Journal of Applied Mathematics

Volume 2014 (2014), Article ID 184098, 7 pages

http://dx.doi.org/10.1155/2014/184098

Research Article

Dividend Problems with a Barrier Strategy in the Dual Risk Model until Bankruptcy

Donghai Liu¹ and Zaiming Liu²

¹Department of Mathematics, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China

²Department of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, China

Received 29 March 2014; Revised 19 July 2014; Accepted 21 July 2014; Published 11 August 2014

Academic Editor: Daqing Jiang

Copyright © 2014 Donghai Liu and Zaiming Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1] 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. [2] studied the expected total discounted dividends in the dual risk model with barrier strategy. Avanzi and Gerber [3] studied the optimal dividends in the dual risk model with diffusion. Ng [4] studied the expected discounted dividend in dual risk model with threshold dividend strategy. Dai et al. [5] studied the optimal dividend strategies in the dual risk model with capital injections.

Albrecher and Lautscham [6] 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 [6] 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,

When ,

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.

Table 1: Influence of and on for , , , , and .

1
2
3
4
5
6

1.7969
1.2439
0.9516
0.7707
0.6476
0.5585

1.9620
1.3630
1.0447
0.8471
0.7124
0.6147

2.1422
1.4934
1.1469
0.9311
0.7837
0.6767

2.3390
1.6363
1.2591
1.0234
0.8621
0.7448

2.5539
1.7929
1.3822
1.1249
0.9484
0.8199

2.7885
1.9644
1.5174
1.2364
1.0433
0.9025

0
3.0446
2.1524
1.6658
1.3590
1.1478
0.9934

1
4.4970
4.0197
3.7594
3.5953
3.4823
3.3997

2
5.5086
5.2523
5.1125
5.0243
4.9636
4.9193

3
6.3049
6.1662
6.0905
6.0428
6.0010
5.9860

4
7.0079
6.9317
6.8901
6.8639
6.8458
6.8326

5
7.6841
7.6410
7.6174
7.6026
7.5924
7.5849

6
8.3710
8.3453
8.3313
8.3224
8.3163
8.3119

7
9.0906
9.0739
9.0649
9.0591
9.0552
9.0523

8
9.8568
9.8447
9.8382
9.8340
9.8312
9.8291

9
10.6795
10.6697
10.6643
10.6609
10.6586
10.6569

Table 2: Influence of and on for , , , , and .

10
11
12
13
14
15

1.7969
1.6610
1.5348
1.4179
1.3097
1.2098

1.9620
1.8136
1.6758
1.5481
1.4300
1.3209

2.1422
1.9802
1.8297
1.6903
1.5614
1.4422

2.3390
2.1621
1.9978
1.8456
1.7048
1.5747

2.5539
2.3607
2.1813
2.0151
1.8614
1.7194

2.7885
2.5776
2.3817
2.2002
2.0324
1.8773

0
3.0446
2.8144
2.6005
2.40230
2.2191
2.0497

1
4.4970
4.1569
3.8409
3.5483
3.2777
3.0275

2
5.5086
5.0919
4.7049
4.3465
4.0150
3.7085

3
6.3049
5.8281
5.3851
4.9749
4.5954
4.2447

4
7.0079
6.4779
5.9855
5.5295
5.1078
4.7179

5
7.6841
7.1029
6.5631
6.0631
5.6006
5.1732

6
8.3710
7.7379
7.1498
6.6051
6.1013
5.6356

7
9.0906
8.4030
7.7644
7.1728
6.6257
6.1201

8
9.8568
9.1113
8.4188
7.77740
7.1842
6.6359

9
10.6795
9.8718
9.1215
8.4266
7.7839
7.1898

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 .

Figure 1: The expected sum of discounted dividend when .

Figure 2: The expected sum of discounted dividend when .

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 .

Table 3: Influence of and on for , , , , and .

1
2
3
4
5
6

0.9593
0.9876
0.9941
0.9965
0.9977
0.9983

0.9413
0.9795
0.9894
0.9934
0.9955
0.9966

0.9153
0.9662
0.9811
0.9876
0.9910
0.9931

0.8779
0.9442
0.9661
0.9765
0.9823
0.9860

0.8240
0.9080
0.9394
0.9555
0.9651
0.9714

0.7462
0.8484
0.8917
0.9158
0.9311
0.9418

0
0.6340
0.7500
0.8063
0.8406
0.8641
0.8813

1
0.3845
0.4549
0.4890
0.5099
0.5241
0.5345

2
0.2332
0.2759
0.2966
0.3092
0.3179
0.3242

3
0.1415
0.1673
0.1799
0.1876
0.1928
0.1966

4
0.0858
0.1015
0.1091
0.1138
0.1169
0.1193

5
0.0520
0.0616
0.0662
0.0690
0.0709
0.0723

6
0.0316
0.0373
0.0401
0.0418
0.0430
0.0439

7
0.0191
0.0226
0.0243
0.0254
0.0261
0.0266

8
0.0116
0.0137
0.0148
0.0154
0.0158
0.0161

9
0.0070
0.0083
0.0090
0.0093
0.0096
0.0098

Figure 3 was produced by the explicit expression in (49); it is plotted to illustrate the impact of and on bankruptcy probability .

Figure 3: The bankruptcy probability when .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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).

References

B. De Finetti, “Su un'impostazione alternativa della teoria collettiva del rischio,” in Proceedings of the Transactions of the 15th International Congress of Actuaries, vol. 2, pp. 433–443, 1957.

B. Avanzi, H. U. Gerber, and E. S.W. Shiu, “Optimal dividends in the dual model,” Insurance: Mathematics and Economics, vol. 41, no. 1, pp. 111–123, 2007.

B. Avanzi and H. U. Gerber, “Optimal dividends in the dual model with diffusion,” ASTIN Bulletin, vol. 38, no. 2, pp. 653–667, 2008.

A. C. Y. Ng, “On a dual model with a dividend threshold,” Insurance: Mathematics & Economics, vol. 44, no. 2, pp. 315–324, 2009.

H. Dai, Z. Liu, and N. Luan, “Optimal dividend strategies in a dual model with capital injections,” Mathematical Methods of Operations Research, vol. 72, no. 1, pp. 129–143, 2010.

H. Albrecher and V. Lautscham, “From ruin to bankruptcy for compound poisson surplus processes,” ASTIN Bulletin, vol. 43, no. 2, pp. 213–243, 2013.