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Optimal dividend strategy with transaction costs foran upward jump modelMing Zhou a & Ka Fai Cedric Yiu ba China Institute for Actuarial Science, Central University of Finance and Economics, 39South College Road, Haidian, Beijing 100081, P.R. Chinab Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, HongKong, P.R. China

Version of record first published: 13 Jan 2012.

To cite this article: Ming Zhou & Ka Fai Cedric Yiu (2012): Optimal dividend strategy with transaction costs for an upwardjump model, Quantitative Finance, DOI:10.1080/14697688.2011.647052

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Quantitative Finance, 2012, 1–10, iFirst

Optimal dividend strategy with transaction costs for

an upward jump model

MING ZHOUy and KA FAI CEDRIC YIU*z

yChina Institute for Actuarial Science, Central University of Finance and Economics, 39 South College Road,Haidian, Beijing 100081, P.R. China

zDepartment of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong, P.R. China

(Received 5 July 2010; revised 3 March 2011; in final form 2 December 2011)

In this paper, we consider the optimal dividend problem with transaction costs when theincomes of a company can be described by an upward jump model. Both fixed andproportional costs are considered in the problem. The value function is defined as the expectedtotal discounted dividends up to the time of ruin. Although the same problem has alreadybeen studied in the pure diffusion model and the spectrally negative Levy process, the optimaldividend problem in an upward jump model has two different aspects in determining theoptimal dividends barrier and in the property of the value function. First, the value function istwice continuous differentiable in the diffusion case, but it is not in the jump model. Second,under the spectrally negative Levy process, downward jumps will not cause any paymentactions; however, it might trigger dividend payments when there are upward jumps. Inderiving the optimal barriers, we show that the value function is bounded by a linear function.Using this property, we establish the verification theorem for the value function. By solvingthe quasi-variational inequalities associated with this problem, we obtain the closed-formsolution to the value function and hence the optimal dividend strategy when the income sizesfollow a common exponential distribution. In the presence of a fixed transaction cost, it isshown that the optimal strategy is a two-barrier policy, and the optimal barriers are onlydependent on the fixed cost and not the proportional cost. A numerical example is used toillustrate how the fixed cost plays a significant role in the optimal dividend strategy and alsothe value function. Moreover, an increased fixed cost results in larger but less frequentdividend payments.

Keywords: Upward jump model; Optimal dividend; Transaction costs; Stochastic jumps;Mathematical finance; Impulsive control

JEL Classification: C4, C6, C44, C61, G3, G35

1. Introduction

In recent years, upward jump models have received

increased attention in the field of finance and insurance.

Under the upward jump model, the company’s incomes

are received randomly and the expenses are paid contin-

uously at a fixed constant rate. Mathematically, the

surplus of a company at time t can be expressed as

XðtÞ ¼ xþ SðtÞ � ct,

where x is the initial surplus, c is the rate of expenses, and

S(t) is the aggregate income by time t, usually modeled by

a compound Poisson process.

There are various applications of this model in practice.

It can be used to describe the surplus of companies such

as oil exploration or firms that make natural resource

investments, in which new fields might be found from

time to time. It can also be applied to model firms

engaging in research and development and filing valuable

product patents along the way, or to model the surplus of

consultancy companies who are constantly bidding for

new projects. In these cases, the Poisson arrival incomes

are gained from a successful bid for a project or striking a

new oil field, and the costs and expenses are assumed to

be paid at a constant rate. Bayraktar and Egami (2008)

use a similar model to describe the capital of a venture

capital investment. In insurance, the upward jump model

has been proposed to describe life annuity or pension*Corresponding author. Email: macyiu@polyu.edu.hk

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online � 2012 Taylor & Francis

http://www.tandfonline.comhttp://dx.doi.org/10.1080/14697688.2011.647052

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insurance, which pays benefits to annuitants and gainsprofits from the released reserve when the insured dies(see, e.g., Grandell (1991, p. 8)). The upward jump modelalso finds applications in inventory management, wherethe supplier supplies products with order arrivals follow-ing a Poisson process. For a company whose surplusprocess can be reasonably described in terms of theupward jump model, it is of interest to study the optimaldividend problem.

Within the framework of stochastic control theory, theoptimal dividend (or consumption) problem has beenstudied extensively in recent decades. In the literature, it isusually assumed that the dividend payments are distrib-uted without any cost (see, e.g., Asmussen and Taksar(1997), Kulenko and Schmidli (2008), Loeffen (2008), andSchmidli (2008) and references therein). This frictionlessassumption usually leads to a barrier or a thresholddividend strategy. With the presence of transaction costs,the optimal dividend problem has been studied by Shreveet al. (1984), Jeanblanc-Picque and Shiryave (1995, caseB), Boguslavskaya (2003, section 4), Cadenillas et al.(2007), and Paulsen (2007, 2008). In these papers, theoptimal dividend problems were established by employingstochastic impulse control theory (Harrison et al. 1983)and the asset processes were usually described by purediffusions without jumps. Bai and Guo (2010) andLoeffen (2009) studied the optimal dividends with trans-action costs for compound Poisson risk processes and thespectrally negative Levy process, respectively. Althoughthe jump-diffusion model was considered, the jump canonly be downward and therefore dividend payments willnot be triggered by jumps.

