Estimating the Gerber-Shiu Function in a Compound Poisson Risk Model with Stochastic Premium Income
In this paper, we consider the compound Poisson risk model with stochastic premium income. We propose a new estimation of Gerber-Shiu function by an efficient method: Fourier-cosine series expansion. We show that the estimator is easily computed and has a fast convergence rate. Some simulation examples are illustrated to show that the estimation has a good performance when the sample size is finite.
In this paper, we consider the following compound Poisson risk model with stochastic premium income:where is the initial surplus and denotes the surplus level of an insurance company at time . The premium number process and the claim number process are homogenous Poisson processes with intensity and , respectively. The individual claim sizes, , are positive independent and identically distributed (i.i.d.) continuous random variables given by generic random variable with density function . The premium sizes, , are positive i.i.d. random variables given by generic random variable with exponential distribution function , . Throughout this paper, we assume that and are mutually independent.
Whenever the surplus process becomes negative, we say that ruin occurs. Defining the ruin time by where if for all , . To avoid that ruin is a certain event, suppose that the following condition holds throughout this paper.
Condition 1 (net profit condition). The above condition guarantees that the expectation of the surplus process will always be positive at any time ; that is to say, the limit of surplus process is almost sure positive. Our research is more meaningful under this assumption.
Let be the interest force and define the expected discounted penalty function bywhere is a measurable penalty function of the surplus before ruin and the deficit at ruin and is an indicator function of the event . This function was first proposed by Gerber and Shiu  to study the time to ruin; the surplus before ruin and the deficit at ruin in the classical risk model. It is also called Gerber-Shiu function in the literature. For recent research progress on the Gerber-Shiu function, we can refer to work by Zhao and Yin , Yin and Wang , Shen et al. , Yin and Yuen , Zhao and Yao , Li et al. [7, 8], Dong et al. , among others. Recently, it has become a standard risk measure in ruin theory since many important risk problems can be studied via it by taking different forms of penalty function and different . Several typical examples are presented as follows:(1)If , is the ruin probability when , and it is the Laplace transform of ruin time when .(2)If , is the expected discounted claim size causing ruin.(3)If or for , is the defective distribution of the deficit at ruin when , and it is the discounted defective distribution of the deficit at ruin when .(4)If , is the th moment of the deficit at ruin when , and it is the discounted th moment of the deficit at ruin when .
The classical compound Poisson risk model assumes that the premiums are received by insurance company at a constant rate over time [10–15]. However, the premiums income in the real life, especially for small insurance companies, is volatile. So it is natural to extend the classical risk model by replacing the constant premium income with a compound Poisson process, which was firstly suggested in Boucherie et al. . Since then, this risk model has been widely studied by many scholars. Boikov  derived integral equations and exponential bounds for nonruin probability. Temnov  derived a representation for ruin probability. Assuming that the income rate is 1, defective renewal equations satisfied by the discounted penalty function were, respectively, obtained by Bao  and Yang and Zhang  under different distributions of interclaim times. Supposing premiums and claims follow general compound Poisson processes, a defective renewal equation satisfied by the Gerber-Shiu function was established in Labbé and Sendova . Zhang and Yang  extended their model by assuming that a specific dependence structure exists among the claim sizes, interclaim times, and premium sizes; then the Laplace transforms and defective renewal equations for the Gerber-Shiu function were obtained when the individual premium sizes are exponentially distributed. Besides, the risk model with stochastic income has also been studied in Zhao and Yin , Yu [23, 24], Xie and Zou , Mishura and Ragulina , Cheng and Wang , Yang et al. , Zeng et al. , Deng et al. , Wang and Zhang , and so on.
In all of the above papers, it is assumed that the probability characteristics of the surplus process, such as the densities of claim size and premium size, the Poisson intensities of claim process, and premium process, are known. In reality, these characteristics for an insurance company are usually unknown, but some data information on surplus levels, claim and premium numbers, and claim and premium sizes can be obtained. Thus in recent years, many scholars began to study the estimation of ruin probability and Gerber-Shiu function based on those data information. For the classical compound Poisson risk model with unknown claim size distribution and Poisson intensity, Politis  and Masiello , respectively, proposed the semiparametric estimators of nonruin and ruin probabilities, respectively. Zhang et al.  presented a nonparametric estimator for ruin probability. Zhang  proposed a nonparametric estimation of the finite time ruin probability; Zhang and Su  proposed a new nonparametric estimation of Gerber-Shiu function by Laguerre series expansion. For the risk model perturbed by a Brownian motion, Shimizu [37, 38] estimated the adjustment coefficient and the Gerber-Shiu function by Laplace transform, respectively. Zhang  estimated the Gerber-Shiu function by Fourier-Sinc series expansion. Su et al.  proposed an estimator for Gerber-Shiu function by Laguerre series expansion. For the estimation of ruin probability and Gerber-Shiu function in the Lévy risk model, we refer the interested readers to Wang and Yin , Zhang and Yang [42, 43], Wang et al. , Shimizu and Zhang , Peng and Wang , and Zhang and Su .
