Loading [MathJax]/extensions/TeX/boldsymbol.js
Research article Special Issues

Dividend problem of an investment risk model under random observation

  • These authors contributed equally to this work.
  • We mainly studied the dividend payout with a two-sided jumps risk model under random observation. The two-sided jumps in the model represent random claims and random returns. First, we obtained the integral differential equation of the expected dividend under the boundary conditions. Because the equations cannot be solved directly under normal circumstances, we chose the sinc numerical method here to approximate the solution of the equations. Then the error analysis of the approximate solution was carried out to illustrate the rationality of the numerical method. Finally, some concrete numerical examples were given.

    Citation: Chunwei Wang, Shaohua Li, Jiaen Xu, Shujing Wang. Dividend problem of an investment risk model under random observation[J]. AIMS Mathematics, 2024, 9(9): 24039-24057. doi: 10.3934/math.20241169

    Related Papers:

    [1] Chunwei Wang, Shujing Wang, Jiaen Xu, Shaohua Li . Numerical method for a compound Poisson risk model with liquid reserves and proportional investment. AIMS Mathematics, 2024, 9(5): 10893-10910. doi: 10.3934/math.2024532
    [2] Chunwei Wang, Jiaen Xu, Naidan Deng, Shujing Wang . Two-sided jumps risk model with proportional investment and random observation periods. AIMS Mathematics, 2023, 8(9): 22301-22318. doi: 10.3934/math.20231137
    [3] Chunwei Wang, Jiaen Xu, Shujing Wang, Naidan Deng . An investment risk model with bilateral jumps. AIMS Mathematics, 2024, 9(1): 2032-2050. doi: 10.3934/math.2024101
    [4] Li Deng, Zhichao Chen . Optimal dividends in a discrete-time dual risk model with stochastic expenses. AIMS Mathematics, 2024, 9(11): 31696-31720. doi: 10.3934/math.20241524
    [5] Fuyun Sun, Yuelei Li . On the improved thinning risk model under a periodic dividend barrier strategy. AIMS Mathematics, 2021, 6(12): 13448-13463. doi: 10.3934/math.2021779
    [6] Javed Hussain, Saba Shahid, Tareq Saeed . Pricing forward-start style exotic options under uncertain stock models with periodic dividends. AIMS Mathematics, 2024, 9(9): 24934-24954. doi: 10.3934/math.20241215
    [7] Kung-Chi Chen, Kuo-Shing Chen . Pricing green financial options under the mixed fractal Brownian motions with jump diffusion environment. AIMS Mathematics, 2024, 9(8): 21496-21523. doi: 10.3934/math.20241044
    [8] Andrey Borisov . Filtering of hidden Markov renewal processes by continuous and counting observations. AIMS Mathematics, 2024, 9(11): 30073-30099. doi: 10.3934/math.20241453
    [9] Kai Xiao, Yonghui Zhou . Linear Bayesian equilibrium in insider trading with a random time under partial observations. AIMS Mathematics, 2021, 6(12): 13347-13357. doi: 10.3934/math.2021772
    [10] Yongchang Wei, Zongbin Yin . Long-time dynamics of a stochastic multimolecule oscillatory reaction model with Poisson jumps. AIMS Mathematics, 2022, 7(2): 2956-2972. doi: 10.3934/math.2022163
  • We mainly studied the dividend payout with a two-sided jumps risk model under random observation. The two-sided jumps in the model represent random claims and random returns. First, we obtained the integral differential equation of the expected dividend under the boundary conditions. Because the equations cannot be solved directly under normal circumstances, we chose the sinc numerical method here to approximate the solution of the equations. Then the error analysis of the approximate solution was carried out to illustrate the rationality of the numerical method. Finally, some concrete numerical examples were given.



    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..

    According to the previous research on the bilateral jump r.m., we define

    U(t)=u0+ctS1t+S2t=u0+ctM1(t)i=1Yi+M2(t)i=1Zi,   t0, (2.1)

    where u0 represents the company's initial surplus on the account and u0 is greater than zero. In addition, {U(t)}t0 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++S1jt} and M2(t)=sup{j:S21+S22++S2jt}, 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}t0 satisfies

    dRtRt=rdt, (2.2)

    where r is the interest rate on a risk-free asset, so obviously r should be greater than zero. Risk asset {Qt}t0 is defined as

    Qt=eσWt+at, (2.3)

    where {Wt,t0} 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}t0 satisfies

    dQtQt=(a+12σ2)dt+σdWt. (2.4)

