Research Article | Open Access
Estimating the Gerber-Shiu Expected Discounted Penalty Function for Lévy Risk Model
This paper studies the statistical estimation of the Gerber-Shiu discounted penalty functions in a general spectrally negative Lévy risk model. Suppose that the claims process and the surplus process can be observed at a sequence of discrete time points. Using the observed data, the Gerber-Shiu functions are estimated by the Laguerre series expansion method. Consistent properties are studied under the large sample setting, and simulation results are also presented when the sample size is finite.
In this paper, the cash flow of an insurance company is described by the following spectrally negative Lévy process:Here, denotes the initial reserve and is the rate of premium. The process is a standard Brownian motion and is a constant. The claims process is a Lévy subordinator, whose Laplace exponent is defined byHere, is a Lévy density on satisfying the usual condition and the additional condition . Note that the condition ensures that the expected aggregate claims are finite over any finite time interval. Finally, let us suppose that the processes and are mutually independent.
In insurance risk theory, one object of interest is the ruin time , which is defined bySet if for all . In order to avoid ruin is a deterministic event, it is assumed that the following net profit condition holds true throughout this paper.
Condition 1. The premium rate .
In this paper, we use the Gerber-Shiu expected discounted penalty function to discuss the ruin problems. This function is defined bywhere is a constant interest force, is a nonnegative measurable penalty function of the deficit at ruin, and is an indicator function of an event . Note that ruin can be caused by oscillation or claims. Then we can decompose the Gerber-Shiu function as follows:whereis the Laplace transform of ruin time when ruin is caused by oscillation andis the expected discounted penalty function when ruin is due to claims.
In insurance risk theory, the Gerber-Shiu function introduced by Gerber and Shiu  is a powerful tool for solving ruin problems. It should be emphasized that most of the existing literatures mainly pursue explicit formulae of Gerber-Shiu functions under various models. For example, Gerber and Shiu  studied the classical risk model; Li and Garrido [2, 3] discussed the Sparre Andersen risk models. In risk theory, the Lévy process is often used to model the surplus process of an insurance company, and a large number of results to Gerber-Shiu function have been made by researchers. Garrido and Morales  used Laplace transform to investigate the classical Gerber-Shiu function. Biffis and Morales  generalized the Gerber-Shiu function to path-dependent penalties. Chau et al.  used the Fourier-cosine method to evaluate the Gerber-Shiu function. For more studies on Gerber-Shiu function, the interested readers are referred to Yin and Wang [7, 8], Asmussen and Albrecher , Chi , Wang et al. , Chi and Lin , Zhao and Yin [13, 14], Shen et al. , Yu [16–18], Yin and Yuen [19, 20], Zhao and Yao , Zheng et al. , Huang et al. , Li et al. , Zhang et al. , Yu et al. , Zeng et al. [27, 28], Li et al. , and Dong et al. .
Contributing to Gerber-Shiu function mentioned above, it is commonly assumed that the probabilistic law of the surplus process is known, so that some analytic formulae can be derived. However, from the practical point of view, the insurance company does not know the probabilistic law, instead the surplus data and claim sizes data are often known. Hence, it is of importance to develop some methods for estimating the Gerber-Shiu type risk measures from the past data.
Over the past decade, the importance of statistical estimation of Gerber-Shiu function has advanced rapidly. As a special type of Gerber-Shiu functions, the ruin probability is estimated by Mnatsakanov et al. , Masiello , and Zhang et al.  under the classical compound Poisson risk model. Further, Zhang  and Yang et al.  estimated the finite ruin probability by double Fourier transform. For the general Gerber-Shiu function, Shimizu  estimated it by Laplace inversion in the compound Poisson insurance risk model. Zhang [37, 38] proposed an estimator by Fourier-sinc and Fourier-cosine series expansion. Zhang and Su [39, 40] and Su et al.  proposed a more efficient estimator by Laguerre series expansion method. In addition, the estimation of ruin probability and Gerber-Shiu function in the Lévy risk model has also attracted the attention of scholars. For example, Shimizu  estimated the Gerber-Shiu function in a general spectrally negative Lévy process. Zhang and Yang [43, 44] estimated the ruin probability in a pure jump Lévy risk model. Shimizu and Zhang  estimated the Gerber-Shiu function in the pure jump Lévy risk model. Moreover, the study is a useful tool not only for the estimation of ruin probability by invasive exotic species on habitat and natural systems; see Wang and Yin , Yuen and Yin , Dong and Yin , Yin et al. , Wang et al. , Zhao et al. , Pavone et al. , Zhou et al. , Costa and Pavone [54, 55], and Yin , but also for explaining the dynamics of the disappearance of the plants of the past and changing of biodiversity and wavelet analysis, see Costa et al. , Pulvirenti et al. [58, 59], Lemarié and Meyer , and Daubechies .
The remainder of this paper is organized as follows. In Section 2, we provide some known results on Gerber-Shiu function, which are useful for constructing the estimator. In Section 3, we give an estimator by the Laguerre series expansion method, where both the surplus data and the aggregate claims data are used. Some consistent properties of the estimator are investigated in Section 4. Finally, in Section 5, we provide numerical examples to illustrate the efficiency of the method.
