1.
Introduction
The insurance company is a common financial institution in our real life. Its profit mainly comes from two aspects: premium income and investment income, and the risks it needs to face mainly include: compensation risk and investment risk. In the past decade, more and more scholars have focused on building appropriate risk models (r.m.) to describe the various situations that insurance companies may face[1,2,3]. At first, the r.m. studied by the researchers was a classical risk model that only considered a company's claims as a negative jump. For example, Zhang et al.[4] studied a new method to estimate the Gerber-Shiu discount penalty function (p.f.) under the classical r.m., and Peng et al.[5] studied a r.m. of dividend payment with perturbations. But in reality, an insurance company's random returns should also be taken into account. To better fit the actual situation, Boucheire et al.[6] first proposed the two-sided jumps r.m., which is used to extend the r.m. of a single jump. Here, it is considered that the company's revenue is random, which is also a random variable, then the random revenue is a non-negative jump, and a negative jump is a claim. Since then, this model has been paid much attention by many researchers. For example, E.C.K. Cheung[7] studied a renewal model with continuous expenses and bidirectional jumps, where the amplitude of the jumps and the time intervals of arrival time are random. From this, E.C.K. Cheung obtained the updated equation of the discounted penalty funtion (e.d.p.f.) with defects. Zhang[8] considered the problem of e.d.p.f. for a two-sided jumps r.m. with dividend payout and obtained some explicit expressions. Wang et al. [9] considered the investment r.m. under the bilateral jump and tried to obtain the maximum surplus through the appropriate investment proportion. Xu et al.[10] studied the problem of ruin probability under bilateral jumps with random observations. For more research on two-side jump r.m., we can refer to references [11,12,13,14,15].
Subsequently, some scholars put forward the dividend barrier strategy, that is, they set a threshold value b>0, and pay dividends to shareholders when the company's earnings are greater than b. The strategy was first proposed by De Finetti. Then, Gerber et al.[16] studied the threshold dividend strategy, and Yin et al.[17] and Cossette et al.[18] put forward the horizontal barrier strategy. The multi-tier dividend strategy can be learned from Xie and Zou[19]. To make the r.m. more realistic, some scholars have added dividend barriers to the study of bilateral jump risk models. For example, Bo et al. [20] studied the Lévy model with bilateral jumps under the dividend barrier strategy and Chen et al.[21] studied the dividend payment and the reward and e.d.p.f. of the dividend strategy with a threshold under the compound Poisson (c.p.) model. The integral differential equation (IDE) is derived under the boundary conditions and the approximate solution (a.s.) is approximated by the sinc numerical method. When studying the c.p. model with proportional investment, Chen and Ou[21] added the dividend with threshold value. Inspired by the above research, we propose a bilateral jumps model with a threshold strategy under random observation.
We introduce our work in the following parts. In the second section, we construct the two-sided jumps risk model with investment interest under random observation, and the observational intervals obey a same exponential distribution. In the third section, we obtain the IDE of the expected discounted dividend payment (e.d.d.p.) function. To solve this equation, in the fourth section, we introduce an excellent numerical method to the solution of the IDEs and get the upper boundary of the error between the a.s. and the real solution. This numerical method is called the sinc numerical method. In the last section, we give some numerical examples to explore the effects of the included parameters on the e.d.d.p..
2.
The model
According to the previous research on the bilateral jump r.m., we define
where u0 represents the company's initial surplus on the account and u0 is greater than zero. In addition, {U(t)}t≥0 stands for the surplus process, while c represents the premium rate paid by the insured, so obviously c>0. Here the two stochastic processes S1t=M1(t)∑i=1Yi and S2t=M2(t)∑i=1Zi, are both c.p. processes, representing the total claims and returns until time t, respectively, and M1(t) and M2(t) are homogeneous Poisson processes with parameters λ1>0 and λ2>0. The claim size is determined by the cumulative distribution function (c.d.f.) FY(⋅) and the probability density function (p.d.f.) fY(⋅) of independent and identically distributed (i.i.d.) positive random variables (r.v.s) {Yi}∞i=1. The random return is given by the c.d.f. FZ(⋅) and the p.d.f. fZ(⋅) of the positive r.v.s {Zi}∞i=1. Define M1(t)=sup{j:S11+S12+⋯+S1j≤t} and M2(t)=sup{j:S21+S22+⋯+S2j≤t}, where inter-claim times {S1j}∞j=1 and inter-return times {S2j}∞j=1 follow the exponential distribution of intensity λ1 and λ2, respectively.
