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
Since De Finetti [
Albrecher and Lautscham [
Let us now consider some of the basic definition and notation for the risk model. Let
The dividends are discounted at a constant force of interest
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
In this section, we derive integrodifferential equations for
In addition,
When
when
We differentiate (
When
Using a similar method of deriving (
Letting
the continuity of
we know that if
We assume in this subsection that
For simplicity, we will assume throughout the rest of the paper that the gain size is exponentially distributed
In this case, (
Applying the operator
Hence the solution of (
From
Analogously, we rewrite (
From (
The condition (
Substituting (
Therefore, we have a system of linear equations (
So we can obtain
As an illustration of the results of the previous subsection, we will give some numerical examples about the expectation of the discounted dividend payments
We set
We set
Tables
Influence of

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 
Influence of

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
The expected sum of discounted dividend
The expected sum of discounted dividend
The probability of bankruptcy
In addition,
When
When
Using a similar argument, we can also derive the corresponding bankruptcy probability
In this subsection, we assume that
Applying the operators
We know the solution of (
For
Similar to deriving
We can perform analysis for the probability of bankruptcy
In Table
Influence of

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
The bankruptcy probability
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 (13