In addition, Fang and Oosterlee  proposed a novel method for pricing European options by Fourier-cosine series expansion. This method is also called the COS method in the literature, and it can be easily used to approximate an integrable function as long as the corresponding Fourier transform has closed-form expression. Now the COS method has been widely used for pricing options and other financial derivatives. See, e.g., Fang and Oosterlee , Ruijter and Oosterlee , and Zhang and Oosterlee  to name a few. Recently, the COS method has also been used in risk theory to compute and estimate some risk measures by some actuarial researchers. For example, Chau et al. [52, 53] used the COS method to compute the ruin probability and Gerber-Shiu function in the Lévy risk models. Zhang  applied the COS method to compute the density of the time to ruin in the classical risk model. Yang et al.  used a two-dimensional COS method to estimate the discounted density function of the deficit at ruin in the classical risk model.
Inspired by the work Yang et al. , in this paper we shall use the COS method to estimate the Gerber-Shiu function in the compound Poisson risk model with stochastic income. We note that this problem has also been considered by  later on, where the Laguerre series expansion method is used to estimate the Gerber-Shiu function. In particular, it can be found that the convergence rate obtained in Theorem 6 in this paper is also used by  to make error analysis of their estimator. We would like to remark that the convergence rate in Theorem 6 is one of the main contributions of our paper, since it plays an important role in studying the consistency property of the estimator.
The remainder of this paper is organised as follows. In Section 2, we first briefly introduce the Fourier-cosine series expansion method and then derive the Fourier transform of the Gerber-Shiu function. In Section 3, an estimator of the Gerber-Shiu function is proposed by the observed sample of the surplus process. The consistent property is studied in Section 4 under large sample size setting. Finally, in Section 5 we present some simulation results to show that the estimator behaves well under finite sample size setting.
2.1. The Fourier Transform of Gerber-Shiu Function
In this subsection, we derive the Fourier transform of the Gerber-Shiu function. Let and denote the class of integrable and square integrable functions on the positive axis, respectively. Let and denote the Fourier transform and Laplace transform of a . For any complex number , we denote its real part and imaginary part by and , respectively.
For convenience, we introduce the Dickson-Hipp operator (see, e.g., Dickson and Hipp  and Li and Garrido ), which for any integrable function on and any complex number with is defined as The Dickson-Hipp operator has been widely used in ruin theory to simplify the expression of ruin related functions. For properties on this operator, we refer the interested readers to Li and Garrido .
By Theorem 5.8 in Labbé and Sendova , we know that when , the Gerber-Shiu function satisfies the following renewal equation:where It follows from Lemma 5.3 in Labbé and Sendova  that appearing in the above formulae is the unique nonnegative root of the following equation (w.r.t. ), known as the Lundberg’s fundamental equation:
Remark 1. Let It is clear that is a continuous function such that , and if . In addition, since and , we have which shows that is an increasing function. Thus we conclude that is the unique nonnegative root of the equation , and it is located in the interval . In particular, we have when .
We assume the following conditions hold true in our literature, which could be satisfied by penalty functions.
Condition 2 (the integrability of ). For the penalty function , it satisfies
Condition 3. For the penalty function , there exist some integers and constant C such that
Proof. It is easily seen that . Under Condition 2 we have and thus Furthermore, under Condition 3 we have so , and then . By Theorem 1.4.5 in Stenger , we have . Using the inequality , we finally derive due to (6).
Now, we compute the Fourier transform of the Gerber-Shiu function. Applying the Fourier transform on both sides of (6) gives leading toFor the Fourier transform , we have where Then we obtainSimilarly, for we obtain
2.2. Fourier-Cosine Series Expansion
In this subsection, we present some known results on the Fourier-cosine series expansion method and give the estimating formula of the Gerber-Shiu function derived by Fourier-cosine series expansion.