In reality, to protect the interests of the manager and the insured, the manager needs to have a reasonable plan for the surplus funds. Under normal circumstances, insurance companies generally take a portfolio of risk and risk-free investments for surplus funds[22]. As investment income becomes a larger share of insurance company's total revenue, we need to take into account investment ratio factors. Therefore, suppose that the manager uses part of the surplus funds for risk-free investment and the other part for risk investment. In that way, risk-free investment {Rt}t≥0 satisfies
where r is the interest rate on a risk-free asset, so obviously r should be greater than zero. Risk asset {Qt}t≥0 is defined as
where {Wt,t≥0} is a standard Brownian motion, and σ and a represent the volatility and expected rate of return of risk assets, respectively, both of which are greater than zero. So the risk asset process {Qt}t≥0 satisfies
Let q∈(0,1) represent the proportion of the insurance company's surplus invested in risky assets, and then 1−q represents the proportion invested in risk-free assets. So U(t) satisfies
where ξ=(a+12σ2)q+(1−q)r, U(t−) is the left limit of U(t) at t, and the loading condition to ensure that the formula holds is c+λ2E[Z1]>λ1E[Y1].
We consider the dividend problems of the above model under the dividend strategy: when U(t) is greater than threshold b, dividends are paid consecutively in α, where α is constant and greater than zero; when U(t) is greater than zero and less than b>0, no dividends are paid; and when U(t) is less than zero, bankruptcy occurs at this time (but, in practice, the state of this moment may not be observed and therefore is still meaningful in the short term). Combined with Eq (2.5), the surplus process with threshold b is represented by {Ub(t),t≥0}, and {Ub(t),t≥0} satisfies
where V(Q,R,q,t)=(1−q)dRtRt+qdQtQt.
Let the cumulative dividend paid until the ruin time t be D(t), and Tb=inf{t:Ub(t)≤0} is the ruin time. The present value of accumulated dividends before the ruin time Tb is Du,b, so
where δ is the interest force and is greater than zero, and I(⋅) stands for the indicator function. According to the above definition, it is not difficult to derive 0<Du,b<αδ, which provides convenience for the subsequent derivation of the boundary of the IDEs. For u∈R, the expectation of Du,b is represented by
It should be emphasized that the surplus can be observed randomly in this paper. In practice, however, the executive director of an insurance company randomly reviews the balance of the company's books to determine whether dividends are being paid or whether it is ruined (e.g., [23,24,25]). Suppose {Tj}∞j=0 is a series of discrete time points of the moments of observing surplus, where Tj is the jth observation time. In addition, we stipulate that T0=0 and Tj∗ is the time when the company goes to ruin, where j∗=inf{j≥1:M(j)≤0}. Suppose {Sj}∞j=0 is an i.i.d sequence, where Sj=Tj−Tj−1 is the jth observation interval and Sj are positive r.v.s, which are subject to an exponential distribution of intensity γ>0. Suppose {Yi}∞i=1, {Zi}∞i=1, {M1(t)}t≥0, {M2(t)}t≥0, {Wt,t≥0}, and {Sj}∞j=0 are independent of each other. Let the surplus level of the jth observation be M(j)=U(Tj), and combine (2.5) to obtain
3.
IDEs of V(u;b)
In this section, our work is to give the IDEs of e.d.d.p. V(u;b). Before we begin, we need to discuss the range of values of u, considering a time interval (0,dt]. If a claim occurred before observation, it is possible that Ub(t)<0 was not observed. Therefore, the range of values of u extends to the entire field of real numbers. In addition, it is not difficult to find that for different initial surplus u, V(u;b) behaves differently. For convenience, let us set
Here are the following conclusions.
