# On Periodic Dividends for the Classical Risk Model with Debit Interest

Authors: Hua Dong; Xianghua Zhao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A periodic dividend problem is studied in this paper. We assume that dividend payments are made at a sequence of Poisson arrival times, and ruin is continuously monitored. First of all, three integro-differential equations for the expected discounted dividends are obtained. Then, we investigate the explicit expressions for the expected discounted dividends, and the optimal dividend barrier is given for exponential claims. A similar study on a generalized Gerber–Shiu function involving the absolute time is also performed. To demonstrate the existing results, we give some numerical examples.


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## Body

## 1. Introduction

Suppose the dynamics of the surplus process of an insurance company at timet is defined as the solution to(1)dRt=cdt−dSt,Rt≥0,c+αRtdt−dSt,−cα≤Rt<0,where c>0 is the premium charged in the unit time and α>0 is the debit interest. St=∑i=1NtXi is a compound Poisson process with intensity γ>0 representing the total claim amounts until time t,andXi is the i−th claim size. Suppose that the claim sizes are independent of each other and have common density function ς. We also assume that Xi and Nt are mutually independent.Model (1) means that the insurer can make loans at the rate α>0 once the process Rt goes below zero. When Rt<−c/α, the insurer can be prohibited to run its business, i.e., the insurer is absolutely ruined. The absolute ruin risk models have been investigated extensively, see Gerber and Yang [1], Cai [2], Yuen et al. [3], Wang and Yin [4], and Cai and Yang [5], among others.In recent years, the research problems in risk theory are more and more closely related to real life. The risk model with debit interest is a good example. The insurer cannot monitor the surplus continuously, and the dividend can be paid at some certain times. Because of its importance in real life, the topic of periodic dividends has become very popular in risk theory during the last twenty years. Considering dividends can only be made at some discrete times in practice, Albrecher et al. [6] put forward the periodic barrier dividends in this type of risk model. They assumed that both barrier dividends and ruin can only be observed at some randomized times. Considering the insurer monitors bankruptcy more closely than dividends, Avanzi et al. [7] investigated periodic dividend barrier strategy in the dual model where the solvency is monitored continuously. For those risk models with periodic dividends and bankruptcy, the reader can refer to Zhang and Cheung [8, 9], Avanzi et al. [10], Dong et al. [11], and Peng et al. [12], among others. Different to the papers mentioned above, the periodic dividend strategy in Peng et al. [12] is threshold strategy.Now, we study surplus process (1) under periodic dividend strategy. We assume the ruin is continuously monitored as usual. Let Zi be the sequence of dividend observation times with Z0=0 and Wi=Zi−Zi−1i=1,2,… be an exponentially random variable with Wi=1/β. In addition, we assume that Wkk=1,2,… are independent of Nt and Xi. Furthermore, we assume that no dividends are made at time 0. The modified surplus process is given by Rtb with b>0 being the constant level of the dividend barrier. Denote by T=inft>0:Rtb<−c/α the time of absolute ruin for Rtb with the convention that infϕ=∞.Let(2)νx,b=x∑Zk≤Te−σZkRZkb−b+,be the expected discounted dividend before time T, and σ≥0. The generalized Gerber–Shiu function is also our concern:(3)Φx=xe−σTsNTωRT−b,RTb1T<∞,0<s≤1,where 1A denotes the indicator function of event A. ωx,y≥0 is a bounded measurable function of x≥−c/α,y>c/α, RT−b is the surplus prior to time T, and RTb is the deficit at time T. When s=1,Φx is the Gerber–Shiu function proposed by Gerber and Shiu [13]. When ω⋅,⋅=1,Φx describes the joint distribution of the absolute time T and NT:(4)ϕx=xsNTe−σT1T<∞.Whenω⋅,⋅=1,σ=0,Φx is the probability generating function of NT. The topic on the number of claims has been studied by many scholars in various forms. Some scholars focus on some generalized Gerber–Shiu functions, see Li and Lu [14] and Wang et al. [15]; some only considered the Laplace transform, see Egidio dos Reis [16]; and some studied the density function, see Dickson [17] and Czarna et al. [18].We arrange the paper as follows. In Section2, we first derive a system of integro-differential equations for νx,b. For exponential claims, explicit results for νx,b are given. The generalized Gerber–Shiu function is discussed in Section 3. In Section 4, some numerical results are shown to illustrate the given results.