    Let q(0,1) represent the proportion of the insurance company's surplus invested in risky assets, and then 1q represents the proportion invested in risk-free assets. So U(t) satisfies

    dU(t)=qU(t)dQtQt+(1q)U(t)dRtRt+cdtdS1t+dS2t=qσU(t)dWt+(c+ξU(t))dtdM1(t)i=1Yi+dM2(t)i=1Zi, (2.5)

    where ξ=(a+12σ2)q+(1q)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),t0}, and {Ub(t),t0} satisfies

    dUb(t)={Ub(t)V(Q,R,q,t)+cdtdS1t+dS2t,           <Ub(t)0,Ub(t)V(Q,R,q,t)+cdtdS1t+dS2t,            0<Ub(t)b,Ub(t)V(Q,R,q,t)+(cα)dtdS1t+dS2t,  b<Ub(t)<, (2.6)

    where V(Q,R,q,t)=(1q)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

    Du,b=Tb0eδtdD(t)=αTb0I(Ub(t)>b)eδtdt, (2.7)

    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 uR, the expectation of Du,b is represented by

    V(u;b)=E[Du,b|Ub(0)=u]. (2.8)

    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{j1:M(j)0}. Suppose {Sj}j=0 is an i.i.d sequence, where Sj=TjTj1 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)}t0, {M2(t)}t0, {Wt,t0}, 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

    M(j)=M(j1)+TjTj1qσM(t)dWt+TkTk1(ξM(t)+c)dtTjTj1dM1(t)i=1Yi+TjTj1dM2(t)i=1Zi. (2.9)

    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

    V(u;b)={V1(u;b),  u(,0],V2(u;b),  u(0,b],V3(u;b),  u(b,).

    Here are the following conclusions.

    Theorem 3.1. For u(,0], V1(u;b) satisfies

    12q2u2σ2V1(u;b)+(ξu+c)V1(u;b)(λ1+λ2+δ)V1(u;b)+λ10V1(uy;b)dFY(y)+λ2[u0V1(u+z;b)dFZ(z)+u+buV2(u+z;b)dFZ(z)+u+bV3(u+z;b)dFZ(z)]=0. (3.1)

    For u(0,b], V2(u;b) satisfies

    12q2u2σ2V2(u;b)+(ξu+c)V2(u;b)(δ+λ1+λ2)V2(u;b)+λ1[u0V2(uy;b)dFY(y)+uV1(uy;b)dFY(y)]+λ2[bu0V2(u+z;b)dFZ(z)+buV3(u+z;b)dFZ(z)]=0, (3.2)

    and for u(b,), V3(u;b) satisfies

    12q2u2σ2V3(u;b)+(ξu+cα)V3(u;b)(δ+λ1+λ2)V3(u;b)+λ1[ub0V3(uy;b)dFY(y)+uubV2(uy;b)dFY(y)+uV1(uy;b)dFY(y)]+λ20V3(u+z;b)dFZ(z)+α=0. (3.3)

    The following boundary conditions are satisfied

    limuV1(u;b)=0; (3.4)
    limu+V3(u;b)=αδ; (3.5)
    V2(b;b)=V3(b+;b); (3.6)
    V2(b;b)=V3(b+;b). (3.7)

    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],

    V1(u,b)=eδdt{γdtP0E[V1(h1t;b)]+(1γdt)P0E[V1(h1t;b)]+(1γdt)P1E[E[V1(h1t+Z1;b)|0<Z1<u]+E[V2(h1t+Z1;b)|u<Z1<bh1t]+E[V3(h1t+Z1;b)|bh1t<Z1]]+γdtP1E[V1(h1t+Z1;b)]+(1γdt)P2E[V1(h1tY1;b)]+γdtP2E[V1(h1tY1;b)]}, (3.8)

    and for u(0,b],

      V2(u,b)=eδdt{γdtP0E[V2(h1t;b)]+(1γdt)P0E[V2(h1t;b)]+(1γdt)P1E[E[V2(h1t+Z1;b)|u<Z1<bh1t]+E[V3(h1t+Z1;b)|bh1t<Z1]]+γdtP1E[V2(h1t+Z1;b)]+(1γdt)P2E[E[V2(h1tY1;b)|h1tb<Y1<u]+E[V1(h1tY1;b)|u<Y1]]+γdtP2E[V2(h1tY1;b)]}, (3.9)

    amd for u(b,),

    V3(u,b)=eδdt{αdt+γdtP0E[V3(h2t;b)]+(1γdt)P0E[V3(h2t;b)]+(1γdt)P2E[E[V1(h1tY1;b)|u<Y1]+E[V2(h1tY1;b)|h1tb<Y1<u]+E[V3(h2tY1;b)|0<Y1<h1tb]]+γdtE[V3(h2tY1;b)]+(1γdt)P1E[V3(h2t+Z1;b)]+γdtP1E[V3(h2t+Z1;b)]}, (3.10)

    where

    P0=P(S11>dt,S21>dt)=1(λ1+λ2)dt+o(dt), (3.11)
    P1=P(S11>dt,S21dt)=λ2dt+o(dt), (3.12)
    P2=P(S11dt,S21>dt)=λ1dt+o(dt). (3.13)

