# Dividend Problems with a Barrier Strategy in the Dual Risk Model until Bankruptcy

Authors: Donghai Liu; Zaiming Liu
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

The paper studies the dual risk model with a barrier strategy under the concept of
bankruptcy, in which one has a positive probability to continue business despite temporary
negative surplus. Integrodifferential equations for the expectation of the discounted
dividend payments and the probability of bankruptcy are derived. Moreover, when
the gain size distribution is exponential, explicit solutions for the expected dividend
payments and the bankruptcy probability are obtained for constant bankruptcy rate
function. It also provided some numerical examples to illustrate the applications of the
explicit solutions.


---

## Body

## 1. Introduction

In the continuous time dual risk model, the company's surplus process{U(t),t≥0} with initial surplus U(0)=u is given by
(1)U(t)=u-ct+S(t)=u-ct+∑i=1N(t)‍Xi,t≥0,
where c>0 is the constant rate of expenses per unit time, {N(t),t≥0} is a Poisson process with intensity λ>0, and the gain size {Xi,i≥1} is a sequence of independent and identically distributed positive continuous random variables with finite mean, independent of {N(t),t≥0}. Assume all Xi have the same distribution as a generic random variable X, which has probability density function f(x) and cumulative distribution function F(x). Under model (1), the expected increase in surplus per unit time is μ=λEX-c, and it is assumed that μ>0.Since De Finetti [1] proposed dividend strategies for an insurance risk model, the risk model in the presence of dividend payments has become a more and more popular topic in risk theory. In recent years, quite a few interesting papers have been written on the dual risk model with dividend strategy. Avanzi et al. [2] studied the expected total discounted dividends in the dual risk model with barrier strategy. Avanzi and Gerber [3] studied the optimal dividends in the dual risk model with diffusion. Ng [4] studied the expected discounted dividend in dual risk model with threshold dividend strategy. Dai et al. [5] studied the optimal dividend strategies in the dual risk model with capital injections.Albrecher and Lautscham [6] made a distinction between ruin and bankruptcy, in the traditional actuarial model; if the surplus is negative, the company is ruined and has to go out of business; in particular, no dividends are paid after ruin. They consider a relaxation of the ruin concept to the concept of bankruptcy, in which the company with a negative surplus is assumed to be able to continue doing business as usual until bankruptcy takes place, and bankruptcy means that the company goes out of business. Concretely, a suitable bankruptcy rate function ω(u) depending on the size of the negative surplus, which is defined on -a≤u<0, zero for u>0, and ω(u)=∞ for u<-a. This is a nonincreasing function; whenever the negative surplus is u, ω(u)dt is the probability of bankruptcy within dt time units. Albrecher and Lautscham [6] considered the optimal dividend barrier in diffusion risk model until bankruptcy. In this paper, we extend the idea of ruin to more general bankruptcy concept in dual risk model.Let us now consider some of the basic definition and notation for the risk model. Letτ be the bankruptcy time of the surplus process with dividend payments and define the overall probability of bankruptcy as
(2)ψ(u,b)=P{τ<∞∣U(0)=u}.
A barrier dividend strategy is given by a parameter b≥0; if at a potential dividend payment time, the surplus is above b; the excess is paid as a dividend and then the aggregate dividends D(t) till time t is
(3)D(t)=(U(t-)-b)+,
where (U(t-)-b)+=max⁡{U(t-)-b,0}.The dividends are discounted at a constant force of interestδ≥0, the total discounted dividends until bankruptcy are
(4)Du,b=∫0τ‍e-δtdD(t),
and the expected discounted value of dividends by V(u,b)=E[Du,b].The purpose of this paper is to present the expected value of the discounted sum of all dividend payments until bankruptcy and the probability of bankruptcy in the dual risk model. In Section2, integrodifferential equations for the expected dividend payments until bankruptcy are derived; moreover, explicit solutions are also obtained under constant bankruptcy rate function and exponential gain size; finally, we provide some numerical examples with illustrations for the expected dividend payments under the concept of bankruptcy. In Section 3, we derive equations for the probability of bankruptcy ψ(u,b), which are solved explicitly for constant bankrupt rate function and exponential gain size and we also provide numerical examples with illustrations for the probability of bankruptcy.


## 2.  The Expectation of the Discounted Dividend Payments

In this section, we derive integrodifferential equations forV(u,b); the results are summarized in the following theorem. At first, we define
(5)V(u,b)={V1(u,b),u<0,V2(u,b),0≤u<b,u-b+V2(b,b),u≥b.
Similarly, we can define ψi(u,b) in next section as Vi(u,b)(i=1,2).


