# The Perturbed Dual Risk Model with Constant Interest and a Threshold Dividend Strategy

Authors: Fanzi Zeng; Jisheng Xu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the perturbed dual risk model with constant interest and a threshold dividend strategy. Firstly, we investigate the moment-generation function of the present value of
total dividends until ruin. Integrodifferential equations with certain boundary conditions are derived for the present value of total dividends. Furthermore, using techniques of sinc numerical methods, we obtain the approximation results to the expected present value of total dividends. Finally, numerical examples are presented to show the impact of interest on the expected present value of total dividends and the absolute ruin probability.


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## Body

## 1. Introduction

In insurance mathematics, the classical risk model has been the center of focus for decades. The surplus in the classical model at timet can be expressed as
(1)U(t)=u+ct-S(t),t≥0,
where u≥0 is the initial surplus, c>0 is the premium rate, and S(t) is the aggregate claims by time t, usually modeled by a compound Poisson process. In recent years, quite a few interesting papers have been written on a model which is dual to the classical insurance risk model. See, for example, Avanzi et al. [1], Avanzi and Gerber [2]. The surplus of a dual classical risk model at time t is
(2)U(t)=u-ct+S(t),t≥0.In this model, the premium rate should be viewed as an expense rate and claims should be viewed as profits or gains. While not very popular in insurance mathematics, this model has appeared in various literature (see Cramér [3, Section  5.13]  and Seal [4, pages 116–119] and Takács [5, pages 152–154]). There were many possible interpretations for this model. For example, we can treat the surplus as the amount of the capital of a business engaged in research and development. The company paid expenses for research, and occasional profit of random amounts (such as the award of a patent or a sudden increase in sales) arises according to a Poisson process. A similar model was used by Bayraktar and Egami [6] to model the capital of a venture capital investment. Another model was an annuity business. The company issues payments continuously to annuitants, while the gross reserve of an annuitant was released as emerging profit when he died. Yang and Zhu [7] generalized the dual model into a regime-switching setting and calculated bounds for ruin probabilities. One of the current topics of interest in insurance mathematics is the problem of maximizing the expected total discounted dividends until ruin, which goes back to de Finetti [8], and it had also been studied by Bühlmann [9, Section  6.4], Gerber [10, Sections  7 and 8], and Gerber [11, Section 10.1]. In Avanzi et al. [1], the authors studied the expected total discounted dividends until ruin for the dual model under the barrier strategy by means of integrodifferential equations. They derived explicit formulas when profits or gains followed an exponential or a mixture of exponential distributions and showed that the optimal value of the dividend barrier under the dual model was independent of the initial surplus. Avanzi and Gerber [2] studied a dual model perturbed by diffusion and discussed how the optimal value of the dividend barrier can be determined. Albrecher et al. [12] studied a dual model that also paid taxes when the surplus was at a running maximum and calculated the expected total discounted dividends before ruin for exponentially distributed profits.Considering the perturbed dual risk model, the surplus of an insurer has the following form(3)U(t)=u-ct+∑i=1N(t)Xi+σB(t),t≥0,
where u≥0 is the initial surplus and c>0 is the constant rate. {Xi,i=1,2,…} is a sequence of independent income size random variables with a common distribution function F(x) with F(0)=0 and density function f(x). {N(t),t≥0} is the Poisson income-number process with an intensity λ>0 and is defined as N(t)=sup{k:T1+⋯+Tk≤t}, where the i.i.d. interincome times {Ti}i=1∞ have a common exponential distribution with a parameter λ. {B(t),t≥0} is a standard Brownian motion with B(0)=0 and σ>0 is a constant, representing the diffusion volatility parameter. In addition, {Xi,i=1,2,…}, {N(t),t≥0} and {B(t),t≥0}, are mutually independent.Dividend strategy for insurance risk models was first proposed by de Finetti [8] to reflect more realistically the surplus cash flowed in an insurance portfolio. Various threshold strategies have been studied by many authors, including Gerber and Landry [13], Ng [14], and Lin and Pavlova [15]. Among them, Ng [14] showed that the threshold strategy was optimal when the dividend rate was bounded and the individual claim distribution was exponential. Recently, the threshold dividend strategy has been considered in the class of compound Poisson process perturbed by diffusion and its generalizations; readers may refer to Avanzi and Gerber [2], Gao and Liu [16], Wan [17], and the references therein.Then, we consider the modification of the surplus process by a threshold strategy with a threshold levelb. When U(t) is below b, no dividends are paid and the surplus decreases at the original rate c1. But when U(t) is above b, the surplus would decrease at a different rate c2>c1 and dividends are paid at rate c2-c1. Incorporating the threshold strategy into (3) yields the surplus process {Ub(t),t≥0} which can be expressed by
(4)dUb(t)={-c1dt+dS(t)+σdB(t),Ub(t)<b,-c2dt+dS(t)+σdB(t),Ub(t)≥b,
where Ub(0)=u, 0<b<∞, and S(t)=∑i=1N(t)‍Xi.In this paper, we consider that the insurance company earns credit interest with a constant forcer>0. In the mean time, the insurer will receive interest. Incorporating interest into (4) yields the surplus process {Ub(t),t≥0} which can be expressed by
(5)dUb(t)={-c1dt+rUb(t)dt+dS(t)+σdB(t),Ub(t)<b,-c2dt+rUb(t)dt+dS(t)+σdB(t),Ub(t)≥b.LetD(t) denote the cumulative amount of dividends paid out up to time t and δ>0 the force of interest; then
(6)Du,b=∫0Tbe-δtdD(t)
is the present value of all dividends until Tb, where Tb denoted by Tb=inf{t≥0:Ub(t)≤0} is the time of ruin. An alternative expression for Du,b is
(7)Du,b=(c2-c1)∫0Tbe-δtI(Ub(t)>b)dt,
with I(·) denoting the indicator function. It is obvious that 0<Du,b≤(c2-c1)/δ.In the sequel, we will be interested in the following moment generating function:(8)M(u,y;b)=E[eyDu,b]
(for those values of y where it exist) and the expected total discounted dividends until ruin is
(9)V(u;b)=E[Du,b∣Ub(0)=u].
Throughout this paper, we assume that M(u,y;b) and V(u;b) are sufficiently smooth functions in u and y, respectively.


