In this paper, we consider that the financial surplus of an insurance company evolves as a compound Poisson risk model perturbed by a Wiener process

U t = u + c t − ∑ i = 1 N t     X i + σ W ( t ) ,   t ≥ 0 (1)

where u ≥ 0 is the initial capital and c > 0 is the constant premium per time. The aggregate claims ∑ i = 1 N t     X i follows a compound Poisson process, where the number of claims { N t } t ≥ 0 is an homogeneous Poisson process with intensity λ > 0 , and the individual claim sizes { X i } i ≥ 1 is a sequence of positive i.i.d. random variables generated by a generic variable X with density f X and mean μ . Finally, { W ( t ) } t ≥ 0 is a standard Brownian motion with W ( 0 ) = 0 , and σ > 0 is the diffusion volatility parameter. We suppose that { N t } , { X i } and { W ( t ) } are mutually independent. In this paper, we shall assume throughout the safety loading condition c > λ μ , so that ruin is an uncertain event.

The ruin time is defined by τ = inf { t ≥ 0 : U ( t ) < 0 } with the convention τ = ∞ if U ( t ) ≥ 0 for all t ≥ 0 . Problems related to ruin are a hot topic in risk theory literature. Starting with Cramér and Lundberg’s fundamental collective insurance risk model which was established at the turn of the century [1] [2] , actuarial scholars are still investigating into specific examples of it or tweaking some of its aspects to make it more useful. [3] proposed three equivalent equations for ruin probability in a Cramér Lundberg model, and the solutions were means of inverse Laplace transform. To extend the ruin problem, [4] was the first to propose the joint distribution of the time to ruin, the surplus before ruin, and the deficit at ruin, named the Gerber-Shiu function.

ϕ ( u ) = E [ e − δ τ w ( U ( τ − ) , | U ( τ ) | ) ↿ { τ < ∞ } | U 0 = u ] ,   u ≥ 0 (2)

where w : [ 0, ∞ ) ∗ [ 0, ∞ ) ↦ [ 0, ∞ ) is a measurable penalty function of the surplus prior to ruin and the deficit at ruin, δ ≥ 0 represents the force of interest and ↿ { A } is the indicator function of the event A. Set w 0 = w ( 0,0 ) . The Gerber-Shiu function is a powerful tool for studying ruin related problems. To make it simple, when w ≡ 1 , ϕ becomes the ruin probability when δ = 0 or the Laplace transform of the time to ruin when δ > 0 . For w ( x , y ) ≡ y , ϕ becomes the expected discounted deficit at ruin when ruin is caused by a claim. Finally, for w ( x , y ) ≡ x + y , ϕ becomes the expected discounted claim size causing ruin. Over the last decades, Gerber-Shiu has drawn the attention of numerous actuarial scholars in various risk models. [5] [6] [7] [8] [9]

The Wiener-Poisson risk model was initially proposed in the literature of actuarial science by [10] to extend the classical compound Poisson risk model, where the diffusion perturbation is used to describe the uncertainty of the premium income and aggregate claims. Since then, many scholars have made substantial contributions to this model. For instance, [11] showed that the probability of ruin satisfied certain renewal equations; [12] studied the ruin problems; [13] derived and solved boundary conditions for the n-th moment of the discounted dividend payments; [14] investigated some explicit solutions of the expected discounted penalty function; [15] considered a model with two-sided jumps. For further investigation, we refer the interested readers to [16] [17] and [18] . The explicit formula of the Gerber-Shiu function can be determined when it is assumed that the Poisson intensity, claim size density and diffusion volatility parameter are known. However, insurance companies do not have these probability distributions, instead they have the data information on the surplus flow levels, individual claim sizes and claim numbers. So the big challenge is to figure out how to estimate the Gerber-Shiu function based on the data information.