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

Theorem 1.
V
1
(
u
,
b
) and V2(u,b) satisfy the following system of integrodifferential equations:
(6)cV1′(u,b)+(λ+δ+ω(u))V1(u,b)-λ∫u0‍V1(x,b)fX(x-u)dx-λ∫0b‍V2(x,b)·fX(x-u)dx-λV2(b,b)F¯(b-u)-λ∫b-u∞‍(1-FX(x))dx=0,u<0,(7)cV2′(u,b)+(λ+δ)V2(u,b)-λ∫ub‍V2(x,b)fX(x-u)dx-λ∫b-u∞‍(1-FX(x))dx-λV2(b,b)F¯(b-u)=0,0≤u<b.In addition,V1(u,b) and V2(u,b) satisfy
(8)V1(0-,b)=V2(0+,b),(9)lim⁡u→-∞V1(u,b)=0,(10)V2′(0+,b)-V1′(0-,b)=ω(0-)cV1(0-,b).Proof.
Whenu>b, the surplus drops to level b immediately due to the initial payment of dividends and thus
(11)V2(u,b)=u-b+V2(b,b);
whenu<0, conditioning on the first occurrence time of either a gain or an event of bankruptcy up to time t yields that
(12)V1(u,b)=(1-λt)(1-ω(u)t)e-δtV1(u-ct,b)+(1-λt)ω(u)t·0+(1-ω(u)t)λt·[∫0-u+ct‍V1(u-ct+x,b)fX(x)dx+∫-u+ctb-u+ct‍V2(u-ct+x,b)fX(x)dx+∫b-u+ct∞‍(u-ct+x-b+V2(b,b))fX(x)dx]+o(t)=0.
We differentiate (12) with respect to t, and by taking the limit t→0 we can get
(13)cV1′(u,b)+(λ+δ+ω(u))V1(u,b)-λ∫0-u‍V1(u+x,b)fX(x)dx-λ∫-ub-u‍V2(u+x,b)·fX(x)dx-λ∫b-u∞‍(u+x-b)fX(x)dx-λV2(b,b)F¯(b-u)=0.
That is,
(14)cV1′(u,b)+(λ+δ+ω(u))V1(u,b)-λ∫u0‍V1(x,b)fX(x-u)dx-λ∫0b‍V2(x,b)·fX(x-u)dx-λ∫b-u∞‍(1-FX(x))dx-λV2(b,b)F¯(b-u)=0.
When0≤u<b,
(15)V2(u,b)=(1-λt)e-δtV2(u-ct,b)+λt∫0b-u+ct‍V2(u-ct+x,b)fX(x)dx+λt∫b-u+ct∞‍(u-ct+x-b+V2(b,b))fX(x)dx+o(t)=0.
Using a similar method of deriving (6), we can also obtain (7).
Lettingu↓0 in (12) and u↑0 in (15), it follows that V(u,b) is continuous at u=0 as long as ω(0) is bounded; that is,
(16)V1(0-,b)=V2(0+,b);
the continuity ofV(u,b) at u=0; we can deduce from (6) and (7) for u→0 that
(17)V2′(0+,b)-V1′(0-,b)=ω(0-)cV1(0-,b);
we know that ifu→-∞, the bankruptcy takes place, so lim⁡u→-∞V1(u,b)=0 is obvious.


### 2.2. Explicit Expressions forV(u,b)

We assume in this subsection thatω(u) is constant; the positive constant is denoted as ω; that is,
(18)ω(u)=ω,-a≤u<0.For simplicity, we will assume throughout the rest of the paper that the gain size is exponentially distributedF(x)=1-e-αx.In this case, (6) can be rewritten as
(19)cV1′(u,b)+(λ+δ+ω)V1(u,b)-λ∫u0‍V1(x,b)αe-α(x-u)dx-λ∫0b‍V2(x,b)αe-α(x-u)dx-λV2(b,b)e-α(b-u)-λ∫b-u∞‍e-αxdx=0.
That is,
(20)cV1′(u,b)+(λ+δ+ω)V1(u,b)-λ∫u0‍V1(x,b)αe-αxeαudx-λ∫0b‍V2(x,b)αe-αx·eαudx-λV2(b,b)e-α(b-u)-λαe-α(b-u)=0.Applying the operator(d/du-α) to (20), we obtain the differential equation
(21)cV1′′(u,b)+(λ+δ+ω-cα)V1′(u,b)-α(δ+ω)V1(u,b)=0.Hence the solution of (21) is of the form
(22)V1(u,b)=A1e-r1u+B1es1u,
where A1,B1 are arbitrary coefficients and -r1<0 and s1>0 are the two solutions to the characteristic equation about η:
(23)cη2+(λ+δ+ω-cα)η-δ(α+ω)=0.Fromlim⁡u→-∞V1(u,b)=0, it follows that A1=0.Analogously, we rewrite (7) as
(24)cV2′(u,b)+(λ+δ)V2(u,b)-λ∫ub‍V2(x,b)αe-α(x-u)dx-λαe-α(b-u)-λV2(b,b)e-α(b-u)=0.
That is,
(25)cV2′(u,b)+(λ+δ-cα)V2′(u,b)-αδV2(u,b)=0.
The solution of (25) is of the form
(26)V2(u,b)=A2er2u+B2es2u,
where A2, B2 are constants and r2, s2 are the solutions of the equation about η′(27)c(η′)2+(λ+δ-cα)η′-δα=0.From (8), we obtain that
(28)B1=A2+B2.The condition (10) gives the equation
(29)A2r2+B2s2-B1s1=ωcB1.Substituting (26) into (7), we have
(30)A2(cr2+δ)er2b+B2(cs2+δ)es2b=λα.Therefore, we have a system of linear equations (28)–(30) for the constants A2, B1, and B2. Solving the system of linear equations, we have
(31)A2=λα(s2-ωc-s1)×((cr2+δ)(s2-ωc-s1)er2b-(cs2+δ)(r2-ωc-s1)es2b)-1,B1=λα(s2-r2)×((cr2+δ)(s2-ωc-s1)er2b-(cs2+δ)(r2-ωc-s1)es2b)-1,B2=λα(r2-ωc-s1)×((cs2+δ)(r2-ωc-s1)es2b-(cr2+δ)(s2-ωc-s1)er2b)-1.So we can obtain(32)V1(u,b)=λα(-(cs2+δ)(r2-ωc-s1)es2b)-1(s2-r2)×((cr2+δ)(s2-ωc-s1)er2b-(cs2+δ)(r2-ωc-s1)es2b)-1)es1u,V2(u,b)=λα(s2-ωc-s1)er2u-(r2-ωc-s1)es2u×((cr2+δ)(s2-ωc-s1)er2b-(cs2+δ)(r2-ωc-s1)es2b)-1.


