We consider the perturbed dual risk model with constant interest and a threshold dividend strategy. Firstly, we investigate the moment-generation function of the present value of total dividends until ruin. Integrodifferential equations with certain boundary conditions are derived for the present value of total dividends. Furthermore, using techniques of sinc numerical methods, we obtain the approximation results to the expected present value of total dividends. Finally, numerical examples are presented to show the impact of interest on the expected present value of total dividends and the absolute ruin probability.
In insurance mathematics, the classical risk model has been the center of focus for decades. The surplus in the classical model at time
In this model, the premium rate should be viewed as an expense rate and claims should be viewed as profits or gains. While not very popular in insurance mathematics, this model has appeared in various literature (see Cramér [
Considering the perturbed dual risk model, the surplus of an insurer has the following form
Dividend strategy for insurance risk models was first proposed by de Finetti [
Then, we consider the modification of the surplus process by a threshold strategy with a threshold level
In this paper, we consider that the insurance company earns credit interest with a constant force
Let
In the sequel, we will be interested in the following moment generating function:
In this section, we will give the integrodifferential equations satisfied by the moment generating function
In the following, we firstly derive the integrodifferential equations satisfied by
When
When
Now we consider the case of
Set
Let
When
For
Similarly, using the above method, we get (
When
The second order system of integrodifferential equations such as in Theorem
To construct an approximation on the interval
The function
In order to adopt the sinc method procedure, we arrange the systems in Theorem
Using the properties of convolution, the above equation can be further written as
Set
Let
Then, by using Theorems
Since
Having replaced the integral term on the left-hand side of system (
Then, by Theorem
Replacing (
Having multiplied the resulting equations by
Now, since
We set
The above linear system contains
In this section, we consider some numerical samples to illustrate the performance of sinc method and investigate how much the values of
Let
From Figure
The expected present value of total dividends
The sinc method is a highly efficient numerical method developed by Stenger, the pioneer of this field, people in his school, and others [
sinc methods are based on the use of the cardinal function,
For any
The sinc functions are cardinal for the interpolating points
Let
Given
Observe that
Corresponding to positive numbers
Another important family of functions is
Let
Given three positive integers
Let
Let
Let
Let
Let
This work was supported in part by the National Natural Science Foundation of China under Grant nos. 61370096 and 61173012 and the Key Project of Natural Science Foundation of Hunan Province under Grant no. 12JJA005.