## 2. Integrodifferential Equations

### 2.1. Integrodifferential Equations forM(u,y;b)

In this section, we will give the integrodifferential equations satisfied by the moment generating functionM(u,y;b), respectively. It is easy to see that M(u,y;b) behaves differently with different initial surplus. Hence, for notation convenience, we set
(10)M(u,y;b)={M1(u,y;b),if0<u<b,M2(u,y;b),ifb≤u<∞.
In the case of σ=0, we write M(u,y;b) as M0(u,y;b), and the other one is similar.In the following, we firstly derive the integrodifferential equations satisfied byM0(u,y;b).Theorem 1.
When0<u<b,
(11)(ru-c1)∂M10(u,y;b)∂u=yδ∂M10(u,y;b)∂y+λM10(u,y;b)-λ∫0b-uM10(u+x,y;b)f(x)dx-λ∫b-u∞M20(u+x,y;b)f(x)dx,
and  for  b≤u<∞,
(12)(ru-c2)∂M20(u,y;b)∂u=yδ∂M20(u,y;b)∂y+λM20(u,y;b)-λ∫0∞M10(u+x,y;b)f(x)dx,
with the following boundary conditions:
(13)M10(b-,y;b)=M20(b,y;b),(14)limu→∞M20(u,y;b)=e((c2-c1)/y)δ,(15)(rb-c1)∂M10(u,y;b)∂u|u=b-=(rb-c2)∂M20(u,y;b)∂u|u=b.Proof.
When0<u<b, consider t>0 such that the surplus cannot reach level b by time; that is, hδ(u,τ)=erτ(u-(c1/r))+(c1/r)>0. By conditioning on the time and amount of first claim, if it occurs by t, and whether the claim causes ruin, one gets
(16)M10(u,y;b)=(1-λτ)M10(hδ(u,τ),ye-δτ;b)+λτE[M10(hδ(u,τ)+x,ye-δτ;b)].
By Taylor’s expansion,
(17)M10(hδ(u,τ),ye-δτ;b)=M10(u,y;b)+(ru-c1)τ∂M10(u,y;b)∂u-yδτ∂M10(u,y;b)∂y+o(τ).
Also, due to
(18)E[M10(hδ(u,τ),ye-δτ;b)]=∫0∞‍M10(hδ(u,τ)+x,ye-δτ;b)f(x)dx.
Substituting the above expansion into (16), dividing both sides of (16) by t, and letting t→0, we can get (11). Using the same way as (11), we can easily prove (12) when u→∞, Tb=∞ so the condition (12) is correct, by the method used in Gao and Liu [16], and it is easy to check that the boundary conditions hold. This ends the proof of Theorem 1.Now we consider the case ofσ≠0. For t≥0, define W(t)=σ∫0t‍e-rsdB(s). Thus, W(t) is an Ito^ stochastic integral. Denote by {〈W〉(t),t≥0} the variance process of {W(t),t≥0}. We have 〈W〉(t)=σ2∫0t‍e-2rsds=(2r)-1σ2(1-e-2rs) for t≥0. Let v(s)=inf{t:〈W〉(t)>s}; then
(19)v(s)=12lnσ2σ2-2rs,0<s<σ22r.SetW(t)=B(v(t)) for t≥0, i=1,2. By the time change of Brownian motion, we have that Wi is local standard Brownian motion with Wi(0)=0 running for an amount of time σ2/2r.Letϕ(u,s)=ers(u+c1∫0s‍e-rtdt) and ϕ1(u,s)=ers(u+c2∫0∞‍e-rtdt). Then, we have the following results.Theorem 2.