### 2.3. Numerical Examples forV(u,b)

As an illustration of the results of the previous subsection, we will give some numerical examples about the expectation of the discounted dividend paymentsV(u,b).Example 2.
We setδ=0.1,c=2,λ=3,α=1, and b=10; it is easy to check that the net profit condition holds; we can discuss impact of the model parameters u and ω on V(u,b).Example 3.
We setδ=0.1,c=2,λ=3,α=1, and ω=1; it is easy to check that the net profit condition holds; we can discuss impact of the model parameters u and b on V(u,b).Tables1 and 2 provide numerical results for V(u,b). We find that, for a fixed ω, as can be expected, V(u,b) increases with u, The results obtained in Table 1 also illustrate the effect of the bankruptcy parameter ω for V(u,b). From Table 2, we find that the expectation of the discounted dividend payments V(u,b) decreases with dividend barrier b, which makes sense intuitively.Table 1
Influence ofu and ω on V(u,b) for δ=0.1, c=2, λ=3, α=1, and b=10.
u
∖
ω
1
2
3
4
5
6
-
6
1.7969
1.2439
0.9516
0.7707
0.6476
0.5585
-
5
1.9620
1.3630
1.0447
0.8471
0.7124
0.6147
-
4
2.1422
1.4934
1.1469
0.9311
0.7837
0.6767
-
3
2.3390
1.6363
1.2591
1.0234
0.8621
0.7448
-
2
2.5539
1.7929
1.3822
1.1249
0.9484
0.8199
-
1
2.7885
1.9644
1.5174
1.2364
1.0433
0.9025
0
3.0446
2.1524
1.6658
1.3590
1.1478
0.9934
1
4.4970
4.0197
3.7594
3.5953
3.4823
3.3997
2
5.5086
5.2523
5.1125
5.0243
4.9636
4.9193
3
6.3049
6.1662
6.0905
6.0428
6.0010
5.9860
4
7.0079
6.9317
6.8901
6.8639
6.8458
6.8326
5
7.6841
7.6410
7.6174
7.6026
7.5924
7.5849
6
8.3710
8.3453
8.3313
8.3224
8.3163
8.3119
7
9.0906
9.0739
9.0649
9.0591
9.0552
9.0523
8
9.8568
9.8447
9.8382
9.8340
9.8312
9.8291
9
10.6795
10.6697
10.6643
10.6609
10.6586
10.6569Table 2
Influence ofu and b on V(u,b) for δ=0.1, c=2, λ=3, α=1, and ω=1.
u
∖
b
10
11
12
13
14
15
-
6
1.7969
1.6610
1.5348
1.4179
1.3097
1.2098
-
5
1.9620
1.8136
1.6758
1.5481
1.4300
1.3209
-
4
2.1422
1.9802
1.8297
1.6903
1.5614
1.4422
-
3
2.3390
2.1621
1.9978
1.8456
1.7048
1.5747
-
2
2.5539
2.3607
2.1813
2.0151
1.8614
1.7194
-
1
2.7885
2.5776
2.3817
2.2002
2.0324
1.8773
0
3.0446
2.8144
2.6005
2.40230
2.2191
2.0497
1
4.4970
4.1569
3.8409
3.5483
3.2777
3.0275
2
5.5086
5.0919
4.7049
4.3465
4.0150
3.7085
3
6.3049
5.8281
5.3851
4.9749
4.5954
4.2447
4
7.0079
6.4779
5.9855
5.5295
5.1078
4.7179
5
7.6841
7.1029
6.5631
6.0631
5.6006
5.1732
6
8.3710
7.7379
7.1498
6.6051
6.1013
5.6356
7
9.0906
8.4030
7.7644
7.1728
6.6257
6.1201
8
9.8568
9.1113
8.4188
7.77740
7.1842
6.6359
9
10.6795
9.8718
9.1215
8.4266
7.7839
7.1898Figure1 is plotted to illustrate the impact of bankruptcy rate function ω(x) to dividend payments V(u,b) for various parameter choices ω. Figure 2 is plotted to illustrate the impact of dividend barrier b to dividend payments V(u,b) for various b.Figure 1
The expected sum of discounted dividendV(u,b) when b=10.Figure 2
The expected sum of discounted dividendV(u,b) when ω=1.


