Optimal Layer Reinsurance for Compound Fractional Poisson Model

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 ofmaximizing 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.


Introduction
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 [1].Musson et al. [2] studied the earthquake interarrival times for several regions in Japan and Greece and found that a lognormal distribution provided a good fit.Salim and Pawitan [3] investigated the hourly rainfall data in the southwest of Ireland by a generalized Bartlett-Lewis model with Pareto storm interarrival time.Stoynov et al. [4] proposed an approach for modeling the flood arrivals on Chinese rivers Yangtze and Huanghe by switch-time distributions, which can be considered as distributions of sums of random number exponentially distributed random variables.
Considering the importance of quantifying the stochastic behavior of extreme events in actuarial sciences, Beghin and Macci [5] deal with a fractional Poisson model for insurance, in which the interarrival times between claims are assumed to have Mittag-Leffler distribution instead of the exponential distribution as in the classical Poisson model.Inspired by this work and motivated by the use of the fractional Poisson process in modeling extreme events, such as earthquakes and storms, Biard and Saussereau [6] initiatively described surplus processes of insurance companies by compound fractional Poisson processes, and some results for ruin probabilities are also presented under various assumptions on the distribution of the claim sizes.Different from the case in the classical compound Poisson process (CPP), the compound fractional Poisson process (CFPP) becomes nonstationary [6] and is no longer Markovian [7].The long-range dependence and the short-range dependence of the CFPP are studied in [6,8], the estimation of parameters is given by [9], and the convergence of quadratic variation is investigated by [10].To complete the review of the existing literature on the CFPP, we refer the reader to [11][12][13][14][15][16][17][18].
In this paper, we model the surplus process of an insurance company by the abovementioned CFPP proposed by [6], which can be expressed as where  is the initial capital,  is the constant premium rate, and   ,  = 1, 2, 3, ⋅ ⋅ ⋅ represents the size of the th claim and the claim sizes are assumed to be independent and identically distributed nonnegative variables with a common distribution function .The counting process  ℎ () is the fractional Poisson process that was first defined in [11,19] as a renewal process with Mittag-Leffler waiting time.Specifically, it has independent and identically distributed interarrival times (  ) between two claims with distribution given by for  > 0 and 0 < ℎ ≤ 1, where is the Mittag-Leffler function (Γ denotes the Euler gamma function) defined for any complex number .With   =  1 + 2 +⋅ ⋅ ⋅   , the time of the th jump, the process ( ℎ ()) ≥0 defined by is the so-called fractional Poisson process of parameter ℎ.It includes the usual Poisson process when ℎ = 1.This paper supposes the insurer reinsures his or her risk by a layer reinsurance treaty.As in [20,21], we assume that the common distribution function () of   is such a continuous function that (0) = 0, 0 < () < 1 for 0 <  <  and () = 1 for  ≥ , here  = inf{ : () = 1, and 0 <  ≤ +∞; that the moment generating function of (),   (), exists for  ∈ (−∞,  ∞ ) for some 0 <  ∞ ≤ ∞; and that lim → ∞   () = +∞.Let  be the expected value of   .Denote the decision variables representing the layer retention by  1 and  2 .The ceded loss function is the layer reinsurance in the form of where {} + = max{, 0}, and 0 <  1 ≤  2 ≤ .Thus, the insurer will retain from the ith claim Then {  ( 1 ,  2 )} are i.i.d.strictly positive random variables and independent of the claim counting process  ℎ ().
Assume that the reinsurance premium is charged by the expected value principle, and denote the expected value of   ( 1 ,  2 ) by ( 1 ,  2 ).Then the premium income rate becomes where  = (/)(Γ(1 + ℎ)/ℎ ℎ−1 ) − 1 denotes the security loading of the insurer, and  is the security loading of the reinsurer.As usual, we assume that  > .Note that the following inequality should be held, Otherwise, the insurance company faces ruin with probability one.Thus, the reserve process of the insurer with the layer reinsurance policy can be represented by Now define the ruin time by and define the ruin probability by