When0<u<b,
(20)12σ2∂2M1(u,y;b)∂u2+(ru-c1)∂M1(u,y;b)∂u=yδ∂M1(u,y;b)∂y+λM1(u,y;b)-λ∫0b-uM1(u+x,y;b)f(x)dx+λ∫b-u∞M2(u+x,y;b)f(x)dx,
and when b≤u<∞,
(21)12σ2∂2M2(u,y;b)∂u2+(ru-c2)∂M2(u,y;b)∂u=yδ∂M2(u,y;b)∂y+(λ-(c2-c1))M2(u,y;b)+λ∫0∞‍M2(u+x,y;b)f(x)dx,
with the following boundary conditions:
(22)M1(0,y;b)=1,(23)M1(b-,y;b)=M2(b,y;b),(24)limu→∞M2(u,y;b)=e((c2-c1)/y)δ,(25)(rb-c1)∂M1(u,y;b)∂u∣u=b-=(rb-c2)∂M2(u,y;b)∂u∣u=b.Proof.
For0<u<b, assume that ϵ,t>0 such that ϵ<u<b. Define T0=Ttϵ∧T1 and Ttϵ=inf{s>0:ϕ(u+W(s),s)∉(ϵ,b)}∧t. We have P(T0<∞)=1 for all s∈(0,T0). Then, by the strong Markov property, we get
(26)M1(u,y;b)=Eu[eyDU,b]=Eu[M1(Ub(T0),ye-δT0;b)]=E[I(T1>t)M1(Ub(Ttℰ),ye-δTtℰ;b)]+E[I(T1≤t)M1(Ub(TT1ℰ),ye-δTT1ℰ;b)]=I(t)+II(t).
By the assumption of independence, one gets
(27)I(t)=e-λtE[M1(Ub(Ttℰ),ye-δTtℰ;b)],II(t)=∫0tλe-λsE[I(Ttℰ=s)M1(Ub(s),ye-δTtℰ;b)]ds+∫0tλe-λsE[I(Ttℰ<s)M1(Ub(s),ye-δTtℰ;b)]ds=∫0tλe-λsE[I(Ttℰ=s)gs(ϕ(u+W(s),s))]ds+∫0tλe-λsE[I(Ttℰ<s)M1(Ub(s),ye-δTtℰ;b)]ds,
where
(28)gs(ϕ(u+W(s),s))=∫0b-ϕ(u+W(s),s)M1(ϕ(u+W(s),s)+x,ye-δs;b)ds+∫b-ϕ(u+W(s),s)∞M2(ϕ(u+W(s),s)+x,ye-δs;b)ds.
Noting that limt→0(Ttϵ=t)=1, limt→0(Ttϵ<t)=0 a.s. and using Ito^’s formula, we have
(29)limt→0I(t)-M1(u,y;b)t=12σ2∂2M10(u,y;b)∂u2+(ru-c1)∂M10(u,y;b)∂u-yδ∂M10(u,y;b)∂y-λM10(u,y;b),limt→0II(t)t=λ∫0b-uM10(u+x,y;b)f(x)dx+λ∫b-u∞M20(u+x,y;b)f(x)dx.
From (26) and (29), we can get (20) for all u∈(ϵ,). Hence, (20) holds in (0,b).
Similarly, using the above method, we get (21). Conditions (22) and (23) are obvious. By (14) and (15) and by the weak convergence method used in [17], it is easy to check that the boundary conditions (24) and (25) hold. This ends the proof of Theorem 2.


### 2.2. Integrodifferential Equations forV(u,b)

Theorem 3.
When0<u<b,
(30)12σ2V1′′(u)+(ru-c1)V1′(u)-(λ+δ)V1(u)+λ∫0b-uV1(u+x;b)f(x)dx+λ∫b-u∞V2(u+x;b)f(x)dx=0,
and when b≤u<∞,
(31)12σ2V2′′(u)+(ru-c2)V2′(u)-(λ+δ)V2(u)+λ∫0∞V2(u+x;b)f(x)dx+c2-c1=0,
with the following boundary conditions:
(32)V1(0;b)=0,V1(b-;b)=V2(b;b),limu→∞V2(u;b)=(c2-c1)δ,(rb-c1)∂V1(u;b)∂u|u=b-=(rb-c2)∂V2(u;b)∂u|u=b.