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

Theorem 1.
V
1
(
u
,
b
) and V2(u,b) satisfy the following system of integrodifferential equations:
(6)cV1′(u,b)+(λ+δ+ω(u))V1(u,b)-λ∫u0‍V1(x,b)fX(x-u)dx-λ∫0b‍V2(x,b)·fX(x-u)dx-λV2(b,b)F¯(b-u)-λ∫b-u∞‍(1-FX(x))dx=0,u<0,(7)cV2′(u,b)+(λ+δ)V2(u,b)-λ∫ub‍V2(x,b)fX(x-u)dx-λ∫b-u∞‍(1-FX(x))dx-λV2(b,b)F¯(b-u)=0,0≤u<b.In addition,V1(u,b) and V2(u,b) satisfy
(8)V1(0-,b)=V2(0+,b),(9)lim⁡u→-∞V1(u,b)=0,(10)V2′(0+,b)-V1′(0-,b)=ω(0-)cV1(0-,b).Proof.
Whenu>b, the surplus drops to level b immediately due to the initial payment of dividends and thus
(11)V2(u,b)=u-b+V2(b,b);
whenu<0, conditioning on the first occurrence time of either a gain or an event of bankruptcy up to time t yields that
(12)V1(u,b)=(1-λt)(1-ω(u)t)e-δtV1(u-ct,b)+(1-λt)ω(u)t·0+(1-ω(u)t)λt·[∫0-u+ct‍V1(u-ct+x,b)fX(x)dx+∫-u+ctb-u+ct‍V2(u-ct+x,b)fX(x)dx+∫b-u+ct∞‍(u-ct+x-b+V2(b,b))fX(x)dx]+o(t)=0.
We differentiate (12) with respect to t, and by taking the limit t→0 we can get
(13)cV1′(u,b)+(λ+δ+ω(u))V1(u,b)-λ∫0-u‍V1(u+x,b)fX(x)dx-λ∫-ub-u‍V2(u+x,b)·fX(x)dx-λ∫b-u∞‍(u+x-b)fX(x)dx-λV2(b,b)F¯(b-u)=0.
That is,
(14)cV1′(u,b)+(λ+δ+ω(u))V1(u,b)-λ∫u0‍V1(x,b)fX(x-u)dx-λ∫0b‍V2(x,b)·fX(x-u)dx-λ∫b-u∞‍(1-FX(x))dx-λV2(b,b)F¯(b-u)=0.
When0≤u<b,
(15)V2(u,b)=(1-λt)e-δtV2(u-ct,b)+λt∫0b-u+ct‍V2(u-ct+x,b)fX(x)dx+λt∫b-u+ct∞‍(u-ct+x-b+V2(b,b))fX(x)dx+o(t)=0.
Using a similar method of deriving (6), we can also obtain (7).
Lettingu↓0 in (12) and u↑0 in (15), it follows that V(u,b) is continuous at u=0 as long as ω(0) is bounded; that is,
(16)V1(0-,b)=V2(0+,b);
the continuity ofV(u,b) at u=0; we can deduce from (6) and (7) for u→0 that
(17)V2′(0+,b)-V1′(0-,b)=ω(0-)cV1(0-,b);
we know that ifu→-∞, the bankruptcy takes place, so lim⁡u→-∞V1(u,b)=0 is obvious.