    According to the Itô formula, we get

    E[V1(h1t;b)]=E[V1(u;b)+(ξu+c)V1(u;b)dt+12q2u2σ2V1(u;b)dt]+o(dt), (3.14)
    E[V2(h1t;b)]=E[V2(u;b)+(ξu+c)V2(u;b)dt+12q2u2σ2V2(u;b)dt]+o(dt), (3.15)
    E[V3(h2t;b)]=E[V3(u;b)+(ξu+cα)V3(u;b)dt+12q2u2σ2V3(u;b)dt]+o(dt), (3.16)

    where

    h1t=u+quσdWt+(ξu+k)dt, (3.17)
    h2t=u+quσdWt+(ξu+kα)dt, (3.18)

    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 αc, 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).

    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.

    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 RR

    ϕ(z)=z, (4.1)

    where zR. Define the grid point zk of sinc as

    zk=ϕ1(kh)=kh, (k=0,±1,±2,),

    where kZ, h>0. Based on the sinc method, the basis function of zΓ on the interval (,) is given by the following composite function

    Cj(z)=C(j,h)ϕ(z)=sinc(ϕ(z)jhh).

    Following the steps of the sinc method, we arrange Eqs (3.1)–(3.3) into the following integral differential

    12q2u2σ2V(u;b)+(ξu+cαIu>b)V(u;b)(λ1+λ2+δ)V(u;b)+0λ1V(uy;b)dFY(y)+0λ2V(u+z;b)dFZ(z)+αI(u>b)=0. (4.2)

    By the nature of convolution, Eq (4.2) is rewritten as

    12q2u2σ2V(u;b)+(ξu+cαIu>b)V(u;b)(λ1+λ2+δ)V(u;b)+uλ1V(y;b)fY(uy)dy++uλ2V(z;b)fZ(zu)dz+αI(u>b)=0. (4.3)

    According to formulas (3.4) and (3.5), and Definition 1.5.2 in reference[27], we have

    h(u;b)=v(t1;b)+ζ(u)v(t2;b)1+ζ(u),

    where ζ(u)=eϕ(u)=eu, when t1, t2. Set

    W(u)=V(u;b)h(u;b)=V(u;b)eu1+euαδ, (4.4)

    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]).

    V(u;b)=h(u;b)+W(u)=W(u)+eu1+euαδ, (4.5)
    V(u;b)=h(u;b)+W(u)=W(u)+eu(1+eu)2αδ, (4.6)
    V(u;b)=h(u;b)+W(u)=W(u)+eu(1eu)(1+eu)3αδ. (4.7)

    When u, u

    limuW(u)=0;limu+W(u)=0.

    Substituting (4.5)–(4.7) into (4.3), by simple calculation, we have

    μ0(u)W(u)+μ1(u)W(u)+μ2(u)W(u)+λ1uW(y)K1(uy)dy+λ2uW(z)K2(zu)dz+f(u)=0, (4.8)

    where μ0(u)=(quδ)22, μ1(u)=ξu+cαI(u>b), μ2(u)=(δ+λ1+λ2),

    K1(uy)=fY(uy), (4.9)
    K2(zu)=fZ(zu), (4.10)
    f(u)=αIu>b+μ0(u)eu(1eu)(1+eu)3αδ+μ1(u)eu(1+eu)2αδ+μ2(u)eu1+euαδ           +λ1uey1+eyαδK1(uy)dy+λ2uez1+ezαδK2(zu)dz. (4.11)

    When h>0, define the sinc grid point as

    uk=kh, k=±1,±2,. (4.12)

    Then consulting reference [27], according to Theorem 1.5.13, Theorem 1.5.14, and Theorem 1.5.20, we can get

    uK1(uy)W(y)dyn1j=n2n1i=n2ωiAijUj, (4.13)
    uK2(zu)W(z)dzn1j=n2n1i=n2ωiBijUj, (4.14)
    W(u)˜W(u)=n1j=n2UjC(j,h)ϕ(x), (4.15)

    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

    μ0(uk)˜W(uk)+μ1(uk)˜W(uk)+μ2(uk)˜W(uk)+λ1n1j=n2n1i=n2ωi(uk)AijUj+λ2n1j=n2n1i=n2ωi(uk)BijUj=f(uk), (4.16)

    where

    ˜W(uk)=n1j=n2Uj[C(j,h)ϕ(uk)]=n1j=n2Ujδ(0)jk, (4.17)
    ˜W(uk)=n1j=n2Uj[C(j,h)ϕ(uk)]=n1j=n2Ujϕ(uk)δ(1)jk, (4.18)
    ˜W(uk)=n1j=n2Uj[C(j,h)ϕ(uk)]=n1j=n2Uj[ϕ(uk)h1δ(1)jk+[ϕ(uk)]2h2δ(2)jk]. (4.19)