## 2.1. Integrodifferential Equations forM(u,y;b)

In this section, we will give the integrodifferential equations satisfied by the moment generating functionM(u,y;b), respectively. It is easy to see that M(u,y;b) behaves differently with different initial surplus. Hence, for notation convenience, we set
(10)M(u,y;b)={M1(u,y;b),if0<u<b,M2(u,y;b),ifb≤u<∞.
In the case of σ=0, we write M(u,y;b) as M0(u,y;b), and the other one is similar.In the following, we firstly derive the integrodifferential equations satisfied byM0(u,y;b).Theorem 1.
When0<u<b,
(11)(ru-c1)∂M10(u,y;b)∂u=yδ∂M10(u,y;b)∂y+λM10(u,y;b)-λ∫0b-uM10(u+x,y;b)f(x)dx-λ∫b-u∞M20(u+x,y;b)f(x)dx,
and  for  b≤u<∞,
(12)(ru-c2)∂M20(u,y;b)∂u=yδ∂M20(u,y;b)∂y+λM20(u,y;b)-λ∫0∞M10(u+x,y;b)f(x)dx,
with the following boundary conditions:
(13)M10(b-,y;b)=M20(b,y;b),(14)limu→∞M20(u,y;b)=e((c2-c1)/y)δ,(15)(rb-c1)∂M10(u,y;b)∂u|u=b-=(rb-c2)∂M20(u,y;b)∂u|u=b.Proof.
When0<u<b, consider t>0 such that the surplus cannot reach level b by time; that is, hδ(u,τ)=erτ(u-(c1/r))+(c1/r)>0. By conditioning on the time and amount of first claim, if it occurs by t, and whether the claim causes ruin, one gets
(16)M10(u,y;b)=(1-λτ)M10(hδ(u,τ),ye-δτ;b)+λτE[M10(hδ(u,τ)+x,ye-δτ;b)].
By Taylor’s expansion,
(17)M10(hδ(u,τ),ye-δτ;b)=M10(u,y;b)+(ru-c1)τ∂M10(u,y;b)∂u-yδτ∂M10(u,y;b)∂y+o(τ).
Also, due to
(18)E[M10(hδ(u,τ),ye-δτ;b)]=∫0∞‍M10(hδ(u,τ)+x,ye-δτ;b)f(x)dx.
Substituting the above expansion into (16), dividing both sides of (16) by t, and letting t→0, we can get (11). Using the same way as (11), we can easily prove (12) when u→∞, Tb=∞ so the condition (12) is correct, by the method used in Gao and Liu [16], and it is easy to check that the boundary conditions hold. This ends the proof of Theorem 1.Now we consider the case ofσ≠0. For t≥0, define W(t)=σ∫0t‍e-rsdB(s). Thus, W(t) is an Ito^ stochastic integral. Denote by {〈W〉(t),t≥0} the variance process of {W(t),t≥0}. We have 〈W〉(t)=σ2∫0t‍e-2rsds=(2r)-1σ2(1-e-2rs) for t≥0. Let v(s)=inf{t:〈W〉(t)>s}; then
(19)v(s)=12lnσ2σ2-2rs,0<s<σ22r.SetW(t)=B(v(t)) for t≥0, i=1,2. By the time change of Brownian motion, we have that Wi is local standard Brownian motion with Wi(0)=0 running for an amount of time σ2/2r.Letϕ(u,s)=ers(u+c1∫0s‍e-rtdt) and ϕ1(u,s)=ers(u+c2∫0∞‍e-rtdt). Then, we have the following results.Theorem 2.
When0<u<b,
(20)12σ2∂2M1(u,y;b)∂u2+(ru-c1)∂M1(u,y;b)∂u=yδ∂M1(u,y;b)∂y+λM1(u,y;b)-λ∫0b-uM1(u+x,y;b)f(x)dx+λ∫b-u∞M2(u+x,y;b)f(x)dx,
and when b≤u<∞,
(21)12σ2∂2M2(u,y;b)∂u2+(ru-c2)∂M2(u,y;b)∂u=yδ∂M2(u,y;b)∂y+(λ-(c2-c1))M2(u,y;b)+λ∫0∞‍M2(u+x,y;b)f(x)dx,
with the following boundary conditions:
(22)M1(0,y;b)=1,(23)M1(b-,y;b)=M2(b,y;b),(24)limu→∞M2(u,y;b)=e((c2-c1)/y)δ,(25)(rb-c1)∂M1(u,y;b)∂u∣u=b-=(rb-c2)∂M2(u,y;b)∂u∣u=b.Proof.
For0<u<b, assume that ϵ,t>0 such that ϵ<u<b. Define T0=Ttϵ∧T1 and Ttϵ=inf{s>0:ϕ(u+W(s),s)∉(ϵ,b)}∧t. We have P(T0<∞)=1 for all s∈(0,T0). Then, by the strong Markov property, we get
(26)M1(u,y;b)=Eu[eyDU,b]=Eu[M1(Ub(T0),ye-δT0;b)]=E[I(T1>t)M1(Ub(Ttℰ),ye-δTtℰ;b)]+E[I(T1≤t)M1(Ub(TT1ℰ),ye-δTT1ℰ;b)]=I(t)+II(t).
By the assumption of independence, one gets
(27)I(t)=e-λtE[M1(Ub(Ttℰ),ye-δTtℰ;b)],II(t)=∫0tλe-λsE[I(Ttℰ=s)M1(Ub(s),ye-δTtℰ;b)]ds+∫0tλe-λsE[I(Ttℰ<s)M1(Ub(s),ye-δTtℰ;b)]ds=∫0tλe-λsE[I(Ttℰ=s)gs(ϕ(u+W(s),s))]ds+∫0tλe-λsE[I(Ttℰ<s)M1(Ub(s),ye-δTtℰ;b)]ds,
where
(28)gs(ϕ(u+W(s),s))=∫0b-ϕ(u+W(s),s)M1(ϕ(u+W(s),s)+x,ye-δs;b)ds+∫b-ϕ(u+W(s),s)∞M2(ϕ(u+W(s),s)+x,ye-δs;b)ds.
Noting that limt→0(Ttϵ=t)=1, limt→0(Ttϵ<t)=0 a.s. and using Ito^’s formula, we have
(29)limt→0I(t)-M1(u,y;b)t=12σ2∂2M10(u,y;b)∂u2+(ru-c1)∂M10(u,y;b)∂u-yδ∂M10(u,y;b)∂y-λM10(u,y;b),limt→0II(t)t=λ∫0b-uM10(u+x,y;b)f(x)dx+λ∫b-u∞M20(u+x,y;b)f(x)dx.
From (26) and (29), we can get (20) for all u∈(ϵ,). Hence, (20) holds in (0,b).
Similarly, using the above method, we get (21). Conditions (22) and (23) are obvious. By (14) and (15) and by the weak convergence method used in [17], it is easy to check that the boundary conditions (24) and (25) hold. This ends the proof of Theorem 2.