## 2.2. Explicit Expressions forV(u,b)

We assume in this subsection thatω(u) is constant; the positive constant is denoted as ω; that is,
(18)ω(u)=ω,-a≤u<0.For simplicity, we will assume throughout the rest of the paper that the gain size is exponentially distributedF(x)=1-e-αx.In this case, (6) can be rewritten as
(19)cV1′(u,b)+(λ+δ+ω)V1(u,b)-λ∫u0‍V1(x,b)αe-α(x-u)dx-λ∫0b‍V2(x,b)αe-α(x-u)dx-λV2(b,b)e-α(b-u)-λ∫b-u∞‍e-αxdx=0.
That is,
(20)cV1′(u,b)+(λ+δ+ω)V1(u,b)-λ∫u0‍V1(x,b)αe-αxeαudx-λ∫0b‍V2(x,b)αe-αx·eαudx-λV2(b,b)e-α(b-u)-λαe-α(b-u)=0.Applying the operator(d/du-α) to (20), we obtain the differential equation
(21)cV1′′(u,b)+(λ+δ+ω-cα)V1′(u,b)-α(δ+ω)V1(u,b)=0.Hence the solution of (21) is of the form
(22)V1(u,b)=A1e-r1u+B1es1u,
where A1,B1 are arbitrary coefficients and -r1<0 and s1>0 are the two solutions to the characteristic equation about η:
(23)cη2+(λ+δ+ω-cα)η-δ(α+ω)=0.Fromlim⁡u→-∞V1(u,b)=0, it follows that A1=0.Analogously, we rewrite (7) as
(24)cV2′(u,b)+(λ+δ)V2(u,b)-λ∫ub‍V2(x,b)αe-α(x-u)dx-λαe-α(b-u)-λV2(b,b)e-α(b-u)=0.
That is,
(25)cV2′(u,b)+(λ+δ-cα)V2′(u,b)-αδV2(u,b)=0.
The solution of (25) is of the form
(26)V2(u,b)=A2er2u+B2es2u,
where A2, B2 are constants and r2, s2 are the solutions of the equation about η′(27)c(η′)2+(λ+δ-cα)η′-δα=0.From (8), we obtain that
(28)B1=A2+B2.The condition (10) gives the equation
(29)A2r2+B2s2-B1s1=ωcB1.Substituting (26) into (7), we have
(30)A2(cr2+δ)er2b+B2(cs2+δ)es2b=λα.Therefore, we have a system of linear equations (28)–(30) for the constants A2, B1, and B2. Solving the system of linear equations, we have
(31)A2=λα(s2-ωc-s1)×((cr2+δ)(s2-ωc-s1)er2b-(cs2+δ)(r2-ωc-s1)es2b)-1,B1=λα(s2-r2)×((cr2+δ)(s2-ωc-s1)er2b-(cs2+δ)(r2-ωc-s1)es2b)-1,B2=λα(r2-ωc-s1)×((cs2+δ)(r2-ωc-s1)es2b-(cr2+δ)(s2-ωc-s1)er2b)-1.So we can obtain(32)V1(u,b)=λα(-(cs2+δ)(r2-ωc-s1)es2b)-1(s2-r2)×((cr2+δ)(s2-ωc-s1)er2b-(cs2+δ)(r2-ωc-s1)es2b)-1)es1u,V2(u,b)=λα(s2-ωc-s1)er2u-(r2-ωc-s1)es2u×((cr2+δ)(s2-ωc-s1)er2b-(cs2+δ)(r2-ωc-s1)es2b)-1.