    Substituting (4.17)–(4.19) into Eq (4.16), we have

    n1j=n2Uj{μ0(uk)(ϕ(uk)δ(1)jkh+(ϕ(uk))2δ(2)jkh2)+μ1(uk)ϕ(uk)δ(1)jkh+μ2(uk)δ(0)jk+λ1n1i=n2ωi(uk)Aij+λ2n1i=n2ωi(uk)Bij}=f(uk). (4.20)

    Multiplying Eq (4.20) by h2[ϕ(uk)]2, we have

    n1j=n2Uj{μ0(uk)δ(2)jk+h[μ0(uk)ϕ(uk)[ϕ(uk)]2+μ1(uk)ϕ(uk)]δ(1)jk+h2μ2(uk)[ϕ(uk)]2δ(0)jk+λ1h2[ϕ(uk)]2n1i=n2ωi(uk)Aij+λ2h2[ϕ(uk)]2n1i=n2ωi(uk)Bij}=f(uk)h2[ϕ(uk)]2. (4.21)

    Since

    δ(0)jk=δ(0)kj,    δ(1)jk=δ(1)kj,    δ(2)jk=δ(2)kj, and  ϕ(xk)ϕ(uk)2=(1ϕ(uk)),

    formula (4.21) can be turned into

    n1j=n2Uj{μ0(uk)δ(2)kj+h[μ0(uk)ϕ(uk)[ϕ(uk)]2+μ1(uk)ϕ(uk)]δ(1)kj+h2μ2(uk)[ϕ(uk)]2δ(0)kj+λ1h2[ϕ(uk)]2n1i=n2ωi(uk)Aij+λ2h2[ϕ(uk)]2n1i=n2ωi(uk)Bij}=f(uk)h2[ϕ(uk)]2. (4.22)

    Set I(m)=[δ(m)kj](n2+n1+1)×(n2+n1+1), and m=1,0,1,2. We rewrite Eq (4.22) as

    GU=F, (4.23)

    where

    U=[Uj]T=[Un2,,Un1]T,F=[h2f(un2)ϕ(un2)2,,h2f(un1)(ϕ(un1))2],G=μ0I(2)+hDm(μ0(1ϕ)μ1ϕ)I(1)+h2Dm(μ2ϕ2)I(0)+λ1h2Dm(1(ϕ)2)ΩmA   +λ2h2Dm(1(ϕ)2)ΩmB.

    So solving Eq (4.23), we get the expression of the approximate solution (a.s) of (4.5):

    V(u;b)=W(u)+eu1+euαδ˜W(u)+eu1+euαδ=n1j=n2UjC(j,h)ϕ(u)+eu1+euαδ. (4.24)

    The meanings of the symbols mentioned in the above process are shown in Table 1.

    Table 1.  Symbol specification.
    n1 positive integer
    n2 [n1˜β˜α]
    Dm(f) diag[f(Zn2),,f(Zn1)]
    Ωm (ωn2,ωn2+1,,ωn11,ωn1)
    ωn2 (1+en2h)[11+ρn1j=(n21)γj1+ejh]
    ωn1 (1+en1h)[ρ1+ρn11j=n2eijγj1+ejh]
    ωn2 11+ρn1j=(n21)γj1+ejh
    ωn1 ρ1+ρn11j=n2eijγj1+ejh
    ωj C(j,h)ϕ, j=n1+1,,n21
    γj C(j,h)ϕ, j=n1,,n2

     | Show Table
    DownLoad: CSV

    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

    G(u)=λ1μ0(u)uW(y)K1(uy)dyλ2μ0(u)uW(z)K2(zu)dzf(u)μ0(u),

    so we have

    G(u)=~μ2(u)W(u)+~μ1(u)W(u)+W(u), (4.25)

    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)2Lˆα(D) and Eq (4.25) possess a single solution WLˆα(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

    \begin{align} \mathop{\rm{sup}}\limits_{u\in\Gamma}|W(u)-\tilde{W}(u)|\leqslant\tilde{c}N^{5/2}e^{-\sqrt{\pi d \hat{\alpha}N}}. \end{align} (4.26)

    Proof. Let

    \begin{align} \mathscr{O}_{N}(u) = \sum\limits_{k = -{n_2}}^{n_1}W(u_{k})C(k, h)\circ\zeta(u). \end{align} (4.27)

    By using the triangle inequality, it is easily obtained that

    \begin{align} |W(u)-\tilde{W}(u)|\leqslant|W(u)-\mathscr{O}_{N}(u)|+|\mathscr{O}_{N}(u)-\tilde{W}(U)|. \end{align} (4.28)