## 2.2. Integrodifferential Equations forV(u,b)

Theorem 3.
When0<u<b,
(30)12σ2V1′′(u)+(ru-c1)V1′(u)-(λ+δ)V1(u)+λ∫0b-uV1(u+x;b)f(x)dx+λ∫b-u∞V2(u+x;b)f(x)dx=0,
and when b≤u<∞,
(31)12σ2V2′′(u)+(ru-c2)V2′(u)-(λ+δ)V2(u)+λ∫0∞V2(u+x;b)f(x)dx+c2-c1=0,
with the following boundary conditions:
(32)V1(0;b)=0,V1(b-;b)=V2(b;b),limu→∞V2(u;b)=(c2-c1)δ,(rb-c1)∂V1(u;b)∂u|u=b-=(rb-c2)∂V2(u;b)∂u|u=b.


## 3. Numerical Analysis

The second order system of integrodifferential equations such as in Theorem3 does not have an explicit solution except in some special cases. In this section, we propose the approximate solution of this equation via the use of a collocation based on sinc methods. The sinc method is highly efficient numerical method developed by Frank Stenger. It is widely used in various fields of numerical analysis such as interpolation, quadrature, approximation of transforms, and solution of integral, ordinary differential, and partial differential equations. An introduction to the sinc approximation theory can be found in the Appendix.