## 2. Results forνx,b

### 2.1. Integro-differential Equations forνx,b

The integro-differential equation is a conventional method; we start this section with the expression forνx,b.Theorem 1.
For−c/α≤x<0, we have(5)αx+cν′x,b−σ+γνx,b+γ∫0x+c/ανx−z,bςzdz=0,for 0≤x<b, we have(6)cν′x,b−σ+γνx,b+γ∫0x+c/ανx−z,bςzdz=0,and for x>b, we have(7)cν′x,b−σ+β+γνx,b+βx−b+βνb,b+γ∫0x+c/ανx−z,bςzdz=0.In addition, we have the following relations:(8)νb−,b=νb+,b,(9)ν′b−,b=ν′b+,b,(10)ν0−,b=ν0+,b,(11)ν′0−,b=ν′0+,b,(12)ν−cα,b=0.Proof.
A standard method is used here, see also Yuen et al. [3] and Albrecher et al. [6]. For a small interval 0,s, all the possible events are taken into account. Then, we have(13)νx,b=e−β+σ+γsνx+cs,b+βse−σ+γsx+cs−b+νb,b1x+cs>b+βse−σ+γsνx+cs,b10<x+cs<b+γse−σ+γs∫0x+cs+c/ανx+cs−y,bςydy+os,for x≥0, and for −c/α≤x<0,(14)νx,b=e−β+σ+γsνtαs,x,b+βse−σ+γsνtαs,x+γse−σ+βs∫0tαs,x+c/ανtαs,x−y,bςydy+os,where tαs,x=xeαs+ceαs−1/α.
Taking derivative ons in (13) and letting s⟶0, we arrive at (6) and (7). By a similar method, (5) can be obtained from (14).
Continuity condition (8) can be obtained from (13), and (10) can be obtained by comparing (13) with (14). (9) can be obtained by using (6) and (7), while (11) is derived by (5) and (6). Let x=−c/α in (5), and we have boundary condition (12).Remark 1.
Obviously, (6) and (7) are the same as (2.1) and (2.2) in Yuen et al. [3], respectively.