    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

    \begin{align} \mathop{\rm{sup}}\limits_{u\in\Gamma}|W(u)-\mathscr{O}_{N}(u)|\leqslant c^{*}N^{1/2}e^{-\sqrt{\pi d \hat{\alpha}N}}. \end{align} (4.29)

    For inequality (4.28), |\mathscr{O}_{N}(u)-\tilde{W}(u)| fulfills

    \begin{align} |\mathscr{O}_{N}(u)-\tilde{W}(u)|& = \bigg{|}\sum\limits_{j = -{n_2}}^{n_1}[W(u_j)-U_{j}]C(j, h)\circ\zeta(u)-\frac{e^{u}}{(1+e^{u})}\frac{\alpha}{\delta}\bigg{|}\\ &\leq \sum\limits_{j = -{n_2}}^{n_1}|W(u_j)-U_{j}||C(j, h)\circ\zeta(u)|\\ &\leq \sqrt{\bigg( \sum\limits_{j = -{n_2}}^{n_1}|W(u_j)-U_{j}|^2 \bigg)\bigg(\sum\limits_{j = -{n_2}}^{n_1}|C(j, h)\circ\zeta(u)|^2 \bigg)}\\ &\leq \sqrt{\sum\limits_{j = -{n_2}}^{n_1}|W(u_j)-U_{j}|^2} = ||\textbf{W}-\textbf{U}||. \end{align} (4.30)

    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

    \begin{align} ||\textbf{W}-\textbf{U}||& = ||C^{-1}C(\textbf{W}-\textbf{U})||\leq c^{**}N^{5/2}e^{-\sqrt{\pi d \hat{\alpha}N}}, \end{align} (4.31)

    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

    \begin{align} \mathop{\sup}\limits_{u\in\Gamma}|V(u;b)-\tilde{V}(u;b)|\leqslant\tilde{c}N^{5/2}e^{-\sqrt{\pi d \hat{\alpha}N}}. \end{align} (4.32)

    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.

    All numerical examples in this section are assumed to be obtained under

    \begin{align*} f_{Y}(y) = \eta_{1}{\rm{e}}^{-\eta_{1}y}{{I}}_{y > 0}, \end{align*}

    and

    \begin{align*} f_{Z}(z) = \eta_{2}{\rm{e}}^{-\eta_{2}z}{{I}}_{z > 0}. \end{align*}

    Then,

    \begin{align} f_{Y}(u-y) = \eta_{1}{\rm{e}}^{-\eta_{1}(u-y)}{{I}}_{u > y}, \end{align} (5.1)

    and

    \begin{align} f_{Z}(z-u) = \eta_{2}{\rm{e}}^{-\eta_{2}(z-u)}{{I}}_{u < z}. \end{align} (5.2)

    Formulas (4.8) and (4.11) are converted to

    \begin{align} & \mu_0(u)W^{''}(u)+\mu_1(u)W^{'}(u)+\mu_2(u)W(u)+\lambda_1 \int_{-\infty}^u W(y)\eta_{1}e^{-\eta_{1}(u-y)}{\rm{d}}y\\ &+\lambda_2\int_u^\infty W(z)\eta_{2}e^{-\eta_{2}(z-u)}{\rm{d}}z+f(u) = 0, \end{align} (5.3)

    and

    \begin{align} f(u) = &\alpha{{I}}_{u > b}+\mu_0(u)\frac{{\rm{e}}^u(1-{\rm{e}}^u)}{(1+{\rm{e}}^u)^3}\frac{\alpha}{\delta}+\mu_1(u)\frac{{\rm{e}}^u}{(1+{\rm{e}}^u)^2}\frac{\alpha}{\delta}+\mu_2(u)\frac{{\rm{e}}^u}{1+{\rm{e}}^u}\frac{\alpha}{\delta}\\ &+\lambda_1 \int_{-\infty}^u\frac{{\rm{e}}^y}{1+{\rm{e}}^y}\frac{\alpha}{\delta}\eta_{1}{\rm{e}}^{-\eta_{1}(u-y)}{\rm{d}}y+\lambda_2\int_u^{\infty}\frac{{\rm{e}}^z}{1+{\rm{e}}^z}\frac{\alpha}{\delta}\eta_{2}{\rm{e}}^{-\eta_{2}(z-u)}{\rm{d}}z. \end{align} (5.4)

    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.

    Figure 1.  The change of V(u; b) with q .
    Table 2.  The value of V(u; b) when q changes.
    q u=-1 −0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
    0.2 3.430 3.294 3.360 3.378 3.182 3.431 3.059 3.504 3.080 3.640 3.265
    0.3 3.575 3.242 3.442 3.537 3.080 3.733 2.810 3.931 2.830 4.205 3.184
    0.4 3.701 3.057 3.493 3.738 2.882 4.203 2.395 4.604 2.374 5.040 2.898
    0.5 3.806 2.707 3.494 3.982 2.559 4.869 1.773 5.565 1.656 66.198 2.325

     | Show Table
    DownLoad: CSV

    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.