Theorem 3.1. For u∈(−∞,0], V1(u;b) satisfies
For u∈(0,b], V2(u;b) satisfies
and for u∈(b,∞), V3(u;b) satisfies
The following boundary conditions are satisfied
Proof. Consider an infinitesimal interval (0, dt], and discuss whether claims and benefits occur or not. The cumulative distribution function of Yi and Zi is continuous. For u∈(−∞,0],
and for u∈(0,b],
amd for u∈(b,∞),
where
According to the Itô formula, we get
where
and o(dt) stands for the infinitesimal of higher order dt.
Substitute Eqs (3.11)–(3.14) into Eqs (3.8)–(3.10), respectively. Divide both sides of the equation by dt and let dt approach zero infinitely. According to the properties of higher order infinitesimals and some careful calculation, we can get the IDEs (3.1)–(3.3).
With further analysis, if the initial surplus Ub<0, the ruin occurs immediately, at which time no dividend is paid; then Tb=0. If 0<Ub<b, then the ruin did not occur and the dividend is always paid at rate α. If Ub>b, then the shares are always paid at rates α−c, so Tb=∞. □
Remark 3.1. Referring to the analysis of Albrecher [26], we can also find that V(u;b) is not differentiable when u=0 in general. Similarly, to fully describe the solution of Theorem 3.1, we also use V1(0−;b)=V2(0+;b) and V1(b−;b)=V2(b+;b), and the boundary conditions (3.4) and (3.5).
4.
Sinc asymptotic analysis
The sinc numerical method was proposed by James H. Wilkinson in the 1950s and developed by Frank Stenger in the 1990s. Frank Stenger summarized his work results in [27], which caused a great response in various fields (e.g., [28,29]). The real solutions to Eqs (3.1)–(3.3) are theoretically difficult to obtain. Therefore, we changed the angle, tried to obtain the a.s. by a numerical method, and then carried out an error analysis. Nowadays, the commonly used numerical methods for solving integral differential equations include the RK-Fehlberg method, the sinc method, the Runge-Kutta method, the Adams method, and so on. The sinc numerical method has high accuracy and good convergence when the sampling interval is small enough, which makes it perform well in high-precision numerical results. At the same time, the sinc method has an adaptive sampling interval. When the sampling interval is small, the sinc method can accurately reflect the details of the original function, to achieve high-precision numerical calculation. When the sampling interval is large, the sinc method can effectively smooth the function and avoid the ringing effect [30] in the interpolation process. Therefore, we also use this numerical method here.
4.1. Approximate solution of V(u;b)
Since the domain of u is the entire real axis, in order to construct approximations on R, we consider conformal mappings. According to Algorithm 1.5.18 of Stenger[27], we define an injective mapping from R→R
where z∈R. Define the grid point zk of sinc as
where k∈Z, h>0. Based on the sinc method, the basis function of z∈Γ on the interval (−∞,∞) is given by the following composite function
Following the steps of the sinc method, we arrange Eqs (3.1)–(3.3) into the following integral differential
By the nature of convolution, Eq (4.2) is rewritten as
According to formulas (3.4) and (3.5), and Definition 1.5.2 in reference[27], we have
where ζ(u)=eϕ(u)=eu, when t1→−∞, t2→∞. Set
and then W(u)∈L˜α,˜β(δ), where L˜α,˜β(δ) is the function space for the sinc approximation over the finite interval (˜α,˜β) (p. 72 in [27]).
When u→−∞, u→∞
Substituting (4.5)–(4.7) into (4.3), by simple calculation, we have
where μ0(u)=(quδ)22, μ1(u)=ξu+c−αI(u>b), μ2(u)=−(δ+λ1+λ2),
When h>0, define the sinc grid point as
Then consulting reference [27], according to Theorem 1.5.13, Theorem 1.5.14, and Theorem 1.5.20, we can get
where A and B are resemble diagonal matrices Λ, with Aij and Bij denoting the elements at (i,j) in A and B, respectively. The approximate value of W(uj) is expressed by Uj.