### 3.1. Numerical Solution of the Expected Present Value of Total DividendsV(u;b)

To construct an approximation on the interval(0,∞), we consider the following conformal map:
(33)ϕ(z)=logz.The functionϕ also provides a one-to-one transformation of (0,∞) onto the real line ℝ. The sinc grid points zk are defined for h>0 and k=0, ±1,±2,… by
(34)zk=ϕ-1(kh)=ekh.In order to adopt the sinc method procedure, we arrange the systems in Theorem3 into the following integrodifferential equation:
(35)12σ2V′′(u)+[(ru-c2)I(u≥b)+(ru-c1)I(0<u<b)]×V′(u)-(λ+δ)V(u)+λ∫0∞V(u+x;b)×f(x)dx+(c2-c1)I(u>b)=0.Using the properties of convolution, the above equation can be further written as(36)12σ2V′′(u)+[(ru-c2)I(u≥b)22σ2V′′(u)++(ru-c1)I(0<u<b)]V′(u)-(λ+δ)V(u)+λ∫u∞V(x;b)f(x-u)dx+(c2-c1)I(u>b)=0,
with the following boundary conditions:
(37)V(0;b)=0,limu→∞V(u;b)=c2-c1δ.SetW(u)=V(u;b)-LV(u;b)=V(u;b)-(u/(1+u))((c2-c1)/δ); then W(u)∈Lα~,β~,d(ϕ) and satisfies
(38)W′′(u)+μ(u)W′(u)-2σ2(λ+δ)W(u)+2σ2λ∫u∞W(x)f(u-x)dx+R(u)=0,
with the following boundary conditions:
(39)W(0)=0,limu→∞W(u)=0,
where
(40)μ(u)=2σ2[(ru-c2)I(u≥b)+(ru-c1)I(0<u<b)],R(u)=2σ2(c2-c1)I(u≥b)-2(1+u)3c2-c1δ+μ(u)1(1+u)2c2-c1δ-2σ2(λ+δ)u1+uc2-c1δ+λ2σ2c2-c1δ∫u∞x1+xf(x-u)dx.Letϕ be defined by (33). The sinc grid points are defined for h>0 by
(41)uk=ekh,k=0,±1,±2,….Then, by using TheoremsA.3 and A.6 in the Appendix, we have
(42)∫u∞f(x-u)W(x)dx≈∑l=-MN∑i=-MNwi*AilWl,(43)W(x)≈W~(x)=∑l=-MNwlWl,
where A=YF(S)Y-1, Ail is the (i,l)th element of matrix A. Wl denotes an approximate value of W(ul), and ϕ(x)=logx, ϕ(0)=-∞, ϕ(+∞)=+∞, and ϕ′(x)=1/x. wj and wj* for j=-M,…,N are defined as Definition A.2 in the Appendix.SinceW(x)∈Lα~,β~,d(ϕ), by Theorem A.3 in the Appendix, then it is convenient to take wj=γj=S(j,h)∘ϕ, j=-M,…,N. Moreover, by some simple calculations, we have wj*(uk)=S(j,h)∘ϕ(uk), j,k=-M,…,N with ϕ  and uk being defined in (33) and (41), respectively.Having replaced the integral term on the left-hand side of system (38) with the right-hand side of (42) and having substituted u=uk for k=-M,…,N, where uk are sinc grid points and also by replacing W(u) by W~(u) as in (43), we obtain the following collocation result:
(44)W~′′(uk)+μ(uk)W~′(uk)-2σ2(λ+δ)W~(uk)+2σ2λ∑l=-MN‍∑i=-MN‍S(j,h)∘ϕ(uk)AilWl+R(uk)=0,
where W~(uk)=∑l=-MN‍WlS(l,h)∘ϕ(uk).Then, by TheoremA.7, we have
(45)W~(uk)=∑l=-MNWlδlk(0),W~′(uk)=∑l=-MNWlϕ′(uk)h-1δlk(1),W~′′(uk)=∑l=-MN‍Wl[(ϕ′(uk))2ϕ′′(uk)h-1δlk(1)hhhhhhhhhh+(ϕ′(uk))2h-2δlk(2)].Replacing (45) in (44), we rewrite (44) as
(46)∑l=-MNWl{ϕ′′(uk)h-1δlk(1)∑i=-MNS(j,h)∘ϕ(uk)Ailhhhhhhhhh+(ϕ′(uk))2h-2δlk(2)+μ(uk)ϕ′(uk)h-1δlk(1)hhhhhhhhh-2σ2(λ+δ)δlk(0)+2σ2λhhhhhhhhh×∑i=-MNS(j,h)∘ϕ(uk)Ailϕ′′}=-R(uk).Having multiplied the resulting equations byh2/(ϕ′(uk))2, we have
(47)∑l=-MNWl{δlk(2)+h[ϕ′′(uk)(ϕ′(uk))2+μ(uk)ϕ′(uk)]δlk(1)hhhhhhhhh-2h2(λ+δ)σ2(ϕ′(uk))2δlk(0)hhhhhhhhh+2λh2σ2(ϕ′(uk))2∑i=-MNS(j,h)∘ϕ(uk)Ailϕ′′(uk)(ϕ′(uk))2}=-h2R(uk)(ϕ′(uk))2.Now, sinceδkl(0)=δlk(0), δkl(1)=-δlk(1), δkl(2)=δlk(2), and ϕ′′(uk)/(ϕ′(uk))2=-(1/ϕ′(uk))′, we obtain the collocation result as
(48)∑l=-MNWl{δkl(2)+h[(1ϕ′(uk))′-μ(uk)ϕ′(uk)]δkl(1)hhhhhhhhh-2h2(λ+δ)σ2(ϕ′(uk))2δkl(0)+2λh2σ2(ϕ′(uk))2hhhhhhhhh×∑i=-MNS(j,h)∘ϕ(uk)Ail(1ϕ′(uk))′}=-h2R(uk)(ϕ′(uk))2,k=-M,…,N.We setI(m)=[δkl(m)] and m=-1,0,1,2, where δkl(m) denotes the (k,l)th element of the matrix I(m). Also, we denote Δ¯=[S(i,h)∘ϕ(uk)] and I(m), m=-1,0,1,2 are square matrices of order (N+M+1)×(N+M+1). So, the system (48) can be given in matrix form as
(49)BW=F,
where W=[W-M,…,WN]T,  F=[-(h2H(u-M)/(ϕ′(u-M))2),…,-(h2H(uN)/(ϕ′(uN))2)]T, and
(50)B=I(2)+hDm[(1ϕ′)′-μϕ′]I(1)-2h2σ2(λ+δ)Dm1(ϕ′)2I(0)+2h2λσ2Dm1(ϕ′)2Δ¯A.The above linear system contains(N+M+1) equations with (N+M+1) unknown coefficient Wk, k=-M,…,N. Solving this linear system, we obtain the approximate solution of the system as follows:
(51)W(u)≈W~(u)=∑l=-MNWlS(l,h)∘ϕ(u),V(u;b)=W(u)+u1+uc2-c1δ.