Recently, non-parametric estimation of risk measures has emerged as a hot topic in risk theory. For the classical model, [19] proposed a semi parametric estimator of the ruin probability, [20] constructed a non parametric estimator for ruin probability by Fourier inversion and kernel density estimation method, [21] estimated the finite time ruin by double Fourier transform. For the Wiener-Poisson risk model, [22] presented the estimator of the Gerber-Shiu function based on a regularized inversion of the Laplace transform, [23] presented an alternative method, the Fourier sinc series expansion to ameliorate the regularized inversion of Laplace transform, [24] proposed a new method for the estimation of the Gerber-Shiu function, Laguerre series expansion method which is not based on the Fourier transform and the Laplace transform.

In this paper, we implement the Fourier cosine series expansion to estimate the Gerber-Shiu function in the risk model (1). The Fourier cosine series expansion has some applications in pricing financial products and it is also named the COS method in the literature. In the field of risk theory, [25] approximated the Gerber-Shiu function under the Levy subordinator model, [26] studied the density of the time to ruin in the classical compound Poisson risk model and [27] estimated the expected discounted penalty function in the Levy risk model based on the Fourier cosine series expansion.

The remainder of this paper is organized as follows. In Section 2, we introduce the Fourier cosine series expansion. In Section 3, we show how to construct our estimator based on the COS method. The convergence rate of the estimator is derived in Section 4. Finally, some numerical simulations are presented in Section 5 to show that the estimator performs well when the sample size is finite and Section 6 is the conclusion.

This section presents some known results of the COS method. For an integrable function g with a finite support [ a 1 , a 2 ] , we have the following Fourier Cosine series expansion,

g ( x ) = ∑ k = 0 ∞   '   A g , k cos ( k π x − a 1 a 2 − a 1 ) (3)

where ∑   ' indicates that the first term in the summation is weighted by one-half, and the cosine coefficients are given by

A g , k = 2 a 2 − a 1 R e { ∫ a 1 a 2     g ( x ) exp ( i k π x − a 1 a 2 − a 1 ) d x } (4)

Let define F g ( s ) = ∫ e i s x g ( x ) d x as the Fourier transform of g. Assume that

∫ a 1 a 2     e i s x g ( x ) d x = ∫ ℝ     e i s x g ( x ) d x = F g ( s ) (5)

Then

A g , k ≈ B g , k : = 2 a 2 − a 1 R e { F g ( k π a 2 − a 1 ) exp ( − i k π a 1 a 2 − a 1 ) } (6)

Finally, the approximation of our function is given by

g ( x ) ≈ g ˜ ( x ) : = ∑ k = 0 K   '   B g , k cos ( k π x − a 1 a 2 − a 1 ) (7)

The approximation error for the COS method is provided in [28] by the following Lemma.

Lemma 1. For the real-value integrable function g supported on ( 0, ∞ ] , suppose that | g ′ ( 0 + ) | < ∞ , | g ′ ( a ) | < ∞ and ∫ 0 ∞ | g ″ ( y ) d y | < ∞ . Then, for some positive constants C 1 and C 2 , we have

sup x ∈ [ 0, a ] | g ( x ) − g ˜ ( x ) | ≤ C 1 a K − 1 + C 2 K a − 1 ∫ a ∞ | g ( y ) | d y . (8)

Remark. Lemma 8 shows that COS method has two types of approximation errors depending on the parameter a which is in fact an integration domain truncation parameter in the COS method and the parameter K which is used in the approximation of the of the COS coefficients by the Fourier transform F g . The first error term a K − 1 means that larger K yields better approximation, but large a may slowdown the convergence rate. The second error term C 2 K a − 1 ∫ a ∞ | g ( y ) | d y means that larger a can result in more accurate approximation of the COS coefficient. However, this error term is increasing with respect to K since larger K means more COS coefficients have to be approximated.

In this section, we shall first approximate the expected discounted penalty function by the COS method. Afterward, we shall replace the estimate values in the approximate function to have our estimator.