Substituting (4.13)–(4.15) into Eq (4.8), replacing the integral term on the right side of Eq (4.8) with Eqs (4.13)–(4.15), and replacing u with uk for k=n2,⋯,n1, where uk is the sinc grid point, we have
where
Substituting (4.17)–(4.19) into Eq (4.16), we have
Multiplying Eq (4.20) by h2[ϕ′(uk)]2, we have
Since
formula (4.21) can be turned into
Set I(m)=[δ(m)kj](n2+n1+1)×(n2+n1+1), and m=−1,0,1,2. We rewrite Eq (4.22) as
where
So solving Eq (4.23), we get the expression of the approximate solution (a.s) of (4.5):
The meanings of the symbols mentioned in the above process are shown in Table 1.
4.2. Error analysis
In the previous subsection, we obtained an inexact solution (e.s.) of the IDEs by using the sinc method. Therefore, in this section, we need to analyze the discrepancy between the a.s.s and the actual solutions. According to references [27,31], we find an upper bound of the error. Moreover, in reality, u is non-negative. Therefore, in this subsection, our discussion takes place under the condition u>0. Multiply 1μ0(u) by both sides of Eq (4.8), and we set
so we have
where ~μ1(u)=μ1(u)μ0(u), ~μ1(u)=μ2(u)μ0(u).
Assumption 4.1. Let ~μ1(u)/ζ′, 1/(ζ′)′, and ~μ2(u)/(ζ′)2 be elements of W∞(D), and we are given that G/(ζ′)2∈Lˆα(D) and Eq (4.25) possess a single solution W∈Lˆα(D).
In the above assumption, W∞(D) represents the family of all functions of W(u) that are analytically and uniformly bounded by D, and Lˆα(D)=Lˆα,ˆα(D).
Theorem 4.2. If the aforementioned assumption is true, W represents the e.s. of Eq (4.25), ˜W represents the a.s. of Eq (4.24), and \boldsymbol{U} = (U_{-n_2}, \ \cdots, \ U_{n_1})^{T} represents the e.s. of Eq (4.23). So there is a constant \tilde{c} > 0 , and different from N , such that
Proof. Let
By using the triangle inequality, it is easily obtained that
Based on Theorem 4.4 in [31], there is a constant c^* > 0 , and different from N , that according to Assumption 3.1, W \in \mathscr{L}_{\hat{\alpha}(\mathscr{D})} , and we have
For inequality (4.28), |\mathscr{O}_{N}(u)-\tilde{W}(u)| fulfills
Similar to Theorem 3.8 in [31], if u\in \Gamma , then \sum_{k\in\ \mathbb{Z}}|C(j, h)\circ\zeta(u)|^2 = 1 , and we can obtain
where \textbf{W} = (W_{-n_2}, \ \cdots, \ W_{n_1})^{T} and c^{**} > 0 that is not dependent on N . Let us take \tilde{c} = \rm{max}\{c^{*}, c^{**}\} , and therefore, inequality (4.25) is obtained by formulas (4.27) − (4.31). □
Through formulas (4.4), (4.24), and (4.25), we get
5.
Numerical example
In this subsection, we provide specific numerical examples to demonstrate the effectiveness of the sinc method, and study the effects of investment ratio q and fluctuation parameter \sigma on the expected discounted dividend payout under exponential and lognormal distributions, respectively.
5.1. The exponential distribution
All numerical examples in this section are assumed to be obtained under
and
Then,
and
Formulas (4.8) and (4.11) are converted to
and
Next, we examine how parameters q and \sigma affect V(u; b) . If not specified, the following example parameters are set as follows: \delta = 0.06, \ \tilde{\alpha} = \frac{\pi}{4}, \ \tilde{\beta} = \frac{\pi}{4}, a = 0.6, \ c = 0.3, \ r = 0.05, \ \alpha = 0.2 , d = \frac{\pi}{4}, \ N = 15, \ \lambda_1 = 1, \ \lambda_2 = 2, \ \eta_1 = 3, \ \eta_2 = 1 .
Example 5.1. The effect of the investment ratio q on the e.d.d.p. is considered in the case of the exponential distribution of claims and returns. Set parameter \sigma = 0.2 . As depicted in Figure 1, it becomes evident that as the proportion of surplus invested in risk assets increases, the corresponding fluctuation of V(u; b) also increases. The value of V(u; b) when q changes is presented in Table 2 partially.