## 3.1. Numerical Solution of the Expected Present Value of Total DividendsV(u;b)

To construct an approximation on the interval(0,∞), we consider the following conformal map:
(33)ϕ(z)=logz.The functionϕ also provides a one-to-one transformation of (0,∞) onto the real line ℝ. The sinc grid points zk are defined for h>0 and k=0, ±1,±2,… by
(34)zk=ϕ-1(kh)=ekh.In order to adopt the sinc method procedure, we arrange the systems in Theorem3 into the following integrodifferential equation:
(35)12σ2V′′(u)+[(ru-c2)I(u≥b)+(ru-c1)I(0<u<b)]×V′(u)-(λ+δ)V(u)+λ∫0∞V(u+x;b)×f(x)dx+(c2-c1)I(u>b)=0.Using the properties of convolution, the above equation can be further written as(36)12σ2V′′(u)+[(ru-c2)I(u≥b)22σ2V′′(u)++(ru-c1)I(0<u<b)]V′(u)-(λ+δ)V(u)+λ∫u∞V(x;b)f(x-u)dx+(c2-c1)I(u>b)=0,
with the following boundary conditions:
(37)V(0;b)=0,limu→∞V(u;b)=c2-c1δ.SetW(u)=V(u;b)-LV(u;b)=V(u;b)-(u/(1+u))((c2-c1)/δ); then W(u)∈Lα~,β~,d(ϕ) and satisfies
(38)W′′(u)+μ(u)W′(u)-2σ2(λ+δ)W(u)+2σ2λ∫u∞W(x)f(u-x)dx+R(u)=0,
with the following boundary conditions:
(39)W(0)=0,limu→∞W(u)=0,
where
(40)μ(u)=2σ2[(ru-c2)I(u≥b)+(ru-c1)I(0<u<b)],R(u)=2σ2(c2-c1)I(u≥b)-2(1+u)3c2-c1δ+μ(u)1(1+u)2c2-c1δ-2σ2(λ+δ)u1+uc2-c1δ+λ2σ2c2-c1δ∫u∞x1+xf(x-u)dx.Letϕ be defined by (33). The sinc grid points are defined for h>0 by
(41)uk=ekh,k=0,±1,±2,….Then, by using TheoremsA.3 and A.6 in the Appendix, we have
(42)∫u∞f(x-u)W(x)dx≈∑l=-MN∑i=-MNwi*AilWl,(43)W(x)≈W~(x)=∑l=-MNwlWl,
where A=YF(S)Y-1, Ail is the (i,l)th element of matrix A. Wl denotes an approximate value of W(ul), and ϕ(x)=logx, ϕ(0)=-∞, ϕ(+∞)=+∞, and ϕ′(x)=1/x. wj and wj* for j=-M,…,N are defined as Definition A.2 in the Appendix.SinceW(x)∈Lα~,β~,d(ϕ), by Theorem A.3 in the Appendix, then it is convenient to take wj=γj=S(j,h)∘ϕ, j=-M,…,N. Moreover, by some simple calculations, we have wj*(uk)=S(j,h)∘ϕ(uk), j,k=-M,…,N with ϕ  and uk being defined in (33) and (41), respectively.Having replaced the integral term on the left-hand side of system (38) with the right-hand side of (42) and having substituted u=uk for k=-M,…,N, where uk are sinc grid points and also by replacing W(u) by W~(u) as in (43), we obtain the following collocation result:
(44)W~′′(uk)+μ(uk)W~′(uk)-2σ2(λ+δ)W~(uk)+2σ2λ∑l=-MN‍∑i=-MN‍S(j,h)∘ϕ(uk)AilWl+R(uk)=0,