2. Preliminaries on Gerber-Shiu Functions
2.1. The Laguerre Basis
In this paper, we use and to denote the spaces of absolutely integrable functions and square integrable functions on the positive half line, respectively. For any , the inner product and the -norm are defined by
For the Laguerre functions, they are defined bywhere are the Laguerre polynomials given by
It is known that is an orthonormal basis of such thatHence, each can be developed on the Laguerre basis, i.e.,where .
The following properties of Laguerre functions are useful, which can be found in Abrumowitz and Stegun .(1)(2)
2.2. Review on Gerber-Shiu Function
In this subsection, we present some necessary results on the Gerber-Shiu function, which are useful for constructing the estimator. First, we consider the root of the following equation (in s):It is known that (13) has a positive root, say , and we have as . Set .
By Corollary 4.1 in Biffis and Morales  we havewhere denotes the convolution operator, , andwhere . Furthermore, from the deduction in Biffis and Morales  one easily obtainsandIt follows from (17) thati.e., satisfies the following renewal equation:Similarly, we can obtainIt should be emphasized that (20) and (21) are both defective renewal equations, since under Condition 1.
For purpose of employing the Laguerre series expansion, we suppose the following condition.
Condition 2. The functions are all square integrable.
Under Condition 2, we have the following Laguerre series expansion formulae: and
The above linear systems can further be rewritten in matrix form as follows:where , , , , and the elements in are given by Furthermore, put , , , , and . Then we can truncate the matrix and vectors in (26) to obtainNote that the matrix is a lower triangular Toeplitz matrix, and all the diagonal elements are positive since . Hence, is nonsingular and explicitly invertible. Solving equations in (28) we get Finally, we can approximate the Gerber-Shiu function by
3. Estimating the Gerber-Shiu Function
Suppose that we can observe the surplus process and the claims process at a sequence of discrete time points, so that the following samples are available:where is a sampling step. Further note that the initial values and and the premium rate are known, but the diffusion volatility parameter and Lvy density are both known. It is convenient to use the following samples:
In this paper, the observation is based on high frequency in a long term, which means that the following condition holds true.
Condition 3. The sampling interval satisfies
We shall use the notation , , provided that such moment is finite. That is to say, we shall use the following condition.
Condition 4 (). , .
The following two conditions are also useful for studying the consistent properties.
Condition 5. The Lvy density is continuous on , and for some ,
Condition 6. For some ,
Now we study how to estimate the Gerber-Shiu function. First, we estimate and . For , we can estimate it bywhich is an unbiased estimator sinceFurthermore, it is easily seen that
For the Lvy density , it is known that under Conditions 3 we have converges weakly to the measure , where denotes the Dirac measure at . Using this result and noting that , we can estimate the Laplace exponent byRecalling that is the positive root of equation, then the estimator of , say , is defined to be the positive root of the following estimating equation:We set as .
Proof. This can be proved using the same arguments as in Shimizu .
Let us consider how to estimate , , and . First, for we havewhich yields the following estimator:where . Next, by changing the order of integrals we havewhereThen we can estimate byFinally, for we havewhere Then we can estimate by
Note that can be explicitly calculated as follows:As for the function , it can also be explicitly calculated for some special cases of penalty function. For example, if , we have
For the vectors , , and , we estimate them by , , and , respectively. As for the matrix , we estimate it by , where the elements in are replaced by . Finally, we estimate the Gerber-Shiu functions by where
4. Consistent Properties
In this section, we investigate the asymptotic properties of the proposed estimators as . In the sequel, we denote by a generic positive constant that may vary at different steps. Our goal is to study the errors of and . By Pythagoras principle, we have and Note that and are statistical errors due to estimating the Laguerre coefficients and and are biases due to series truncation.
First, we consider the biases. To this end, we introduce the Sobolev-Laguerre space (Bongioanni and Torrea ), which is defined bywhere . It follows from Zhang and Su  that, under conditions , , we have
Next, we discuss the statistical errors. For a vector , we denote its 2-norm by . For a square matrix , its spectral norm is defined by , where is the largest eigenvalue of . Furthermore, we denote bythe Frobenius norm. Obviously, we have , and for any two square matrices and with the same dimension,
Lemma 8. Suppose that Condition 1 holds true. Then for all , we have
Proof. Note that the infinite dimensional lower triangular Toeplitz matrix is generated by the sequence , whereFurthermore, by Lemma C.1 in Comte et al.  we know that are Fourier coefficients of the functionFor the function , we havewhere is the complex unit circle. Finally, by Lemma A.1 in Zhang and Su  we obtainThis completes the proof.
Lemma 9. Under Condition 2, for each we have
Proof. The above results hold true sinceandThis completes the proof.
Proof. First, we haveUsing the mean value theorem we obtain , where is a random number between and . Then we haveNext, since and Lemma 7, then we haveand converges to in probability. Hence, (72) leads to (70).
Proof. First, we haveSince and , we haveNext, we can writewhere Then using Jensen inequality we haveFor , using mean value theory we obtainwhere is a random number between and . Then using the upper bound we obtainSince and in probability, we have in probability, leading to . Using Markov inequality we can obtain under Condition 4. Hence, by Lemma 7 we obtainwhich yieldsFor , using mean value theory we havewhere is a random number between and . Then we have