### 2.2. Explicit Expressions ofνx,b

In this section, the density function of claim sizes is supposed to befx=μe−μx,x>0. Applying the operator d/dx+μ to (5), then we have(15)αx+cν″x,b+αμx+cμ+α−γ−σν′x,b−σμνx,b=0,for −c/α≤x<0.By the transformsνx,b=kz and z=−μx+c/α, (15) is reduced to a confluent hypergeometric equation:(16)zk″z+1−σ+γα−zk′z+σαkz=0,z<0.It follows from Abramowitz and Stegun [19] that kx admits the following expression:(17)kz=C1ezU1−γα,1−σ+γα,−z+C2zσ+γ/αMγsα,1+σ+γα,z,where C1 and C2 are two constants, Ma,b,z is the first kind of confluent hypergeometric function, and Ua,b,z is the second kind of confluent hypergeometric function. Hence, the solution of (15) can be expressed as(18)νx,b=C1e−μx+c/αU1−γα,1−σ+γα,μx+cα+C2−μx+cασ+γ/αMγα,1+σ+γα,−μx+cα.Due to boundary condition (13), one immediately deduces C1=0.Using a similar procedure to (15), we deduce(19)cν″x,b−σ+γ−cμν′x,b−σμνx,b=0for 0<x<b and(20)cν″x,b−γ+β+σ−cμν′x,b−σ+βμνx,b+βμνb,b+x−b+β=0,for x≥b. Solving (19) and (20) by the knowledge of the ordinary differential equation, one deduces(21)νx,b=C3eρ0x+C4eR0x,0≤x<b,(22)νx,b=D1eρβx+D2eRβx+D3x+D4,x≥b,where ρβ>0,Rβ<0 are roots of the equation(23)r2+μ−σ+γ+βcr−σ+βμc=0.Clearly,νx,b is bounded. Hence, D1=0. By (18), (21), and (22) and (8)–(11), we get the following equations:(24)C3+C4=C2Δ1,(25)C3ρ0+C4R0=C2Δ2,(26)D2eRβb+D3b+D4=C3eρ0b+C4eR0b,(27)D2RβeRβb+D3=C3ρ0eρ0b+C4R0eR0b,where(28)Δ1=−cμασ+γ/αMγα,1+σ+γα,−cμα,Δ2=σ+γcΔ1−γμσ+α+γ−cμασ+γ/αM1+γα,2+σ+γα,−cμα.Substituting (22) into (7) and equating the coefficients of x lead to(29)D3=ββ+σ,and then equating the constant term yields(30)βD2eRβb−σD4=γμ−c−βbD3+βb.Solving (24)–(30), we get(31)νx,b=R0−ρ0hβ−μx+c/ασ+γ/αMγ/α,1+σ+γ/α,−μx+c/αHβ,b,−cα<x<0,(32)νx,b=R0Δ1−Δ2hβHβ,beρ0x+Δ2−ρ0Δ1hβHβ,beR0x,0≤x≤b,νx,b=D2eRβx+D3x+D4,x>b,where(33)D2=ρ0R0Δ1−Δ2eρ0−Rβb+R0Δ2−ρ0Δ1eR0−RβbhβRβHβ,b−βRββ+σe−Rβb,D3=ββ+σ,D4=1−ρ0/RβR0Δ1−Δ2eρ0b+1−R0/RβΔ2−ρ0Δ1eR0bhβHβ,b−ββ+σb−1Rβ,Hβ,b=σ+βρ0−σRβΔ2−R0Δ1eρ0b+σ+βR0−σRβρ0Δ1−Δ2eR0b,hβ=−βγRβ+μβ+σ−cRβμβ+σ.Remark 2.
Forβ⟶∞, it is easy to check that Rβ⟶−μ. Letting β⟶∞ in (31) and (32) and noticing ρ0+R0=−μ+σ+γ/c, we have(34)Vcx,b=ρ0−R01+αx/cσ+γ/αMγ/α,1+σ+γ/α,−μx+c/αρ0+μΔ1−γμ/γ+α+σM1+γ/α,2+σ+γ/α,−cμ/α,−cα<x<0,Vcx,b=Δ2−R0Δ1eρ0x+ρ0Δ1−R0eR0xρ0+μΔ1−γμ/γ+α+σM1+γ/α,2+σ+γ/α,−cμ/α,0≤x≤b,which are in accordance to (3.4) in Yuen et al. [3].
When the initial surplus−c/α<x≤b, we can investigate the optimal dividend barrier b∗ which can maximize νx,b before absolute ruin. By (31) and (32), we identify that their numerators have nothing to do with the barrier b, and their denominators are Hβ,b. We also find that their numerators are bigger than 0. So, the optimal barrier b∗ solves H′β,b∗=0, i.e.,(35)b∗=1ρ0−R0lnR0σ+βR0−σRβρ0Δ1−Δ2ρ0σ+βρ0−σRβR0Δ1−Δ2.
Whenβ⟶∞,b∗⟶1/ρ0−R0lnR02ρ0Δ1−Δ2/ρ02R0Δ1−Δ2, which is the same as (3.5) in Yuen et al. [3].Remark 3.
When the claim sizes are general distributions, e.g., Erlang(n) or mixture of exponential distribution, the explicit expression for νx,b cannot be provided for −c/α<x<0. Then, some numerical methods will be helpful, e.g., Yu et al. [20] and Zhang et al. [21].


