JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2014/184098 184098 Research Article Dividend Problems with a Barrier Strategy in the Dual Risk Model until Bankruptcy http://orcid.org/0000-0003-3512-5332 Liu Donghai 1 Liu Zaiming 2 Jiang Daqing 1 Department of Mathematics Hunan University of Science and Technology Xiangtan, Hunan 411201 China hut.edu.cn 2 Department of Mathematics and Statistics Central South University Changsha, Hunan 410075 China csu.edu.cn 2014 1182014 2014 29 03 2014 19 07 2014 21 07 2014 11 8 2014 2014 Copyright © 2014 Donghai Liu and Zaiming Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

In the continuous time dual risk model, the company's surplus process {U(t),t0} with initial surplus U(0)=u is given by (1)U(t)=u-ct+S(t)=u-ct+i=1N(t)Xi,t0, where c>0 is the constant rate of expenses per unit time, {N(t),t0} is a Poisson process with intensity λ>0, and the gain size {Xi,i1} is a sequence of independent and identically distributed positive continuous random variables with finite mean, independent of {N(t),t0}. 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  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.  studied the expected total discounted dividends in the dual risk model with barrier strategy. Avanzi and Gerber  studied the optimal dividends in the dual risk model with diffusion. Ng  studied the expected discounted dividend in dual risk model with threshold dividend strategy. Dai et al.  studied the optimal dividend strategies in the dual risk model with capital injections.

Albrecher and Lautscham  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 -au<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  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. Let τ 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 b0; 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 interest δ0, 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 Section 2, 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 for V(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),0u<b,u-b+V2(b,b),ub. Similarly, we can define ψi(u,b) in next section as Vi(u,b)(i=1,2).

2.1. Integrodifferential Equations for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M38"><mml:mi>V</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>u</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>b</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula> 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)-λu0V1(x,b)fX(x-u)dx-λ0bV2(x,b)·fX(x-u)dx-λV2(b,b)F¯(b-u)-λb-u(1-FX(x))dx=0,u<0,(7)cV2(u,b)+(λ+δ)V2(u,b)-λubV2(x,b)fX(x-u)dx-λb-u(1-FX(x))dx-λV2(b,b)F¯(b-u)=0,0u<b.

In addition, V1(u,b) and V2(u,b) satisfy (8)V1(0-,b)=V2(0+,b),(9)limu-V1(u,b)=0,(10)V2(0+,b)-V1(0-,b)=ω(0-)cV1(0-,b).

Proof.

When u>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);

when u<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 t0 we can get (13)cV1(u,b)+(λ+δ+ω(u))V1(u,b)-λ0-uV1(u+x,b)fX(x)dx-λ-ub-uV2(u+x,b)·fX(x)dx-λb-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)-λu0V1(x,b)fX(x-u)dx-λ0bV2(x,b)·fX(x-u)dx-λb-u(1-FX(x))dx-λV2(b,b)F¯(b-u)=0.

When 0u<b, (15)V2(u,b)=(1-λt)e-δtV2(u-ct,b)+λt0b-u+ctV2(u-ct+x,b)fX(x)dx+λtb-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).

Letting u0 in (12) and u0 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 of V(u,b) at u=0; we can deduce from (6) and (7) for u0 that (17)V2(0+,b)-V1(0-,b)=ω(0-)cV1(0-,b);

we know that if u-, the bankruptcy takes place, so limu-V1(u,b)=0 is obvious.

2.2. Explicit Expressions for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M69"><mml:mi>V</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>u</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>b</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

We assume in this subsection that ω(u) is constant; the positive constant is denoted as ω; that is, (18)ω(u)=ω,-au<0.

For simplicity, we will assume throughout the rest of the paper that the gain size is exponentially distributed F(x)=1-e-αx.

In this case, (6) can be rewritten as (19)cV1(u,b)+(λ+δ+ω)V1(u,b)-λu0V1(x,b)αe-α(x-u)dx-λ0bV2(x,b)αe-α(x-u)dx-λV2(b,b)e-α(b-u)-λb-ue-αxdx=0. That is, (20)cV1(u,b)+(λ+δ+ω)V1(u,b)-λu0V1(x,b)αe-αxeαudx-λ0bV2(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.

From limu-V1(u,b)=0, it follows that A1=0.

Analogously, we rewrite (7) as (24)cV2(u,b)+(λ+δ)V2(u,b)-λubV2(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 for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M103"><mml:mi>V</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>u</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>b</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

As an illustration of the results of the previous subsection, we will give some numerical examples about the expectation of the discounted dividend payments V(u,b).

Example 2.

We set δ=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 set δ=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).

Tables 1 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.

Influence of u 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 10.6569

Influence of u 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 7.1898

Figure 1 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.

The expected sum of discounted dividend V(u,b) when b=10.

The expected sum of discounted dividend V(u,b) when ω=1.

3. The Probability of Bankruptcy <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M163"><mml:mi>ψ</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>u</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>b</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula> 3.1. Integrodifferential Equations for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M164"><mml:mi>ψ</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>u</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>b</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula> Theorem 4.

The probability of bankruptcy ψ(u,b) satisfies the following integrodifferential equations: (33)cψ1(u,b)+(λ+ω(u))ψ1(u,b)-ω(u)-λu0ψ1(x,b)fX(x-u)dx-λ0bψ2(x,b)·fX(x-u)dx-λψ2(b,b)F¯(b-u)=0,u<0,(34)cψ2(u,b)+λψ2(u,b)-λubψ2(x,b)fX(x-u)dx-λψ2(b,b)F¯(b-u)=0,u0.

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)limu-ψ1(u,b)=1.

Proof.

When u>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.

When u<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+ctψ2(b,b)fX(x)dx]+o(t)=0. Differentiating (39) with respect to t and taking the limit t0 we can obtain (33).

Using a similar argument, we can also derive the corresponding bankruptcy probability ψ2(u,b) for 0u<b. The conditions (35)–(37) are also obvious.

3.2. Explicit Expressions for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M180"><mml:mi>ψ</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>u</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>b</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

In this subsection, we assume that ω(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-λu0ψ1(x,b)αe-αxeαudx-λ0bψ2(x,b)αe-αxeαudx-λψ2(b,b)e-α(b-u)=0,cψ2(u,b)+λψ2(u,b)-λubψ2(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+A1e-r1u+B1es1u, 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 limu-ψ1(u,b)=1, A1=0 is obvious.

For u0, 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 deriving V(u,b), we have that the conditions (35) and (36) for ψ(u,b) give that (47)1+B1=A1′′,A1′′(α-λc)-B1s1=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+λes1u,ψ2(u,b)=cs1+ωcs1+ω-cα+λe(α-λ/α)u.

Example 5.

We can perform analysis for the probability of bankruptcy ψ(u,b); again we choose δ=0.1,c=2,λ=3,α=1, and b=10.

In Table 3 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).

Influence of u 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 0.0098

Figure 3 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).

The bankruptcy probability ψ(u,b) when b=10.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is fully supported by a Grant from Natural Science Foundation of Hunan (13JJ4083), by Humanities and Social Sciences Project of the Ministry Education in China (10YJC630144), by Hunan Social Science Fund Program (12YBB093), and by Scientific Fund of Hunan Provincial Education Department (13C318) and is also supported by Natural Science Foundation of Anhui Higher Education Institutions (KJ2014ZD21).

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