In this section, we derive the consistency properties of our estimators when the observation interval [ 0, T ] is very large. First, we know that λ ^ − λ = O P ( T − 1 2 ) , σ ^ 2 − σ 2 = O P ( T − 1 2 )

Lemma 2. Suppose that c ≥ λ E ( X ) , E ( X 2 ) < ∞ and a = o ( K ) and E ( ∫ 0 X     w ( X − x ) d x ) ≤ ∞ , E ( ∫ 0 X     x w ( X − x ) d x ) 2 ≤ ∞ , then we have ρ ^ − ρ = O p ( T − 1 / 2 ) and

sup 0 ≤ u ≤ a | ϕ ^ ( u ) − ϕ ˜ ( u ) | = O p ( ( K / a ) log ( K / a ) T )

Proof. The convergence of ρ ^ − ρ = O p ( T − 1 / 2 ) is well known, see [30] .

For the Gerber-Shiu function, we have that

sup 0 ≤ u ≤ a | ϕ ^ ( u ) − ϕ ˜ ( u ) | = sup 0 ≤ u ≤ a | ∑ k = 0 K   '   B ^ ϕ , k cos ( k π u a ) − ∑ k = 0 K   '   B ϕ , k cos ( k π u a ) | ≤ sup 0 ≤ u ≤ a | ∑ k = 0 K   ' ( B ^ ϕ , k − B ϕ , k ) | ≤ sup 0 ≤ u ≤ a | ∑ k = 0 K   ' 2 a [ R e { F ϕ ^ ( k π a ) } − R e { F ϕ ( k π a ) } ] | ≤ sup s ∈ S 2 a ∑ k = 0 K   '   1 | F ϕ ^ ( s ) − F ϕ ( s ) | = 2 ( K + 1 ) a sup s ∈ S | F ϕ ^ ( s ) − F ϕ ( s ) | (16)

where

sup s ∈ S | F ϕ ^ ( s ) − F ϕ ( s ) | ≤ sup s ∈ S | σ ^ 2 2 w 0 ( − i s − ρ ^ ) ψ ^ U ( − i s ) − δ − σ 2 2 w 0 ( − i s − ρ ) ψ U ( − i s ) − δ |     + sup s ∈ S | λ [ L ω ^ ( ρ ^ ) − F ω ^ ( s ) ] ψ ^ U ( − i s ) − δ − λ [ L ω ( ρ ) − F ω ( s ) ] ψ U ( − i s ) − δ | = M 1 ( s ) + M 2 ( s )

It is shown in [31] that M 1 ( s ) = M 2 ( s ) = O p ( ( K / a ) log ( K / a ) T ) Hence,

sup s ∈ S | F ϕ ^ ( s ) − F ϕ ( s ) | = O p ( ( K / a ) log ( K / a ) T ) (17)

Then by formula 16 and 17, we can conclude that

sup 0 ≤ u ≤ a | ϕ ^ ( u ) − ϕ ˜ ( u ) | = O p ( ( K / a ) log ( K / a ) T )

In this section, we present some numerical results to show the effectiveness of our method. All computations are done in MATLAB on a EliteBook, with Intel(R) Core(TM) i5-6300U CPU@2.40GHz 2.50GHz and a RAM of 8GB. Throughout this section, we set c = 8 , λ = 5 , δ = 0.1 and consider claim size density functions, the exponential EXP(1): f X ( x ) = e − x , x > 0 and the Erlang (2, 2): f X ( x ) = 4 x e − 2 x , x > 0 . Then the true value of the Gerber-Shiu function can be found in [23] . Now for these two claim size density functions, closed form of Fourier transforms exist so that the COS method approximation of ϕ ˜ ( u ) can be computed. When using Fourier Cosine Series expansion method to approximate ϕ ( u ) , we apply the cumulant method given in [32] to determine the parameter a. We take L = 10 , K = 2 7 to provide the benchmark.