where W~(uk)=∑l=-MN‍WlS(l,h)∘ϕ(uk).Then, by TheoremA.7, we have
(45)W~(uk)=∑l=-MNWlδlk(0),W~′(uk)=∑l=-MNWlϕ′(uk)h-1δlk(1),W~′′(uk)=∑l=-MN‍Wl[(ϕ′(uk))2ϕ′′(uk)h-1δlk(1)hhhhhhhhhh+(ϕ′(uk))2h-2δlk(2)].Replacing (45) in (44), we rewrite (44) as
(46)∑l=-MNWl{ϕ′′(uk)h-1δlk(1)∑i=-MNS(j,h)∘ϕ(uk)Ailhhhhhhhhh+(ϕ′(uk))2h-2δlk(2)+μ(uk)ϕ′(uk)h-1δlk(1)hhhhhhhhh-2σ2(λ+δ)δlk(0)+2σ2λhhhhhhhhh×∑i=-MNS(j,h)∘ϕ(uk)Ailϕ′′}=-R(uk).Having multiplied the resulting equations byh2/(ϕ′(uk))2, we have
(47)∑l=-MNWl{δlk(2)+h[ϕ′′(uk)(ϕ′(uk))2+μ(uk)ϕ′(uk)]δlk(1)hhhhhhhhh-2h2(λ+δ)σ2(ϕ′(uk))2δlk(0)hhhhhhhhh+2λh2σ2(ϕ′(uk))2∑i=-MNS(j,h)∘ϕ(uk)Ailϕ′′(uk)(ϕ′(uk))2}=-h2R(uk)(ϕ′(uk))2.Now, sinceδkl(0)=δlk(0), δkl(1)=-δlk(1), δkl(2)=δlk(2), and ϕ′′(uk)/(ϕ′(uk))2=-(1/ϕ′(uk))′, we obtain the collocation result as
(48)∑l=-MNWl{δkl(2)+h[(1ϕ′(uk))′-μ(uk)ϕ′(uk)]δkl(1)hhhhhhhhh-2h2(λ+δ)σ2(ϕ′(uk))2δkl(0)+2λh2σ2(ϕ′(uk))2hhhhhhhhh×∑i=-MNS(j,h)∘ϕ(uk)Ail(1ϕ′(uk))′}=-h2R(uk)(ϕ′(uk))2,k=-M,…,N.We setI(m)=[δkl(m)] and m=-1,0,1,2, where δkl(m) denotes the (k,l)th element of the matrix I(m). Also, we denote Δ¯=[S(i,h)∘ϕ(uk)] and I(m), m=-1,0,1,2 are square matrices of order (N+M+1)×(N+M+1). So, the system (48) can be given in matrix form as
(49)BW=F,
where W=[W-M,…,WN]T,  F=[-(h2H(u-M)/(ϕ′(u-M))2),…,-(h2H(uN)/(ϕ′(uN))2)]T, and
(50)B=I(2)+hDm[(1ϕ′)′-μϕ′]I(1)-2h2σ2(λ+δ)Dm1(ϕ′)2I(0)+2h2λσ2Dm1(ϕ′)2Δ¯A.The above linear system contains(N+M+1) equations with (N+M+1) unknown coefficient Wk, k=-M,…,N. Solving this linear system, we obtain the approximate solution of the system as follows:
(51)W(u)≈W~(u)=∑l=-MNWlS(l,h)∘ϕ(u),V(u;b)=W(u)+u1+uc2-c1δ.


## 4. Numerical Example

In this section, we consider some numerical samples to illustrate the performance of sinc method and investigate how much the values ofV(u;b) are affected by interest r. An example is solved under the assumption that the claim size density is given by f(y)=ηe-ηyI(y>0).Example 1.
Letc1=0.1, c2=0.2, μ=5, α=β=π/4, λ=3, b=2, σ=0.05, δ=0.006, α~=1/2, β~=1/4, d=1/4000, and r=0.05, the result is shown in Figure 1.
From Figure1, we see that V(u;2) is an increasing function with respect to u.Figure 1
The expected present value of total dividendsV(u;2).



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