    Figure 2.  The change of V(u; b) with \sigma .
    Table 3.  The value of V(u;b) when \sigma changes.
    \sigma u=-1 −0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
    0.2 3.430 3.294 3.360 3.378 3.182 3.431 3.059 3.504 3.080 3.640 3.265
    0.3 3.575 3.118 3.936 2.999 3.883 2.879 3.712 3.001 3.484 3.484 3.605
    0.4 3.999 3.426 3.697 3.788 3.003 4.041 2.548 4.398 4.964 4.964 3.401
    0.5 4.385 3.476 3.921 4.086 2.852 4.517 2.147 5.091 5.962 5.962 3.444

     | Show Table
    DownLoad: CSV

    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.

    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

    f_{Y}(y) = \frac{1}{2\pi v_{1}y}{\rm{e}}^{-\frac{(\ln y-\eta_{3})}{4v_{1}^2}}{{I}}_{y > 0}, \ \ f_{Z}(z) = \frac{1}{2\pi v_{1}z}{\rm{e}}^{-\frac{(\ln z-\eta_{4})}{4v_{2}^2}}{{I}}_{z > 0}.

    Then,

    \begin{align} f_{Y}(u-y) = \frac{1}{2\pi v_{1}(u-y)}{\rm{e}}^{-\frac{(\ln (u-y)-\eta_{3})}{4v_{1}^2}}{{I}}_{u > y}, \end{align} (5.5)

    and

    \begin{align} f_{Z}(z-u) = \frac{1}{2\pi v_{1}(z-u)}{\rm{e}}^{-\frac{(\ln (z-u)-\eta_{4})}{4v_{2}^2}}{{I}}_{z > u}. \end{align} (5.6)

    Therefore, the formulas (4.8) and (4.11) can be rewritten as:

    \begin{align} & \mu_0(u)W^{''}(u)+\mu_1(u)W^{'}(u)+\mu_2(u)W(u)+\lambda_1 \int_{-\infty}^u W(y)\frac{1}{2\pi v_{1}(u-y)}e^{-\frac{(\ln (u-y)-\eta_{3})}{4v_{1}^2}}{\rm{d}}y\\ &+\lambda_2\int_u^\infty W(z)\frac{1}{2\pi v_{1}(z-u)}e^{-\frac{(\ln (z-u)-\eta_{4})}{4v_{2}^2}}{\rm{d}}z+f(u) = 0, \end{align} (5.7)

    and

    \begin{align} f(u) = &\alpha{{I}}_{u > b}+\mu_0(u)\frac{{\rm{e}}^u(1-{\rm{e}}^u)}{(1+{\rm{e}}^u)^3}\frac{\alpha}{\delta}+\mu_1(u)\frac{{\rm{e}}^u}{(1+{\rm{e}}^u)^2}\frac{\alpha}{\delta}+\mu_2(u)\frac{{\rm{e}}^u}{1+{\rm{e}}^u}\frac{\alpha}{\delta}\\ &+\lambda_1 \int_{-\infty}^u\frac{{\rm{e}}^y}{1+{\rm{e}}^y}\frac{\alpha}{\delta}\frac{1}{2\pi v_{1}(u-y)}{\rm{e}}^{-\frac{(\ln (u-y)-\eta_{3})}{4v_{1}^2}}{\rm{d}}y\\ &+\lambda_2\int_u^{\infty}\frac{{\rm{e}}^z}{1+{\rm{e}}^z}\frac{\alpha}{\delta}\frac{1}{2\pi v_{1}(z-u)}{\rm{e}}^{-\frac{(\ln (z-u)-\eta_{4})}{4v_{2}^2}}{\rm{d}}z. \end{align} (5.8)

    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.

    Figure 3.  The change of V(u; b) with q .
    Table 4.  The value of V(u; b) when q changes.
    q u = 2 2.25 2.5 2.75 3.0 3.25 3.50 3.75 4.0 4.25 4.5
    0.2 1.697 1.873 2.064 2.190 2.273 2.394 2.533 2.565 2.437 2.241 2.052
    0.4 1.768 2.386 3.081 3.490 3.717 4.143 4.732 4.951 4.534 3.880 3.345
    0.6 1.846 3.086 4.541 5.285 5.552 6.349 7.686 8.293 7.466 6.161 5.335
    0.8 1.931 3.936 6.402 7.487 7.583 8.712 11.068 12.278 10.902 8.7263 7.734

     | Show Table
    DownLoad: CSV

    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.

