In this paper, a discrete-time risk model with random income and a constant dividend barrier is considered. Under such a dividend policy, once the insurer’s reserve hits the level

In the actuarial literature, many authors focus their research interests on discrete-time risk models, which can be used as an approximation to continuous time models. Li et al. [

Additionally, problems related to dividends have been considered extensively in the discrete-time setting. Because of the certainty of ruin for a risk model with a constant dividend barrier, the calculation of the expected discounted dividend payment is a major problem of interest in the context. Among the class of discrete-time risk models, Tan and Yang [

All the risk models discussed above are based on a common assumption that the premium is collected with a positive deterministic constant. However, this assumption can be unrealistic and inappropriate in practical contexts because the insurance company may have lump sums of income. Therefore, many authors consider the risk models with stochastic income to capture the uncertainty of the customers’ arrivals, for example, Boikov [

In the present paper, we propose a discrete-time risk model with random income and a constant dividend barrier. A similar model has been discussed by Zhou et al. [

The rest of the paper is organized as follows. A brief description of the discrete-time model and the introduction of the expected present value of total dividends are considered in Section

Throughout, denote by

Suppose that premiums are received at the beginning of each period, and claims are paid out at the end of each period. We introduce a dividend policy to the company that a certain amount of dividends will be paid to the policyholder instantly, as long as the surplus of the company at time

Note that the positive safety loading condition holds if

By considering the occurrence (or not) of the premium income and claims in the next period, we separate the four possible cases as follows: no premium arrival and no claim occurs, a random premium arrival and no claim occurs, a random premium arrival and a claim occurs, and no premium arrival and a claim occurs. For

Now, we show that (

Using the property of the forward difference operator

Substituting (

Letting

Equation (

Multiplying (

To find a set of fundamental solutions to (

We can easily identify the number of zeros on the denominator in (

When

When

In what follows, we assume these zeros

Let

It is easy to see that

Now, we use a discrete operator

Therefore, substituting (

Note that (

For the numerator in (

By inserting (

The direct inversion of the generating functions in (

For

To complete the proof, it remains to show that

In the case of

Differentiating (

Thus, taking the limit

Starting from

In this section, we explain the solution procedure to the difference equation

By substituting (

Upon inversion, we obtain from (

As the explicit expression for

Suppose

Let

Explicit expressions for

Then, we obtain the values of

To finish the calculation of

We depict

Numerical results of

3.48604534795459 | 0.824973674803551 | ||

2.54139402542121 | 0.813367218414144 | ||

1.98618491341006 | 0.797848459477162 |

Numerical results of

0 | 1 | 2 | 3 | |
---|---|---|---|---|

1 | 3.45811 | |||

1 | 4.56061 | |||

4 | 5 | 6 | 7 | |

16.1795 | 92.6721 | |||

25.5918 | 155.915 | |||

8 | 9 | 10 | 11 | |

568.161 | 3548.14 | |||

972.075 | 6096.64 | |||

12 | 13 | 14 | 15 | |

22262.4 | 139848 | |||

38293.9 | 240620 |

Numerical results of

15 | 16 | 17 | 18 | 19 | |
---|---|---|---|---|---|

0.07130 | 0.05689 | 0.04539 | 0.03621 | 0.02889 | |

0.10387 | 0.08287 | 0.06612 | 0.05275 | 0.04209 |

Numerical results of

0.1 | 0.15 | 0.2 | 0.25 | 0.3 | |
---|---|---|---|---|---|

0.19716 | 0.16288 | 0.12968 | 0.09878 | 0.07130 | |

0.27582 | 0.18453 | 0.14212 |

Impact of

In this paper, we consider the compound binomial model with random income and a constant dividend barrier. Furthermore, we analyze the model with a general premium rate

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

All authors have equal contributions. All authors have read and approved the final manuscript.

This research was supported by the Ministry of Education of Humanities and Social Science Project (20YJA910001), Foundation of Educational Department (W201783664), and Science and Technology Department (20180550196) of Liaoning Province.