## 2.1. Integro-differential Equations forνx,b

The integro-differential equation is a conventional method; we start this section with the expression forνx,b.Theorem 1.
For−c/α≤x<0, we have(5)αx+cν′x,b−σ+γνx,b+γ∫0x+c/ανx−z,bςzdz=0,for 0≤x<b, we have(6)cν′x,b−σ+γνx,b+γ∫0x+c/ανx−z,bςzdz=0,and for x>b, we have(7)cν′x,b−σ+β+γνx,b+βx−b+βνb,b+γ∫0x+c/ανx−z,bςzdz=0.In addition, we have the following relations:(8)νb−,b=νb+,b,(9)ν′b−,b=ν′b+,b,(10)ν0−,b=ν0+,b,(11)ν′0−,b=ν′0+,b,(12)ν−cα,b=0.Proof.
A standard method is used here, see also Yuen et al. [3] and Albrecher et al. [6]. For a small interval 0,s, all the possible events are taken into account. Then, we have(13)νx,b=e−β+σ+γsνx+cs,b+βse−σ+γsx+cs−b+νb,b1x+cs>b+βse−σ+γsνx+cs,b10<x+cs<b+γse−σ+γs∫0x+cs+c/ανx+cs−y,bςydy+os,for x≥0, and for −c/α≤x<0,(14)νx,b=e−β+σ+γsνtαs,x,b+βse−σ+γsνtαs,x+γse−σ+βs∫0tαs,x+c/ανtαs,x−y,bςydy+os,where tαs,x=xeαs+ceαs−1/α.
Taking derivative ons in (13) and letting s⟶0, we arrive at (6) and (7). By a similar method, (5) can be obtained from (14).
Continuity condition (8) can be obtained from (13), and (10) can be obtained by comparing (13) with (14). (9) can be obtained by using (6) and (7), while (11) is derived by (5) and (6). Let x=−c/α in (5), and we have boundary condition (12).Remark 1.
Obviously, (6) and (7) are the same as (2.1) and (2.2) in Yuen et al. [3], respectively.


