^{1}

In this paper, we study the optimal retentions for an insurer with a compound fractional Poisson surplus and a layer reinsurance treaty. Under the criterion of maximizing the adjustment coefficient, the closed form expressions of the optimal results are obtained. It is demonstrated that the optimal retention vector and the maximal adjustment coefficient are not only closely related to the parameter of the fractional Poisson process, but also dependent on the time and the claim intensity, which is different from the case in the classical compound Poisson process. Numerical examples are presented to show the impacts of the three parameters on the optimal results.

In various geophysical applications, it is observed that the interarrival times between extreme events are power-law distributed, and the exponentially distributed interarrivals cannot be applied [

Considering the importance of quantifying the stochastic behavior of extreme events in actuarial sciences, Beghin and Macci [

In this paper, we model the surplus process of an insurance company by the abovementioned CFPP proposed by [

This paper supposes the insurer reinsures his or her risk by a layer reinsurance treaty. As in [

Assume that the reinsurance premium is charged by the expected value principle, and denote the expected value of

Thus, the reserve process of the insurer with the layer reinsurance policy can be represented by

Now define the ruin time by

In this section, we devote to get the explicit expressions for the optimal retentions in the layer reinsurance treaty. It is difficult to derive the explicit expression of the ruin probability in the CPP and even more difficult in CFPP. We consider the optimal retentions to maximize the adjustment coefficient, i.e., to maximize the coefficient

If

Assume (

Since

Next we adopt the method used by [

Denote the maximizer of

By differentiating

Then, differentiating

By replacing

Since

According to Lemma

Equation (

For any given

The function

Rewrite (

The equation

Differentiating

Now, we can conclude the main result of this paper.

Let

Since the CFPP degenerates into the classical CPP when

Let

To illustrate the impact of replacing the exponential distributed interarrivals by the general Mittag-Leffler distributed interarrivals, as well as the claim intensity

Assume that the insurer has an initial capital

By applying numerical method, the results for different cases are given in Table

Optimal retention levels and the upper bounds with different parameters.

| | | | | | | | |
---|---|---|---|---|---|---|---|---|

| | | ||||||

| 0.1489 | 0.0001 | | 0.2508 | 0.0162 | | 0.2514 | 0.0166 |

1.8234 | 0.8241 | 0.8021 | ||||||

| ||||||||

| 0.1659 | 0.0014 | | 0.2521 | 0.0210 | | 0.2523 | 0.0220 |

1.3195 | 0.7728 | 0.7629 | ||||||

| ||||||||

| 0.1878 | 0.0063 | | 0.2531 | 0.0259 | | 0.2532 | 0.0263 |

1.0144 | 0.7308 | 0.7273 | ||||||

| ||||||||

| 0.2163 | 0.0174 | | 0.2540 | 0.0309 | | 0.2540 | 0.0310 |

0.8099 | 0.6955 | 0.6946 | ||||||

| ||||||||

| 0.2548 | 0.0360 | | 0.2548 | 0.0360 | | 0.2548 | 0.0360 |

0.6650 | 0.6649 | 0.6649 | ||||||

| ||||||||

| 0.3084 | 0.0606 | | 0.2555 | 0.0411 | | 0.2555 | 0.0412 |

0.5606 | 0.6383 | 0.6377 | ||||||

| ||||||||

| 0.3861 | 0.0876 | | 0.2561 | 0.0462 | | 0.2562 | 0.0467 |

0.4870 | 0.6150 | 0.6128 | ||||||

| ||||||||

| 0.5002 | 0.1088 | | 0.2567 | 0.0513 | | 0.2569 | 0.0525 |

0.4436 | 0.5940 | 0.5895 | ||||||

| ||||||||

| 0.6557 | 0.1109 | | 0.2572 | 0.0564 | | 0.2574 | 0.0584 |

0.4399 | 0.5752 | 0.5682 | ||||||

| ||||||||

| 0.8053 | 0.0807 | | 0.2576 | 0.0613 | | 0.2579 | 0.0644 |

0.5035 | 0.5583 | 0.5486 |

From Table

To characterize and to disperse the extreme event risk that the insurer may face in practice, this paper models the underwriting risk as a compound fractional Poisson process and studies the optimal retentions with a layer reinsurance treaty. At first, the equation that the adjustment coefficient of the compound fractional Poisson process should satisfy is given and proved. Secondly, to overcome the difficulties caused by the newly adopted model, some lemmas are given, and the closed form expressions of the optimal retention levels are obtained. It is found that the optimal retention level and the maximal adjustment coefficient here relate to the parameter

There is a numerical example in this article; the parameter data used to support the findings of this study are included within the article. For the software (Matlab) code data to obtain the numerical results in the example, it will be available upon request by contact with the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by Anhui Provincial Natural Science Foundation (No. 1608085QG169) and Humanity and Social Science Youth Foundation of Ministry of Education (No. 17YJC630212).