Example 5.2. The effect of volatility parameter \sigma on the e.d.d.p. is considered in the case of the exponential distribution of claims and returns. Set parameter q = 0.2 . As depicted in Figure 2, the greater the change of parameter \sigma , the greater the fluctuation of the curve corresponding to V(u; b) . Partial data is presented in Table 3.
As can be seen from Examples 5.1 and 5.2, the impact of two factors on the e.d.d.p. is considered: the proportion of risk investment q and the volatility of risk assets \sigma . First, when a company invests a higher proportion of its surplus in risky assets, the dividend payout is higher, but also more volatile, while the dividend payout is more stable when the investment ratio is lower. This means high risk, high reward, danger, and opportunity. In addition, if the proportion of risk investment is fixed, choosing investment products with more volatile risk assets will bring higher profits, but also bear higher risks. On the contrary, they will earn lower profits and take lower risks. This is in line with reality.
5.2. The lognormal distribution
In this section, it is assumed that f_{Y}(y) and f_{Z}(z) obey a lognormal distribution of parameter (\eta_{3}, \ 2v_{1}^2) and (\eta_{4}, \ 2v_{2}^2) , respectively, where \eta_{3} = \ln y and \eta_{4} = \ln z , and 2v_{1}^2 and 2v_{2}^2 represent the variance, so that f_{Y}(y) and f_{Z}(z) are defined as
Then,
and
Therefore, the formulas (4.8) and (4.11) can be rewritten as:
and
The next example is given in real condition: \delta = 0.06, \ \tilde{\alpha} = \frac{\pi}{4}, \ \tilde{\beta} = \frac{\pi}{4}, \ a = 0.5, \ c = 0.4, \ r = 0.06, \ \alpha = 0.1, \ d = \frac{\pi}{4}, \ N = 10, \ \lambda_1 = 1 , \lambda_2 = 2 , \eta_3 = 3 , \eta_4 = 1 , v_2 = 0.03 , v_1 = 0.03 .
Example 5.3. In the case of a lognormal distribution of claims and returns, let us discuss the effect of investment ratio q on V(u; b) . Set parameter \sigma = 0.2 . It is not difficult to see from Figure 3 that when a company invests more surplus into risk assets, the growth of its expected discounted dividend curve experiences significant fluctuations. Partial data is presented in Table 4.
Example 5.4. In the case of a lognormal distribution of claims and returns, let us discuss the effect of investment ratio \sigma on V(u; b) . Set parameter q = 0.2 . It is not difficult to see from Figure 4 that when the company chooses a product investment with greater risk fluctuation, the growth of its expected discounted dividend curve exhibits substantial variability. Partial data is presented in Table 5.
From Examples 5.3 and 5.4, it can be seen that parameters q and \sigma have different effects on the e.d.d.p. V(u; b) under a lognormal distribution of claims and returns. Other parameters being equal, the expected discounted dividend payout curve fluctuates more when a company invests a larger proportion of its earnings or invests in risky products with a higher freezing rate. It should be noted that when the claim amount and income follow the lognormal distribution, V(u; b) shows a higher sensitivity to the above parameter changes.
6.
Conclusions
We explore a model with two-sided jumps, incorporating random observations and a dividend barrier strategy. By referring to the existing relevant literature, we find that the existing research is the classic model with a dividend strategy or the two-sided jump risk model. We want to know the situation of the dividend barrier strategy under double risk. According to this idea, through the literature review, we find that the model has very important practical significance. At the same time, we find that no scholars have introduced random observation into this model, but this is exactly what is for random observation in real life. In the process of research, we also find that there is no closed solution to the integral differential equation of this model after introducing random observation. To solve this problem, we obtained an a.s. by the sinc numerical method and analyzed the upper limit of the error. Perhaps one day in the future, we will have a better way to find the e.s. to this model.
Author contributions
Chunwei Wang: Methodology, supervision, resources, funding acquisition, writing-review & editing; Shaohua Li: Methodology, software, visualization, writing-original draft; Shujing Wang: Software, visualization; Jiaen Xu: Methodology, validation. All authors have read and agreed to the published version of the manuscript.
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
The research was supported by the National Natural Science Foundation of China (No. 71801085). The authors would like to thank the referees for their valuable comments and suggestions.
Conflict of interest
All authors declare no conflicts of interest in this paper.