## 2.2. Explicit Expressions ofνx,b

In this section, the density function of claim sizes is supposed to befx=μe−μx,x>0. Applying the operator d/dx+μ to (5), then we have(15)αx+cν″x,b+αμx+cμ+α−γ−σν′x,b−σμνx,b=0,for −c/α≤x<0.By the transformsνx,b=kz and z=−μx+c/α, (15) is reduced to a confluent hypergeometric equation:(16)zk″z+1−σ+γα−zk′z+σαkz=0,z<0.It follows from Abramowitz and Stegun [19] that kx admits the following expression:(17)kz=C1ezU1−γα,1−σ+γα,−z+C2zσ+γ/αMγsα,1+σ+γα,z,where C1 and C2 are two constants, Ma,b,z is the first kind of confluent hypergeometric function, and Ua,b,z is the second kind of confluent hypergeometric function. Hence, the solution of (15) can be expressed as(18)νx,b=C1e−μx+c/αU1−γα,1−σ+γα,μx+cα+C2−μx+cασ+γ/αMγα,1+σ+γα,−μx+cα.Due to boundary condition (13), one immediately deduces C1=0.Using a similar procedure to (15), we deduce(19)cν″x,b−σ+γ−cμν′x,b−σμνx,b=0for 0<x<b and(20)cν″x,b−γ+β+σ−cμν′x,b−σ+βμνx,b+βμνb,b+x−b+β=0,for x≥b. Solving (19) and (20) by the knowledge of the ordinary differential equation, one deduces(21)νx,b=C3eρ0x+C4eR0x,0≤x<b,(22)νx,b=D1eρβx+D2eRβx+D3x+D4,x≥b,where ρβ>0,Rβ<0 are roots of the equation(23)r2+μ−σ+γ+βcr−σ+βμc=0.Clearly,νx,b is bounded. Hence, D1=0. By (18), (21), and (22) and (8)–(11), we get the following equations:(24)C3+C4=C2Δ1,(25)C3ρ0+C4R0=C2Δ2,(26)D2eRβb+D3b+D4=C3eρ0b+C4eR0b,(27)D2RβeRβb+D3=C3ρ0eρ0b+C4R0eR0b,where(28)Δ1=−cμασ+γ/αMγα,1+σ+γα,−cμα,Δ2=σ+γcΔ1−γμσ+α+γ−cμασ+γ/αM1+γα,2+σ+γα,−cμα.Substituting (22) into (7) and equating the coefficients of x lead to(29)D3=ββ+σ,and then equating the constant term yields(30)βD2eRβb−σD4=γμ−c−βbD3+βb.Solving (24)–(30), we get(31)νx,b=R0−ρ0hβ−μx+c/ασ+γ/αMγ/α,1+σ+γ/α,−μx+c/αHβ,b,−cα<x<0,(32)νx,b=R0Δ1−Δ2hβHβ,beρ0x+Δ2−ρ0Δ1hβHβ,beR0x,0≤x≤b,νx,b=D2eRβx+D3x+D4,x>b,where(33)D2=ρ0R0Δ1−Δ2eρ0−Rβb+R0Δ2−ρ0Δ1eR0−RβbhβRβHβ,b−βRββ+σe−Rβb,D3=ββ+σ,D4=1−ρ0/RβR0Δ1−Δ2eρ0b+1−R0/RβΔ2−ρ0Δ1eR0bhβHβ,b−ββ+σb−1Rβ,Hβ,b=σ+βρ0−σRβΔ2−R0Δ1eρ0b+σ+βR0−σRβρ0Δ1−Δ2eR0b,hβ=−βγRβ+μβ+σ−cRβμβ+σ.Remark 2.
Forβ⟶∞, it is easy to check that Rβ⟶−μ. Letting β⟶∞ in (31) and (32) and noticing ρ0+R0=−μ+σ+γ/c, we have(34)Vcx,b=ρ0−R01+αx/cσ+γ/αMγ/α,1+σ+γ/α,−μx+c/αρ0+μΔ1−γμ/γ+α+σM1+γ/α,2+σ+γ/α,−cμ/α,−cα<x<0,Vcx,b=Δ2−R0Δ1eρ0x+ρ0Δ1−R0eR0xρ0+μΔ1−γμ/γ+α+σM1+γ/α,2+σ+γ/α,−cμ/α,0≤x≤b,which are in accordance to (3.4) in Yuen et al. [3].
When the initial surplus−c/α<x≤b, we can investigate the optimal dividend barrier b∗ which can maximize νx,b before absolute ruin. By (31) and (32), we identify that their numerators have nothing to do with the barrier b, and their denominators are Hβ,b. We also find that their numerators are bigger than 0. So, the optimal barrier b∗ solves H′β,b∗=0, i.e.,(35)b∗=1ρ0−R0lnR0σ+βR0−σRβρ0Δ1−Δ2ρ0σ+βρ0−σRβR0Δ1−Δ2.
Whenβ⟶∞,b∗⟶1/ρ0−R0lnR02ρ0Δ1−Δ2/ρ02R0Δ1−Δ2, which is the same as (3.5) in Yuen et al. [3].Remark 3.
When the claim sizes are general distributions, e.g., Erlang(n) or mixture of exponential distribution, the explicit expression for νx,b cannot be provided for −c/α<x<0. Then, some numerical methods will be helpful, e.g., Yu et al. [20] and Zhang et al. [21].


## 3. Results forΦx

Using a similar method for Theorem1, we can obtain the following results for the generalized Gerber–Shiu function Φx.Theorem 2.
The generalized Gerber–Shiu functionΦx admits the following expressions:(36)cΦ′x−γ+β+σΦx+βΦb+γs∫0x+c/αΦx−zςzdz+γs∫x+c/x∞ωx,z−xςzdz=0,x≥b,(37)cΦ′x−σ+γΦx+γs∫0x+c/αΦx−zςzdz+γs∫x+c/x∞ωx,z−xςzdz=0,0≤x<b,(38)αx+cΦ′x−σ+γΦx+γs∫0x+c/αΦx−zςzdz+γs∫x+c/x∞ωx,z−xςzdz=0,−cα≤x<0,with the following conditions:(39)Φ0+=Φ0−,(40)Φb+=Φb−,(41)Φ′b+=Φ′b−,(42)Φ′0+=Φ′0−,(43)Φ−cα=γsγ+σ∫0∞ω−cα,−cα−yςydy.


