We consider the Ornstein-Uhlenbeck-type model. We first introduce the model and then find the ordinary differential equations and boundary conditions satisfied by the dividend functions; closed-form solutions for the dividend value functions are given. We also study the distribution of the time value of ruin. Furthermore, the moments and moment-generating functions of total discounted dividends until ruin are discussed.

In recent years, the dividend problem has gained a lot of attention in the actuarial literature. Dividend strategies for insurance risk model were first proposed by de Finetti [

Recently, the multilayer dividend strategy as an extension of the threshold dividend strategy has drawn many authors attention. For example, the perturbed Sparre Andersen and compound Poisson risk models with multilayer dividend strategy have been studied by Yang and Zhang [

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

Consider the following surplus process:

For

Define the random times

with the convention

Assume that

By virtually the same arguments as in Yin and Wen [

The ordinary differential equation (

Denote

The expressions of the expected discounted dividend payments are given by Theorem

(i) For

(ii) For

When

For

So we have

Similarly, when

For

Similarly,

For

For

Using

With some careful calculations, we obtain

In this section, we consider the limit of dividends level. Let

Substituting the above expressions into (

Let

Then dividing numerator and denominator of (

Similarly, when

In this section, we focus on the Laplace transform of the time value of ruin. We assume that dividends are paid according to threshold strategy with parameters

Let

For

For

Similarly, for

It can be verified that

From (

(1) Let

(2) Let

From (

Taking limit in (

In this section, the moment-generating function of the hybrid dividend payments is discussed. We adopt a similar approach to that of Gao and Yin in Section

Let

The moment-generating function

(i) We first provide the solution of

Let

Now the solution for

(ii) Now we derive the integrodifferential equations for

For

By Taylor expansion, we have

Subtracting

Similarly, for

By Taylor expansion, we get

Subtracting

The proof of boundary conditions is routine. This ends the proof of Theorem

For

Recall that

When

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

The authors would like to thank the editor and the anonymous referees for their helpful comments, which have led to this improved version of the paper. The research was supported by the National Natural Science Foundation of China (no. 11171179), the Research Fund for the Doctoral Program of Higher Education of China (no. 20133705110002), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.