## 2.3. Numerical Examples forV(u,b)

As an illustration of the results of the previous subsection, we will give some numerical examples about the expectation of the discounted dividend paymentsV(u,b).Example 2.
We setδ=0.1,c=2,λ=3,α=1, and b=10; it is easy to check that the net profit condition holds; we can discuss impact of the model parameters u and ω on V(u,b).Example 3.
We setδ=0.1,c=2,λ=3,α=1, and ω=1; it is easy to check that the net profit condition holds; we can discuss impact of the model parameters u and b on V(u,b).Tables1 and 2 provide numerical results for V(u,b). We find that, for a fixed ω, as can be expected, V(u,b) increases with u, The results obtained in Table 1 also illustrate the effect of the bankruptcy parameter ω for V(u,b). From Table 2, we find that the expectation of the discounted dividend payments V(u,b) decreases with dividend barrier b, which makes sense intuitively.Table 1
Influence ofu and ω on V(u,b) for δ=0.1, c=2, λ=3, α=1, and b=10.
u
∖
ω
1
2
3
4
5
6
-
6
1.7969
1.2439
0.9516
0.7707
0.6476
0.5585
-
5
1.9620
1.3630
1.0447
0.8471
0.7124
0.6147
-
4
2.1422
1.4934
1.1469
0.9311
0.7837
0.6767
-
3
2.3390
1.6363
1.2591
1.0234
0.8621
0.7448
-
2
2.5539
1.7929
1.3822
1.1249
0.9484
0.8199
-
1
2.7885
1.9644
1.5174
1.2364
1.0433
0.9025
0
3.0446
2.1524
1.6658
1.3590
1.1478
0.9934
1
4.4970
4.0197
3.7594
3.5953
3.4823
3.3997
2
5.5086
5.2523
5.1125
5.0243
4.9636
4.9193
3
6.3049
6.1662
6.0905
6.0428
6.0010
5.9860
4
7.0079
6.9317
6.8901
6.8639
6.8458
6.8326
5
7.6841
7.6410
7.6174
7.6026
7.5924
7.5849
6
8.3710
8.3453
8.3313
8.3224
8.3163
8.3119
7
9.0906
9.0739
9.0649
9.0591
9.0552
9.0523
8
9.8568
9.8447
9.8382
9.8340
9.8312
9.8291
9
10.6795
10.6697
10.6643
10.6609
10.6586
10.6569Table 2
Influence ofu and b on V(u,b) for δ=0.1, c=2, λ=3, α=1, and ω=1.
u
∖
b
10
11
12
13
14
15
-
6
1.7969
1.6610
1.5348
1.4179
1.3097
1.2098
-
5
1.9620
1.8136
1.6758
1.5481
1.4300
1.3209
-
4
2.1422
1.9802
1.8297
1.6903
1.5614
1.4422
-
3
2.3390
2.1621
1.9978
1.8456
1.7048
1.5747
-
2
2.5539
2.3607
2.1813
2.0151
1.8614
1.7194
-
1
2.7885
2.5776
2.3817
2.2002
2.0324
1.8773
0
3.0446
2.8144
2.6005
2.40230
2.2191
2.0497
1
4.4970
4.1569
3.8409
3.5483
3.2777
3.0275
2
5.5086
5.0919
4.7049
4.3465
4.0150
3.7085
3
6.3049
5.8281
5.3851
4.9749
4.5954
4.2447
4
7.0079
6.4779
5.9855
5.5295
5.1078
4.7179
5
7.6841
7.1029
6.5631
6.0631
5.6006
5.1732
6
8.3710
7.7379
7.1498
6.6051
6.1013
5.6356
7
9.0906
8.4030
7.7644
7.1728
6.6257
6.1201
8
9.8568
9.1113
8.4188
7.77740
7.1842
6.6359
9
10.6795
9.8718
9.1215
8.4266
7.7839
7.1898Figure1 is plotted to illustrate the impact of bankruptcy rate function ω(x) to dividend payments V(u,b) for various parameter choices ω. Figure 2 is plotted to illustrate the impact of dividend barrier b to dividend payments V(u,b) for various b.Figure 1
The expected sum of discounted dividendV(u,b) when b=10.Figure 2
The expected sum of discounted dividendV(u,b) when ω=1.


## 3. The Probability of Bankruptcyψ(u,b)

### 3.1. Integrodifferential Equations forψ(u,b)

Theorem 4.
The probability of bankruptcyψ(u,b) satisfies the following integrodifferential equations:
(33)cψ1′(u,b)+(λ+ω(u))ψ1(u,b)-ω(u)-λ∫u0‍ψ1(x,b)fX(x-u)dx-λ∫0b‍ψ2(x,b)·fX(x-u)dx-λψ2(b,b)F¯(b-u)=0,u<0,(34)cψ2′(u,b)+λψ2(u,b)-λ∫ub‍ψ2(x,b)fX(x-u)dx-λψ2(b,b)F¯(b-u)=0,u≥0.In addition,ψ1(u,b) and ψ2(u,b) satisfy
(35)ψ1(0-,b)=ψ2(0+,b),(36)ψ2′(0+,b)-ψ1′(0-,b)=ω(0-)c(ψ1(0-,b)-1),(37)lim⁡u→-∞ψ1(u,b)=1.Proof.
Whenu>b, the surplus drops to level b immediately due to the initial payment of all dividends and thus
(38)ψ2(u,b)=ψ2(b,b),u>b.
Whenu<0, by conditioning on the first occurrence time and amount of the gain or an event of bankruptcy up to time t,
(39)ψ1(u,b)=(1-λt)(1-ω(u)t)ψ1(u-ct,b)+(1-λt)ω(u)t+λt(1-ω(u)t)·[∫0-u+ct‍ψ1(u-ct+x,b)fX(x)dx+∫-u+ctb-u+ct‍ψ2(u-ct+x,b)fX(x)dx+∫b-u+ct∞‍ψ2(b,b)fX(x)dx]+o(t)=0.
Differentiating (39) with respect to t and taking the limit t→0 we can obtain (33).
Using a similar argument, we can also derive the corresponding bankruptcy probabilityψ2(u,b) for 0≤u<b. The conditions (35)–(37) are also obvious.