Optimal Results
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  which satisfies the following inequality which is an implicit equation with respect to ; then the inequality ( ) follows.
Proof.Assume ( 13) holds; we prove the inequality ( 12) by mathematical induction (see [22] for the CPP case).Let   () be the probability that ruin occurs on the nth claim or before with an initial surplus u.Clearly, and lim Furthermore, from we have To complete the nontrivial part of the mathematical induction, we apply the total probability formula with respect to the arrival time and the size of the first claim.Then, we obtain where the last equation is obtained from equation (4.15) in [23].Thus, the inequality (12) follows immediately from (13). Since is equivalent to Substituting ( 7) into ( 19) yields Our goal is to maximize   ( 1 ,  2 ), i.e., to find the optimal retention ( * 1 ,  * 2 ), such that Note that the left-hand side of ( 19) is a concave function and the right-hand side is a convex function, with respect to r.Therefore, there are at most two solutions to (19), and the left-hand side of ( 20) is nonpositive at  =   , i.e.,   is the solution to sup or, equivalently, sup where Next we adopt the method used by [21] to determine the optimal retention level ( * 1 ,  * 2 ).
Lemma 2. Denote the maximizer of ( 1 ,  2 ) with  1 and  2 being  1 and  2 , respectively.en,  1 is the solution to the following equation with respect to  1 , and  2 = .
Proof.By differentiating ( 1 ,  2 ) with respect to  1 , we have which means that for any fixed  2 .
Then, differentiating ( 1 ,  2 ) with respect to  2 and combining with (27), we obtain holds for any  2 ≥  1 , and we have and thus By replacing  2 =  back into (27), we can derive which completes the proof of Lemma 2. Since and  2 = , by ( 8), we have Denote According to Lemma 2, we know that to solve the optimization problem ( 23) is equivalent to solving the equation: or, alternatively, where  = ( 1 ) is a univariate function of  1 determined by (32).In fact, we have Lemmas 3-5.
Since the CFPP degenerates into the classical CPP when ℎ = 1, we immediately obtain the following corollary from eorem .
Corollary 7. Let  1 be the unique positive root of the equation en the optimal layer reinsurance retention level of the classical compound Poisson surplus to maximize the adjustment coefficient is ( 1 , ), and the maximal adjustment coefficient is the unique positive root of ( ), i.e., Remark.It is not difficult to see that Corollary 7 is in fact Theorem 4.3 of [21].By comparing the obtained Theorem 6 in this paper for CFPP with those in [21] for CPP, we find that the optimal retention level ( 1 , ) and the maximal adjustment coefficient   here not only depend on the parameter ℎ of the fractional Poisson process, but also depend on the claim intensity  and are both relevant to time t, which should be more realistic.In fact, the claim intensity is a very important parameter for estimating the ruin probability and by the dynamic reinsurance strategy the change of the insurer's best risk position is reflected with respect to time.
To illustrate the impact of replacing the exponential distributed interarrivals by the general Mittag-Leffler distributed interarrivals, as well as the claim intensity  and the time , on the optimal results, we give some numerical examples and compare the optimal retention levels and the maximal adjustment coefficient with different parameters ℎ, , and .

Examples
Assume that the insurer has an initial capital  = 5, that the claim size   has a uniform distribution on the interval [0, 4], and that  = 0.4,  = 0.5.We compute the values of  1 ,   and the upper bound of ruin probability with different parameter values of ℎ, , and .To this end, we need to solve the following equations: ))) for different given (ℎ, , ) with  1 ∈ ( 1 , ) = (0.1356, 4.0000) and   > 0.
By applying numerical method, the results for different cases are given in Table 1.
From Table 1, it is not difficult to see that the impacts of the parameter ℎ on the optimal retention level and the upper bound of ruin probability are significant, and the impacts of the parameters  and  are also obvious.Specifically, if the risk process of the insurance company obeys the compound fractional Poisson model and the compound Poisson model is used, then the insurer may take more risk and the ruin probability is overestimated or underestimated.Even with the CFPP, the optimal strategy should vary timely and according to the change of the claim intensity.

Conclusion
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 ℎ of the fractional Poisson process, the time , and the claim intensity , which are all absent in the optimal results for the classical compound Poisson process.Finally, numerical examples demonstrate the impacts of the three parameters on the optimal results, respectively.The obtained results in this paper may help the insurers, especially the ones who underwrite extreme risk, to make more appropriate decisions in reinsurance contracts.

Table 1 :
Optimal retention levels and the upper bounds with different parameters.