    Figure 4.  The change of V(u; b) with \sigma .
    Table 5.  The value of V(u; b) when \sigma changes.
    \sigma u = 2 2.25 2.5 2.75 3.0 3.25 3.50 3.75 4.0 4.25 4.5
    0.2 1.697 1.873 2.064 2.190 2.273 2.394 2.533 2.565 2.437 2.241 2.052
    0.4 1.809 2.505 3.263 3.761 4.091 4.577 5.145 5.297 4.810 4.053 3.334
    0.6 1.968 3.487 5.156 6.226 6.915 7.981 9.280 9.671 8.626 6.996 5.493
    0.8 2.134 4.735 7.612 9.393 10.475 12.291 14.632 15.427 13.677 10.929 8.521

     | Show Table
    DownLoad: CSV

    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.

    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.

    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.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    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.

    All authors declare no conflicts of interest in this paper.



    [1] Z. Zhang, E. C. K. Cheung, H. Yang, On the compound Poisson risk model with periodic capital injections, Astin Bull., 48 (2018), 435–477. https://doi.org/10.1017/asb.2017.22 doi: 10.1017/asb.2017.22
    [2] A. Bazyari, On the ruin probabilities in a discrete time insurance risk process with capital injections and reinsurance, Sankhya A, 85 (2023), 1623–1650. https://doi.org/10.1007/s13171-022-00305-3 doi: 10.1007/s13171-022-00305-3
    [3] J. Li, Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return, Insur. Math. Econ., 71 (2016), 195–204. https://doi.org/10.1016/j.insmatheco.2016.09.003 doi: 10.1016/j.insmatheco.2016.09.003
    [4] Z. Zhang, W. Su, A new efficient method for estimating the Gerber-Shiu function in the classical risk model, Scand. Actuar. J., 5 (2018), 426–449. https://doi.org/10.1080/03461238.2017.1371068 doi: 10.1080/03461238.2017.1371068
    [5] X. Peng, W. Su, Z. Zhang, On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy, J. Ind. Manag. Optim., 16 (2020), 1967–1986. https://doi.org/10.3934/jimo.2019038 doi: 10.3934/jimo.2019038
    [6] R. J. Boucherie, O. J. Boxma, K. Sigman, A note on negative customers, GI/G/1 workload, and risk processes, Prob. Eng. Inform. Sci., 11 (1997), 305–311. https://doi.org/10.1017/S0269964800004848 doi: 10.1017/S0269964800004848
    [7] E. C. K. Cheung, On a class of stochastic models with two-sided jumps, Queueing Syst., 69 (2011), 1–28. https://doi.org/10.1007/s11134-011-9228-z doi: 10.1007/s11134-011-9228-z
    [8] L. Zhang, The Erlang(n) risk model with two-sided jumps and a constant dividend barrier, Commun. Statist. Theory Methods, 50 (2021), 5899–5917. https://doi.org/10.1080/03610926.2020.1737712 doi: 10.1080/03610926.2020.1737712
    [9] C. Wang, J. Xu, S. Wang, N. Deng, An investment risk model with bilateral jumps, AIMS Mathematics, 9 (2024), 2032–2050. https://doi.org/10.3934/math.2024101 doi: 10.3934/math.2024101
    [10] J. Xu, C. Wang, N. Deng, S. Wang, Numerical method for a risk model with two-sided jumps and proportional investment, Mathematics, 11 (2023), 1584. https://doi.org/10.3390/math11071584 doi: 10.3390/math11071584
    [11] J. J. Rebello, K. K. Thampi, Some ruin theory components of two sided jump problems under renewal risk process, Int. Math. Forum, 12 (2017), 311–325. https://doi.org/10.12988/imf.2017.611147 doi: 10.12988/imf.2017.611147
    [12] E. M. Martín-González, A. Murillo-Salas, H. Pantí, Gerber-shiu function for a class of markov-modulated lévy risk processes with two-sided jumps, Methodol. Comput. Appl. Probab., 24 (2022), 2779–2800. https://doi.org/10.1007/s11009-022-09954-1 doi: 10.1007/s11009-022-09954-1
    [13] Z. Palmowski, E. Vatamidou, Phase-type approximations perturbed by a heavy-tailed component for the gerber-shiu function of risk processes with two-sided jumps, Stoch. Models, 36 (2020), 337–363. https://doi.org/10.1080/15326349.2020.1717344 doi: 10.1080/15326349.2020.1717344
    [14] E. C. K. Cheung, H. Liu, G. E. Willmot, Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps, Appl. Math. Comput., 331 (2018), 358–377. https://doi.org/10.1016/j.amc.2018.03.037 doi: 10.1016/j.amc.2018.03.037