### 3.1. Explicit Expressions ofϕx

We also assume that the claim sizes have the density functionfx=μe−μxx>0, similar to Section 2.2, and we have(44)αx+cϕ″x+αμx+cμ+α−γ−σϕ′x+γμs−σ+γμϕx=0,−cα<x<0,cϕ″x−σ+γ−cμϕ′x−σ+γμ−γμsϕx=0,0≤x<b,cϕ″x+cμ−γ−β−σϕ′x−γ+β+σμ−γμsϕx+βμϕb=0,x≥b.Then, solving the above system of equations leads to(45)ϕx=c1M1x,s+c2M2x,s,−cα≤x<0,ϕx=c3eρ0sx+c4eR0sx,0≤x<b,ϕx=d1eρβsx+d2eRβsx+d4,x≥b,where(46)M1x,s=e−μx+c/αU1−γsα,1−σ+γα,μx+cα,M2x,s=−μx+cασ+γ/αMγsα,1+σ+γα,−μx+cα,and ρβs>0,Rβs<0 are solutions of the equation(47)cr2+γ+β+σ−cur−γ+β+σ−γsμ=0.Sinceϕx is a bounded function, then d1=0. By conditions (40)–(43), we have(48)Γσ+γ/αΓγ+α+σ−γs/αc1=γsσ+γ,(49)c3+c4=c1M10,s+c2M20,s,(50)c3ρ0s+c4R0s=c1M1′0,s+c2M2′0,s,(51)c3eρ0sb+c4eR0sb=d2eRβsb+d4,(52)c3ρ0seρ0sb+c4R0seR0sb=d2RβseRβsb.Inserting (45) into (36) and comparing the coefficients of constants yield(53)γs−γ−σd4+βd2eRβsb=0.Solving (48)–(53), one obtains(54)c1=γsΓγ+α+σ−γs/ασ+γΓσ+γ/α,c2=γsΓγ+α+σ−γs/αℓ1+ℓ2M1′0,s+ℓ1ρ0s+ℓ2R0sM10,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,s,c3=γsΓγ+α+σ−γs/αℓ1M1′0,sM20,s−M10,sM2′0,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,s,c4=γsΓγ+α+σ−γs/αℓ2M1′0,sM20,s−M10,sM2′0,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,s,d2=γsΓγ+α+σ−γs/ασ+γ−γsM1′0,sM20,s−M10,sM2′0,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,se−Rβsb,d4=βγsΓγ+α+σ−γs/αM1′0,sM20,s−M10,sM2′0,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,s,where(55)ℓ1=R0s−Rβsσ+γ−γs+βR0sR0s−ρ0se−ρ0sb,ℓ2=ρ0s−Rβsσ+γ−γs+βρ0sρ0s−R0se−R0sb.


