The paper deals with a generalization of the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. First of all, we derive piecewise integro-differential equations for the Gerber–Shiu function and the expected discounted dividend payments until ruin. In addition, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes and find explicit formulas for the ruin probability as well as for the expected discounted dividend payments. Lastly, numerical illustrations for some multi-layer dividend strategies are presented.

The ruin measures such as the ruin probability, the surplus prior to ruin and the deficit at ruin have attracted great interest of researchers recently (see, e.g., [

In particular, a lot of attention has been paid to the study of risk models where shareholders receive dividends from their insurance company. De Finetti [

Applying multi-layer dividend strategies enables to change the dividend payment intensity depending on the current surplus. Albrecher and Hartinger [

The absolute ruin problem in the classical risk model with constant interest force and a multi-layer dividend strategy is investigated in [

The present paper generalizes the risk model with stochastic premiums introduced and investigated in [

The rest of the paper is organized as follows. In Section

Let

In the risk model with stochastic premiums introduced in [

It is worth pointing out that, here and subsequently, a sum is always set to 0 if the upper summation index is less than the lower one. In particular, we have

Next, we denote a non-negative initial surplus of the insurance company by

In contrast to the risk model considered in [

From now on, we suppose that the net profit condition holds, which in this case means that

Let

Next, let

For

For

For simplicity of notation, we also write

Note that although we consider the interval

We now fix any

From these equalities we conclude that for all

It is easily seen that the time of the first jump of

Note that the term

Changing the variable

Changing the variable

In the same manner we change variables in all the outer integrals on the right-hand side of (

Thus, from the above and equality (

Moreover, it is easily seen, e.g. from (

From (

To solve equation (

In the assertion of Theorem

We now fix any

Note that the terms

Next, we set

Thus,

Rearranging terms in the expression for

Taking all the integrals on the right-hand side of (

Changing the variable

Likewise, changing the variable

Next, in the same manner we change variables in all those outer integrals on the right-hand side of (

From (

Next, from the above and equality (

Furthermore, it follows immediately, e.g. from (

By (

To solve equation (

If

In this section, we concentrate on the case where claim and premium sizes are exponentially distributed, i.e.

Let now

We now reduce piecewise integro-differential equation (

It is easily seen that the right-hand side of (

Multiplying (

From (

Finally, multiplying (

For

Taking into account the notation introduced in Section

Therefore,

To determine all the other constants

One more condition is obtained from the equality

Taking into account (

Substituting (

Thus, we get the system of

In particular, if

The proposition below enables us to check whether the system of equations (

From (

Thus, the system of equations (

Multiplying (

Similarly, multiplying (

Substituting (

By Vieta’s theorem applied to (

Substituting these equalities into (

Next, multiplying (

Multiplying (

Thus, if the system of equations (

A standard computation shows that the determinant of the system of equations (

By (

The piecewise integro-differential equation (

The proof of the lemma is similar to the proof of Lemma

For

By Lemma

By Vieta’s theorem, we conclude that (

To determine all the other constants

One more condition is obtained from the equality

Substituting (

In particular, if

To present numerical examples for the results obtained in Section

In addition, we denote by

Moreover, let now

Table

The ruin probabilities without and with dividend payments and the expected discounted dividend payments,

0 | 0.666667 | 1 | 0 |

1 | 0.596560 | 0.777184 | 3.663273 |

2 | 0.533825 | 0.737542 | 4.283457 |

5 | 0.382502 | 0.636926 | 5.911685 |

7 | 0.306284 | 0.575029 | 6.716708 |

10 | 0.219462 | 0.492173 | 7.623108 |

15 | 0.125917 | 0.379750 | 8.612682 |

20 | 0.072245 | 0.293007 | 9.190265 |

50 | 0.002577 | 0.061825 | 9.967986 |

70 | 0.000279 | 0.021912 | 9.996285 |

Next, for

The values of

The ruin probabilities without and with dividend payments and the expected discounted dividend payments,

0 | 0.666667 | 1 | 0 |

1 | 0.596560 | 0.721066 | 2.611525 |

2 | 0.533825 | 0.663275 | 2.930525 |

5 | 0.382502 | 0.506845 | 3.490686 |

7 | 0.306284 | 0.426750 | 3.805635 |

10 | 0.219462 | 0.330912 | 4.178134 |

15 | 0.125917 | 0.216577 | 4.559200 |

20 | 0.072245 | 0.141747 | 4.763582 |

50 | 0.002577 | 0.011141 | 4.994372 |

70 | 0.000279 | 0.002044 | 4.999534 |

The results presented in Tables

The author is deeply grateful to the anonymous referees for careful reading and valuable comments and suggestions, which helped to improve the earlier version of the paper.