    [15] W. Zou, J. Gao, J. Xie, On the expected discounted penalty function and optimal dividend strategy for a risk model with random incomes and interclaimdependent claim sizes, Int. J. Comput. Appl. Math., 255 (2014), 270–281. https://doi.org/10.1016/j.cam.2013.05.004 doi: 10.1016/j.cam.2013.05.004
    [16] H. U. Gerber, E. S. W. Shiu, The time value of ruin in a sparre andersen model, N. Am. Actuar. J., 9 (2005), 49–69. https://doi.org/10.1080/10920277.2005.10596197 doi: 10.1080/10920277.2005.10596197
    [17] C. Yin, Y. Shen, Y. Wen, Exit problems for jump processes with applications to dividend problems, Int. J. Comput. Appl. Math., 245 (2013), 30–52. https://doi.org/10.1016/j.cam.2012.12.004 doi: 10.1016/j.cam.2012.12.004
    [18] H. Cossette, E. Marceau, F. Marri, On a compound Poisson risk model with dependence and in the presence of a constant dividend barrier, Appl. Stoch. Models Bus. Ind., 30 (2014), 82–98. https://doi.org/10.1002/asmb.1928 doi: 10.1002/asmb.1928
    [19] J. Xie, W. Zou, On the expected discounted penalty function for a risk model with dependence under a multi-layer dividend strategy, Commun. Statist. Theory Methods, 46 (2017), 1898–1915. https://doi.org/10.1080/03610926.2015.1030424 doi: 10.1080/03610926.2015.1030424
    [20] L. Bo, R. Song, D. Tang, Y. Wang, X. Yang, Lévy risk model with two-sided jumps and a barrier dividend strategy, Insur. Math. Econ., 50 (2012), 280–291. https://doi.org/10.1016/j.insmatheco.2011.12.002 doi: 10.1016/j.insmatheco.2011.12.002
    [21] X. Chen, H. Ou, A compound Poisson risk model with proportional investment, Int. J. Comput. Appl. Math., 242 (2013), 248–260. https://doi.org/10.1016/j.cam.2012.10.027 doi: 10.1016/j.cam.2012.10.027
    [22] M. Escobar-Anel, A. Lichtenstern, R. Zagst, Behavioral portfolio insurance strategies, Fin. Mark. Portfolio Manage., 34 (2020), 353–399. https://doi.org/10.1007/s11408-020-00353-5 doi: 10.1007/s11408-020-00353-5
    [23] H. Albrecher, E. C. K. Cheung, S. Thonhauser, Randomized observation periods for the compound Poisson risk model: The discounted penalty function, Scand. Actuar. J., 2013 (2013), 424–452. https://doi.org/10.1080/03461238.2011.624686 doi: 10.1080/03461238.2011.624686
    [24] W. Zhuo, H. Yang, X. Chen, Expected discounted penalty function for a phase-type risk model with stochastic return on investment and random observation periods, Kybernetes, 47 (2018), 1420–1434. https://doi.org/10.1108/K-05-2017-0153 doi: 10.1108/K-05-2017-0153
    [25] E. C. K. Cheung, Z. Zhang, Periodic threshold-type dividend strategy in the compound Poisson risk model, Scand. Actuar. J., 2019 (2019), 1–31. https://doi.org/10.1080/03461238.2018.1481454 doi: 10.1080/03461238.2018.1481454
    [26] H. Albrecher, E. C. K. Cheung, S. Thonhauser, Randomized observation periods for the compound Poisson risk model: dividends, Astin Bull., 41 (2011), 645–672. https://doi.org/10.2143/AST.41.2.2136991 doi: 10.2143/AST.41.2.2136991
    [27] F. Stenger, Handbook of sinc numerical methods, Boca Raton: CRC Press, 2011. https://doi.org/10.1201/b10375
    [28] C. Wang, N. Deng, S. Shen, Numerical method for a perturbed risk model with proportional investment, Mathematics, 11 (2022), 43. https://doi.org/10.3390/math11010043 doi: 10.3390/math11010043
    [29] Y. Liu, X. Chen, W. Zhuo, Dividends under threshold dividend strategy with randomized observation periods and capital-exchange agreement, Int. J. Comput. Appl. Math., 366 (2022), 112426. https://doi.org/10.1016/j.cam.2019.112426 doi: 10.1016/j.cam.2019.112426
    [30] Y. Wu, X. Wu, Linearized and rational approximation method for solving non-linear Burgers' equation, Internat. J. Numer. Methods Fluids, 45 (2004), 509–525. https://doi.org/10.1002/fld.714 doi: 10.1002/fld.714
    [31] F. Stenger, Numerical methods based on sinc and analytic functions, New York: Springer, 1993. http://doi.org/10.1007/978-1-4612-2706-9
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(793) PDF downloads(52) Cited by(0)

Figures and Tables

Figures(4)  /  Tables(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog