papers have been written on the dual risk model with dividend strategy. Avanzi et al.  studied the expected total discounted dividends in the dual risk model with barrier strategy. Avanzi and Gerber  studied the optimal dividends in the dual risk model with diffusion. Ng  studied the expected discounted dividend in dual risk model with threshold dividend strategy. Dai et al.  studied the optimal dividend strategies in the dual risk model with capital injections.
Albrecher and Lautscham  made a distinction between ruin and bankruptcy, in the traditional actuarial model; if the surplus is negative, the company is ruined and has to go out of business; in particular, no dividends are paid after ruin. They consider a relaxation of the ruin concept to the concept of bankruptcy, in which the company with a negative surplus is assumed to be able to continue doing business as usual until bankruptcy takes place, and bankruptcy means that the company goes out of business. Concretely, a suitable bankruptcy rate function depending on the size of the negative surplus, which is defined on , zero for , and for . This is a nonincreasing function; whenever the negative surplus is , is the probability of bankruptcy within dt time units. Albrecher and Lautscham  considered the optimal dividend barrier in diffusion risk model until bankruptcy. In this paper, we extend the idea of ruin to more general bankruptcy concept in dual risk model.
Let us now consider some of the basic definition and notation for the risk model. Let be the bankruptcy time of the surplus process with dividend payments and define the overall probability of bankruptcy as A barrier dividend strategy is given by a parameter ; if at a potential dividend payment time, the surplus is above ; the excess is paid as a dividend and then the aggregate dividends till time is where .
The dividends are discounted at a constant force of interest , the total discounted dividends until bankruptcy are and the expected discounted value of dividends by .
The purpose of this paper is to present the expected value of the discounted sum of all dividend payments until bankruptcy and the probability of bankruptcy in the dual risk model. In Section 2, integrodifferential equations for the expected dividend payments until bankruptcy are derived; moreover, explicit solutions are also obtained under constant bankruptcy rate function and exponential gain size; finally, we provide some numerical examples with illustrations for the expected dividend payments under the concept of bankruptcy. In Section 3, we derive equations for the probability of bankruptcy , which are solved explicitly for constant bankrupt rate function and exponential gain size and we also provide numerical examples with illustrations for the probability of bankruptcy.
2. The Expectation of the Discounted Dividend Payments
In this section, we derive integrodifferential equations for ; the results are summarized in the following theorem. At first, we define Similarly, we can define in next section as .
2.1. Integrodifferential Equations for
Theorem 1. and satisfy the following system of integrodifferential equations:
In addition, and satisfy
Proof. When , the surplus drops to level immediately due to the initial payment of dividends and thus
when , conditioning on the first occurrence time of either a gain or an event of bankruptcy up to time yields that
We differentiate (12) with respect to , and by taking the limit we can get That is,
Using a similar method of deriving (6), we can also obtain (7).
Letting in (12) and in (15), it follows that is continuous at as long as is bounded; that is,
the continuity of at ; we can deduce from (6) and (7) for that
we know that if , the bankruptcy takes place, so is obvious.
2.2. Explicit Expressions for
We assume in this subsection that is constant; the positive constant is denoted as ; that is,
For simplicity, we will assume throughout the rest of the paper that the gain size is exponentially distributed .
In this case, (6) can be rewritten as That is,
Applying the operator to (20), we obtain the differential equation
Hence the solution of (21) is of the form where are arbitrary coefficients and and are the two solutions to the characteristic equation about :
From , it follows that .
Analogously, we rewrite (7) as That is, The solution of (25) is of the form where , are constants and , are the solutions of the equation about
From (8), we obtain that
The condition (10) gives the equation
Substituting (26) into (7), we have
Therefore, we have a system of linear equations (28)–(30) for the constants , , and . Solving the system of linear equations, we have
So we can obtain
2.3. Numerical Examples for
As an illustration of the results of the previous subsection, we will give some numerical examples about the expectation of the discounted dividend payments .
Example 2. We set , and ; it is easy to check that the net profit condition holds; we can discuss impact of the model parameters and on .
Example 3. We set , and ; it is easy to check that the net profit condition holds; we can discuss impact of the model parameters and on .
Tables 1 and 2 provide numerical results for . We find that, for a fixed , as can be expected, increases with , The results obtained in Table 1 also illustrate the effect of the bankruptcy parameter for . From Table 2, we find that the expectation of the discounted dividend payments decreases with dividend barrier , which makes sense intuitively.
Figure 1 is plotted to illustrate the impact of bankruptcy rate function to dividend payments for various parameter choices . Figure 2 is plotted to illustrate the impact of dividend barrier to dividend payments for various .
3. The Probability of Bankruptcy
3.1. Integrodifferential Equations for
Theorem 4. The probability of bankruptcy satisfies the following integrodifferential equations:
In addition, and satisfy
Proof. When , the surplus drops to level immediately due to the initial payment of all dividends and thus
When , by conditioning on the first occurrence time and amount of the gain or an event of bankruptcy up to time , Differentiating (39) with respect to and taking the limit we can obtain (33).
Using a similar argument, we can also derive the corresponding bankruptcy probability for . The conditions (35)–(37) are also obvious.
3.2. Explicit Expressions for
In this subsection, we assume that is also constant , and the gain size follows ; then (33) and (34) can be rewritten as
Applying the operators to (40), they can be rewritten as
We know the solution of (41) is of the form where are constants and , are the solutions of the equation about : From , is obvious.
For , we obtain the solution of (42) which is of the form When , we have . As , that is, , so is obvious. Then
Similar to deriving , we have that the conditions (35) and (36) for give that We have So
Example 5. We can perform analysis for the probability of bankruptcy ; again we choose , and .
In Table 3 we find that, for a given , the probability of bankruptcy increases with . And it also can be expected, as a large value of bankruptcy rate implies the large bankruptcy probability .
Figure 3 was produced by the explicit expression in (49); it is plotted to illustrate the impact of and on bankruptcy probability .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is fully supported by a Grant from Natural Science Foundation of Hunan (13JJ4083), by Humanities and Social Sciences Project of the Ministry Education in China (10YJC630144), by Hunan Social Science Fund Program (12YBB093), and by Scientific Fund of Hunan Provincial Education Department (13C318) and is also supported by Natural Science Foundation of Anhui Higher Education Institutions (KJ2014ZD21).
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