By , any real function has a cosine expansion when it is finitely supported. Therefore, for a integrable function defined on , it has the following cosine series expansion:where means the first term of the summation has half weight. For a function , we introduce an auxiliary function then has a finite domain and when . By (22) we haveDue to the existence of the Fourier transform of , the integrand of has to decay to zero at and we can truncate the integration range with a large ; then we have Thus Furthermore, for a large integer , the above summation can be truncated as follows:
We now consider the Gerber-Shiu function . It follows from (27) that, for ,We can easily see that the key of approximating the Gerber-Shiu function by Fourier-cosine series expansion method lies in the calculation of the Fourier transform for . Therefore, we derive a specific expression of Fourier transform in the next section.
3. Estimation Procedure
In this section, we study how to estimate the Gerber-Shiu function by Fourier-cosine series expansion based on the discretely observed information of the surplus process, the aggregate claims and premiums processes. According to (28), we know that the key is to construct an estimation of based on this discrete information. In reality, for an insurance company, even though the relevant probability characteristics are unknown, it is easy to obtain the data sets of surplus levels, the sizes of claims and premiums, and the number of claims and premiums through observation.
Assume that we can observe the surplus process over a long time interval . Let be a fixed interobservation interval. Furthermore, w.l.o.g. we assume is an integer denoted as .
Suppose that the insurer can get the following data sets:(1)Data set of surplus levels:where is the observed surplus level at time .(2)Data set of claim numbers and claim sizes:where is the total claim number up to time .(3)Data set of premium numbers and claim sizes:where is the total premium number up to time .
Next, we study how to estimate the Fourier transform based on the above data sets. To estimate , by (17), (20), and (21) we should first estimate the following characteristics: First, we can estimate by the empirical characteristic function Similarly, can be estimated by Next, for the function , we have Similarly, Then, and can be, respectively, estimated by According to the property of Poisson distribution, and can be estimated by It is easily seen that Since the premium size follows exponential distribution with parameter , we have ; then we can estimate by It is also easily seen that . The estimator of , denoted as , is defined to be the positive root of the following equation (in ): Since as , we set as . Furthermore, we can estimate , respectively, by .
Remark 3. Let It is clear that , , and if . In addition, since and , it follows from the strong law of large numbers that, for any , which shows that the probability that has a unique positive root tends to one as . Thus we conclude that is located in the interval with probability tending to one as .
Proposition 4. Suppose that Condition 1 holds true, then we have .
Proof. By Remark 1, is an increasing function and has unique nonnegative root . Then for any , we have . In addition, by Remark 3, is nondecreasing on event and . Also, we find that, for any , . Thus it follows from Lemma 5.10 in Van der Vaart  that .
Once we have obtained the estimation of the above characteristics, by (17), (20), and (21), the estimation of Fourier transform , denoted as , can be defined bywhere Finally, replacing in (28) by its estimation , the Gerber-Shiu function can be estimated by
4. Asymptotic Properties
In this section, we study the asymptotic properties of the estimation . For any function , its -norm is defined by . Throughout this section, represents a positive generic constant that may take different values at different steps. For two nonnegative functions and with common domain , means that uniformly for every . In addition, we define It is easy to see that
For reader’s convenience, we introduce some definitions in empirical process theory, which are used to study the asymptotic properties. For any measurable function , its -norm is defined by . Given two functions and , the bracket is the set of all functions with . An -bracket in is a bracket with . For a class , the bracketing number is the minimum number of -brackets needed to cover . For , the bracketing integral is defined by .
We shall use the -norm to study the asymptotic properties of the estimator.
Put when . By triangle inequality, we havewhere the first term is the bias caused by Fourier-cosine series approximation, and the second term is the bias caused by statistical estimation.
For the bias , by the similar arguments in Zhang  we obtain the following result.
Next, we study the error . For the estimator , we derive the following result.
Theorem 6. Suppose that Condition 1 holds and , then we have
Proof. By the mean value theorem, we have where is a random number between and . Since , we obtain where Since , and , it is easily seen that .
In addition, we introduce the following set: Since and , we have Furthermore, As a result, since and , we derive that .
The following two theorems give the uniform convergence rates of and .
Theorem 7. Suppose that Condition 1 holds, , and . Then for large and , we have
Proof. By (20) and (44),where We first study . where For , we have thendue to and .
Now we introduce the following two classes of real-valued functions: Then we haveWe only study the convergence rate of the first term , since the second term follows similarly.
For any real-valued function , we have which implies that is contained in the single bracket . For two functions , where , the mean value theorem gives where is a number between and . Under the condition , it follows from Example 19.7 in Van der Vaart  that, for any , there exists a constant such that the bracket number for satisfies As a result, for every , the bracketing integral Furthermore, by Corollary 19.35 in Van der Vaart  we have then Therefore,Similarly,Combining (66), (73), and (74), we obtainAs a result, (64) and (75) giveNext, we consider . We havewhere For , by the mean value theorem we have