First of all, we test the effectiveness of the Fourier Cosine method for approximating ϕ ( u ) . In Table 1 and Table 2, we present some average relative errors and average absolute errors for ϕ ˜ ( u ) respectively for exponential and Erlang distributions which are calculated by the following

Averagerelativeerrors : 1 # U ∑ u ∈ U | ϕ ˜ ( u ) − ϕ ( u ) | ϕ ( u )

Averageabsoluteerrors : 1 # U ∑ u ∈ U | ϕ ˜ ( u ) − ϕ ( u ) |

Here, we take U = { 1 , 2 , ⋯ , 15 } , since when U > 15 , ϕ ( u ) is very small. We consider the truncation parameter K = 2 q , q = 4 , 5 , 6 , 7 . It can be observed that in each column of Table 1 and Table 2, both average relative errors and the average absolute errors are decreasing w.r.t. q, which implies that large truncation parameter can reduce the bias under the model setting. In Table 1 and Table 2, we find that the average relative errors are larger than the average absolute errors, which is due to that ϕ ( u ) is always smaller than 1, and in particular, it is close to zero for large initial surplus.

Next we test the performance of our estimator ϕ ^ ( u ) . We fixe K = 2 7 . As for the observation interval [ 0, T ] , We shall take T = 2 p for p = 0 , 1 , 2 , 3 , 4 , 5 . For the COS parameter a, we also apply the cumulant method given in [32] . We take a ^ = k 1 ^ + L k 2 ^ + k 4 ^ where k j ^ = d j d j log F f ^ ( − i s ) | s = 0 . We repeat 300 simulations, and compute empirical average relative errors and average absolute errors for ϕ ^ ( u ) . Which are defined by

Averagerelativeerrors : 1 # U ∑ u ∈ U 1 300 ∑ j = 1 300 | ϕ ^ j ( u ) − ϕ ( u ) | ϕ ( u )

Averageabsoluteerrors : 1 # U ∑ u ∈ U 1 300 ∑ j = 1 300 | ϕ ^ j ( u ) − ϕ ( u ) |

where ϕ ^ j ( u ) denote the j^th simulation values of ϕ ^ ( u ) . In Table 3 and Table 4, we present the empirical estimation errors for ϕ ^ ( u ) respectively for exponential and Erlang distributions. As expected, both the empirical average errors and empirical absolute errors are decreasing w.r.t. p which is due to that, as p increases (or equivalently T increases), more sample can be used to estimate ϕ ^ ( u ) . Again, we observe that the average relative errors are larger than the average absolute errors.

Finally, we plot 300 consecutive estimators (green curves) on the same picture together with the true curve (red curve) to illustrate variability bands and show the stability of the procedures. In Figure 1, we observe that the beams of estimators are much closed to the true curves. In particular, it follows that the variances for estimating ϕ ( u ) are very small for a large observation interval. We find that ϕ ( u ) is a decreasing function of the initial surplus u, which means that the ruin is more likely to happen when u is small. At the same time, we can also observe that as T increases, the estimator tends to be stable and converges to ϕ ( u ) . We plot the corresponding curves. Again, we can observe that the estimator becomes better as T becomes larger.

In this paper, we have proposed a new estimator of the expected discounted penalty function in the perturbed compound Poisson risk model. Our estimator is based on the COS method. The COS coefficient is easily derived based on the Fourier transform of the Gerber-Shiu function which is useful when approximating the function by the COS method. Suppose we have the surplus flow level, the claim number, and the claim sizes over a long-term interval. Then, we construct our estimator based on the COS approximation we have derived, by replacing the different functions with their estimates. We have derived theoretical errors and presented some simulation results to show the effectiveness of our estimator. We have shown that our estimator has an accurate convergence rate. Since the COS method does not utilize the Fast Fourier transform algorithm, our estimator is easier to compute compared to the Fourier sinc method. Since this method and the Fourier sinc method use the Fourier transform of the Gerber-Shiu, the comparison of the two methods could be future research.

The authors would like to thank the Pan African University Institute of Basic Sciences Technology and Innovation (PAUSTI) for their financial assistance.

The authors declare that there is no conflict of interests.

Kenmoe, M.R.N., Aduda, J.A. and Le Doux, M.B.M. (2023) Estimating the Gerber-Shiu Function by Fourier Cosine Series Expansion in the Wiener-Poisson Risk Model. Journal of Mathematical Finance, 13, 271-283. https://doi.org/10.4236/jmf.2023.133017