## 3.1. Explicit Expressions ofϕx

We also assume that the claim sizes have the density functionfx=μe−μxx>0, similar to Section 2.2, and we have(44)αx+cϕ″x+αμx+cμ+α−γ−σϕ′x+γμs−σ+γμϕx=0,−cα<x<0,cϕ″x−σ+γ−cμϕ′x−σ+γμ−γμsϕx=0,0≤x<b,cϕ″x+cμ−γ−β−σϕ′x−γ+β+σμ−γμsϕx+βμϕb=0,x≥b.Then, solving the above system of equations leads to(45)ϕx=c1M1x,s+c2M2x,s,−cα≤x<0,ϕx=c3eρ0sx+c4eR0sx,0≤x<b,ϕx=d1eρβsx+d2eRβsx+d4,x≥b,where(46)M1x,s=e−μx+c/αU1−γsα,1−σ+γα,μx+cα,M2x,s=−μx+cασ+γ/αMγsα,1+σ+γα,−μx+cα,and ρβs>0,Rβs<0 are solutions of the equation(47)cr2+γ+β+σ−cur−γ+β+σ−γsμ=0.Sinceϕx is a bounded function, then d1=0. By conditions (40)–(43), we have(48)Γσ+γ/αΓγ+α+σ−γs/αc1=γsσ+γ,(49)c3+c4=c1M10,s+c2M20,s,(50)c3ρ0s+c4R0s=c1M1′0,s+c2M2′0,s,(51)c3eρ0sb+c4eR0sb=d2eRβsb+d4,(52)c3ρ0seρ0sb+c4R0seR0sb=d2RβseRβsb.Inserting (45) into (36) and comparing the coefficients of constants yield(53)γs−γ−σd4+βd2eRβsb=0.Solving (48)–(53), one obtains(54)c1=γsΓγ+α+σ−γs/ασ+γΓσ+γ/α,c2=γsΓγ+α+σ−γs/αℓ1+ℓ2M1′0,s+ℓ1ρ0s+ℓ2R0sM10,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,s,c3=γsΓγ+α+σ−γs/αℓ1M1′0,sM20,s−M10,sM2′0,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,s,c4=γsΓγ+α+σ−γs/αℓ2M1′0,sM20,s−M10,sM2′0,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,s,d2=γsΓγ+α+σ−γs/ασ+γ−γsM1′0,sM20,s−M10,sM2′0,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,se−Rβsb,d4=βγsΓγ+α+σ−γs/αM1′0,sM20,s−M10,sM2′0,sσ+γΓσ+γ/αℓ1ρ0s+ℓ2R0sM20,s−ℓ1+ℓ2M2′0,s,where(55)ℓ1=R0s−Rβsσ+γ−γs+βR0sR0s−ρ0se−ρ0sb,ℓ2=ρ0s−Rβsσ+γ−γs+βρ0sρ0s−R0se−R0sb.


## 4. Numerical Examples

In this last section, we give some numerical examples to illustrate the results. In Figures1–4, we assume that the claim size follows exponential distribution with mean 2, the premium rate c=4, the Poisson intensity γ=0.8, the discounted factor σ=0.04, and the debit interest α=0.06. Figure 1 depicts the profiles of νx,b for x,b∈0,30×0,30. We find that νx,b is not a monotone function of b. When b=8, the curves of xe−σT1T<∞ and νx,b are given in Figures 2 and 3 for several different β, respectively. It is shown that νx,b is a decreasing function of β, while xe−σT1T<∞ is a increasing function of β. The bigger β means the more frequent observation, which can lead to the earlier ruin. By Figure 3, we also find that νx,b is more and more close to Vcx,b as β becomes more and more large. The profiles of νx,b are given in Figure 4 for some different initial surplus x. By Figure 4, we know that the optimal barrier is b∗=16.2789 which has nothing to do with the initial surplus.Figure 1
Influence ofx and b on the expected discounted dividend νx,b.Figure 2
Influence ofβ on xexp−σT.Figure 3
Influence ofβ on νx,b.Figure 4
ν
x
,
b as a function of b.The numerical results forb∗ are given in Table 1 for various observation intensity β and debit interest α. The optimal dividend barrier b∗ is an increasing function of β, which is also an increasing function of α. More frequent observation will give rise to more dividend payments in the initial stage and earlier absolute ruin. In order to avoid the earlier absolute ruin, a bigger dividend barrier will be needed to decrease the dividend payments at the beginning.Table 1
Impact ofβ and α on b∗ when c=4,γ=0.8,μ=0.5,andσ=0.04.
α
β
=
1
β
=
5
β
=
10
β
=
20
β
=
30
β
=
50
β
=
100
β
⟶
∞
0.04
14.4939
15.9856
16.2774
16.4446
16.5045
16.5541
16.5925
16.6320
0.06
14.4954
15.9870
16.2789
16.4460
16.5059
16.5556
16.5940
16.6335
0.07
14.4960
15.9877
16.2795
16.4467
16.5066
16.5563
16.5947
16.6341
0.084
14.5125
16.0042
16.2960
16.4632
16.5231
16.5727
16.6111
16.6506



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*Source: 6395717-2020-03-17.xml*