### 3.2. Explicit Expressions forψ(u,b)

In this subsection, we assume thatω(u) is also constant ω, and the gain size follows f(x)=αe-αx; then (33) and (34) can be rewritten as
(40)cψ1′(u,b)+(λ+ω)ψ1(u,b)-ω-λ∫u0‍ψ1(x,b)αe-αxeαudx-λ∫0b‍ψ2(x,b)αe-αxeαudx-λψ2(b,b)e-α(b-u)=0,cψ2′(u,b)+λψ2(u,b)-λ∫ub‍ψ2(x,b)αe-αxeαudx-λψ2(b,b)e-α(b-u)=0.Applying the operators(d/dx-α) to (40), they can be rewritten as
(41)cψ1′′(u,b)+(λ+ω-cα)ψ1′(u,b)-ωαψ1(u,b)+ωα=0.(42)cψ2′′(u,b)+(λ-cα)ψ2′(u,b)=0.We know the solution of (41) is of the form
(43)ψ1(u,b)=1+A1′e-r1′u+B1′es1′u,
where A1′,B1′ are constants and -r1′<0, s1′>0 are the solutions of the equation about R:
(44)cR2+(λ+ω-cα)R-ωα=0.
From lim⁡u→-∞ψ1(u,b)=1, A1′=0 is obvious.Foru≥0, we obtain the solution of (42) which is of the form
(45)ψ2(u,b)=A1′′e(α-(λ/c))u+B1′′.
When u→∞, we have ψ2(u,b)→0. As λ/α-c>0, that is, α-λ/α<0, so B1′′=0 is obvious. Then
(46)ψ2(u,b)=A1′′e(α-λ/α)u.Similar to derivingV(u,b), we have that the conditions (35) and (36) for ψ(u,b) give that
(47)1+B1′=A1′′,A1′′(α-λc)-B1′s1′=ωcB1′.
We have
(48)A1′′=cs1′+ωcs1′+ω-cα+λ,B1′=cα-λcs1′+ω-cα+λ.
So
(49)ψ1(u,b)=1+cα-λcs1′+ω-cα+λes1′u,ψ2(u,b)=cs1′+ωcs1′+ω-cα+λe(α-λ/α)u.Example 5.
We can perform analysis for the probability of bankruptcyψ(u,b); again we choose δ=0.1,c=2,λ=3,α=1, and b=10.In Table3 we find that, for a given u, the probability of bankruptcy ψ(u,b) increases with u. And it also can be expected, as a large value of bankruptcy rate ω implies the large bankruptcy probability ψ(u,b).Table 3
Influence ofu and ω on ψ(u,b) for δ=0.1, c=2, λ=3, α=1, and b=10.
u
∖
ω
1
2
3
4
5
6
-
6
0.9593
0.9876
0.9941
0.9965
0.9977
0.9983
-
5
0.9413
0.9795
0.9894
0.9934
0.9955
0.9966
-
4
0.9153
0.9662
0.9811
0.9876
0.9910
0.9931
-
3
0.8779
0.9442
0.9661
0.9765
0.9823
0.9860
-
2
0.8240
0.9080
0.9394
0.9555
0.9651
0.9714
-
1
0.7462
0.8484
0.8917
0.9158
0.9311
0.9418
0
0.6340
0.7500
0.8063
0.8406
0.8641
0.8813
1
0.3845
0.4549
0.4890
0.5099
0.5241
0.5345
2
0.2332
0.2759
0.2966
0.3092
0.3179
0.3242
3
0.1415
0.1673
0.1799
0.1876
0.1928
0.1966
4
0.0858
0.1015
0.1091
0.1138
0.1169
0.1193
5
0.0520
0.0616
0.0662
0.0690
0.0709
0.0723
6
0.0316
0.0373
0.0401
0.0418
0.0430
0.0439
7
0.0191
0.0226
0.0243
0.0254
0.0261
0.0266
8
0.0116
0.0137
0.0148
0.0154
0.0158
0.0161
9
0.0070
0.0083
0.0090
0.0093
0.0096
0.0098Figure3 was produced by the explicit expression in (49); it is plotted to illustrate the impact of ω and u on bankruptcy probability ψ(u,b).Figure 3
The bankruptcy probabilityψ(u,b) when b=10.


## 3.1. Integrodifferential Equations forψ(u,b)

Theorem 4.
The probability of bankruptcyψ(u,b) satisfies the following integrodifferential equations:
(33)cψ1′(u,b)+(λ+ω(u))ψ1(u,b)-ω(u)-λ∫u0‍ψ1(x,b)fX(x-u)dx-λ∫0b‍ψ2(x,b)·fX(x-u)dx-λψ2(b,b)F¯(b-u)=0,u<0,(34)cψ2′(u,b)+λψ2(u,b)-λ∫ub‍ψ2(x,b)fX(x-u)dx-λψ2(b,b)F¯(b-u)=0,u≥0.In addition,ψ1(u,b) and ψ2(u,b) satisfy
(35)ψ1(0-,b)=ψ2(0+,b),(36)ψ2′(0+,b)-ψ1′(0-,b)=ω(0-)c(ψ1(0-,b)-1),(37)lim⁡u→-∞ψ1(u,b)=1.Proof.
Whenu>b, the surplus drops to level b immediately due to the initial payment of all dividends and thus
(38)ψ2(u,b)=ψ2(b,b),u>b.
Whenu<0, by conditioning on the first occurrence time and amount of the gain or an event of bankruptcy up to time t,
(39)ψ1(u,b)=(1-λt)(1-ω(u)t)ψ1(u-ct,b)+(1-λt)ω(u)t+λt(1-ω(u)t)·[∫0-u+ct‍ψ1(u-ct+x,b)fX(x)dx+∫-u+ctb-u+ct‍ψ2(u-ct+x,b)fX(x)dx+∫b-u+ct∞‍ψ2(b,b)fX(x)dx]+o(t)=0.
Differentiating (39) with respect to t and taking the limit t→0 we can obtain (33).
Using a similar argument, we can also derive the corresponding bankruptcy probabilityψ2(u,b) for 0≤u<b. The conditions (35)–(37) are also obvious.


## 3.2. Explicit Expressions forψ(u,b)

In this subsection, we assume thatω(u) is also constant ω, and the gain size follows f(x)=αe-αx; then (33) and (34) can be rewritten as
(40)cψ1′(u,b)+(λ+ω)ψ1(u,b)-ω-λ∫u0‍ψ1(x,b)αe-αxeαudx-λ∫0b‍ψ2(x,b)αe-αxeαudx-λψ2(b,b)e-α(b-u)=0,cψ2′(u,b)+λψ2(u,b)-λ∫ub‍ψ2(x,b)αe-αxeαudx-λψ2(b,b)e-α(b-u)=0.Applying the operators(d/dx-α) to (40), they can be rewritten as
(41)cψ1′′(u,b)+(λ+ω-cα)ψ1′(u,b)-ωαψ1(u,b)+ωα=0.(42)cψ2′′(u,b)+(λ-cα)ψ2′(u,b)=0.We know the solution of (41) is of the form
(43)ψ1(u,b)=1+A1′e-r1′u+B1′es1′u,
where A1′,B1′ are constants and -r1′<0, s1′>0 are the solutions of the equation about R:
(44)cR2+(λ+ω-cα)R-ωα=0.
From lim⁡u→-∞ψ1(u,b)=1, A1′=0 is obvious.Foru≥0, we obtain the solution of (42) which is of the form
(45)ψ2(u,b)=A1′′e(α-(λ/c))u+B1′′.
When u→∞, we have ψ2(u,b)→0. As λ/α-c>0, that is, α-λ/α<0, so B1′′=0 is obvious. Then
(46)ψ2(u,b)=A1′′e(α-λ/α)u.Similar to derivingV(u,b), we have that the conditions (35) and (36) for ψ(u,b) give that
(47)1+B1′=A1′′,A1′′(α-λc)-B1′s1′=ωcB1′.
We have
(48)A1′′=cs1′+ωcs1′+ω-cα+λ,B1′=cα-λcs1′+ω-cα+λ.
So
(49)ψ1(u,b)=1+cα-λcs1′+ω-cα+λes1′u,ψ2(u,b)=cs1′+ωcs1′+ω-cα+λe(α-λ/α)u.Example 5.
We can perform analysis for the probability of bankruptcyψ(u,b); again we choose δ=0.1,c=2,λ=3,α=1, and b=10.In Table3 we find that, for a given u, the probability of bankruptcy ψ(u,b) increases with u. And it also can be expected, as a large value of bankruptcy rate ω implies the large bankruptcy probability ψ(u,b).Table 3
Influence ofu and ω on ψ(u,b) for δ=0.1, c=2, λ=3, α=1, and b=10.
u
∖
ω
1
2
3
4
5
6
-
6
0.9593
0.9876
0.9941
0.9965
0.9977
0.9983
-
5
0.9413
0.9795
0.9894
0.9934
0.9955
0.9966
-
4
0.9153
0.9662
0.9811
0.9876
0.9910
0.9931
-
3
0.8779
0.9442
0.9661
0.9765
0.9823
0.9860
-
2
0.8240
0.9080
0.9394
0.9555
0.9651
0.9714
-
1
0.7462
0.8484
0.8917
0.9158
0.9311
0.9418
0
0.6340
0.7500
0.8063
0.8406
0.8641
0.8813
1
0.3845
0.4549
0.4890
0.5099
0.5241
0.5345
2
0.2332
0.2759
0.2966
0.3092
0.3179
0.3242
3
0.1415
0.1673
0.1799
0.1876
0.1928
0.1966
4
0.0858
0.1015
0.1091
0.1138
0.1169
0.1193
5
0.0520
0.0616
0.0662
0.0690
0.0709
0.0723
6
0.0316
0.0373
0.0401
0.0418
0.0430
0.0439
7
0.0191
0.0226
0.0243
0.0254
0.0261
0.0266
8
0.0116
0.0137
0.0148
0.0154
0.0158
0.0161
9
0.0070
0.0083
0.0090
0.0093
0.0096
0.0098Figure3 was produced by the explicit expression in (49); it is plotted to illustrate the impact of ω and u on bankruptcy probability ψ(u,b).Figure 3
The bankruptcy probabilityψ(u,b) when b=10.



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*Source: 184098-2014-08-11.xml*