Furthermore, several dividend strategies have beenproposed in recent years by including jumps in the model.In particular, Avanzi et al. (2007) and Avanzi and Gerber(2008) studied the expectation of discounted dividends upto the time of ruin using a common barrier strategy. Ng(2009) considered the same problem under a generalthreshold dividend strategy. Gerber and Smith (2008)investigated the optimal dividend barrier with incompleteinformation under the model with diffusion. Albrecheret al. (2008) studied the expected discounted taxes beforeruin in which the taxes are paid at a constant rate whenthe surplus is at a running maximum. Zhu and Yang(2008) generalized the model to a regime-switching settingand studied the ruin probabilities instead of the dividendproblem.

In this paper, inspired by the above studies, we considerthe optimal impulse dividend problem in an upward jumpmodel under the framework of stochastic impulse control.The impulse dividends are paid out, incurring both fixedand proportional costs at the same time. The valuefunction is defined to maximize the expected totaldiscounted dividends payments up to the time of ruin.Because of the fixed cost, we found that the optimalstrategy is no longer a common barrier strategy orthreshold strategy. Instead, for the upward jump modelwith exponentially distributed income sizes, we prove thatthe optimal dividend strategy is a two-barrier policy. Withthe two-barrier strategy, there are no dividends paid out

before the surplus hits or exceeds the upper barrier. Onceit does, a lump sum amount to the surplus in excess of thelower barrier is paid out as dividends together with thetransaction costs. The surplus then starts at the lowerbarrier again. Dividends will not be paid until the time thesurplus hits the upper barrier again, and the same rule isapplied. In addition, we find that the optimal barriers areonly affected by the fixed cost and are independent of theproportional cost. When the fixed cost is zero, the two-barrier strategy degenerates to the common barrierstrategy studied by Avanzi et al. (2007). When the fixedcost is high, the lower barrier in the two-barrier strategywill be close to zero, which means that the investor willpay out all the surplus when it hits the upper barrier.

Finally, we would like to point out that Øksendal andSulem (2007) attempted to seek the classical solution tothe optimal dividend strategy by extending the valuefunction from a pure diffusion case to a jump diffusioncase (see example 6.4). However, the value function given there in equation (6.2.15) is not a full solution to thequasi-integrovariational inequalities presented in theo-rem 6.2. Due to the existence of jumps, the result ofequation (x) in theorem 6.2 involves the whole domain of . Therefore, it is true that 0(x) for �15x51 is thesolution to equation (x); however, after replacement by alinear function when x4x� and by 0 when x50, thecandidate solution in (6.2.15) is no longer a validsolution to equation (x). Similarly, Zou et al. (2009)incurred the same problem in their study. Despite thisslight flaw, their studies are indeed inspiring. In thispaper, we fix the problem for the upward jump modelwhen the jump size follows an exponential distribution.

The rest of the paper is organized as follows. In section2 we present a rigorous mathematical formulation of theoptimal impulse dividend problem. In section 3 weintroduce the quasi-variational inequality with respect tothis stochastic impulse problem and establish the verifi-cation theorem. In section 4 the classical solutions to thevalue function and optimal impulse dividend strategy areobtained when the income size follows a commonexponential distribution. Numerical calculations andeconomic explanations are given in section 5.

2. Formulation of the optimal dividend problem

Let (�,F , {F t, t� 0}, P) be a complete filtered probabil-ity space on which all stochastic objects are well definedthroughout this paper. Here {F t, t� 0} is a filtration thatsatisfies the usual conditions. The surplus process of anasset is described by an upward jump risk model. Withoutdividend payments, the surplus at time t, also denoted byX(t), is

XðtÞ ¼ xþ SðtÞ � ct: ð2:1Þ

Here x is the initial surplus and c is the rate of expenses;{S(t)} is a compound Poisson process of form SðtÞ ¼PNðtÞ

k¼1 Ik, where {N(t)} is a Poisson process with jumpintensity �, and {Ik}, independent of {N(t)}, denotes aseries of incomes with a common distribution function F

2 M. Zhou and K.F.C. Yiu

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and a finite mean �. Clearly, the process X is a Levyprocess with upward jumps, but no diffusion, and S(t) canbe written as SðtÞ ¼

R t0

R10 zNðds, dzÞ, where N(ds, dz) is a

Poisson random measure with finite Levy measure �F.The infinitesimal generator of X, denoted by A, can beexpressed as

AgðxÞ ¼ �cg0ðxÞ þ �

Z 10

½ gðxþ yÞ � gðxÞ�dFð yÞ ð2:2Þ

for any function g in the domain of A. Throughout thispaper we denote {Px, x� 0} as the family of probabilitymeasures corresponding to the surplus process {X(t)} suchthat X(0)¼x, and we denote E

x as the expectation withrespect to P

x.Assume that the dividends are paid out in an impulsive

manner. An impulse dividend strategy � is described by

� ¼ ð�i, i ¼ 1, 2, . . . ; �i, i ¼ 1, 2, . . .Þ, ð2:3Þ

where {�i, i¼ 1, 2, . . .} is a sequence of increasing stoppingtimes with respect to {F t, t� 0} and {�i, i¼ 1, 2, . . .} is asequence of positive random variables. Here, �i is F �imeasurable denoting the amount of money paid out as

dividends to shareholders at time �i. In addition, trans-action costs are required for each payment. For thedividend payment �i, we assume that the insurer will pay afixed cost � � 0 and a proportion of the transaction cost��i with �� 0. Thus, for dividends �i, the amount ofmoney deducted from the surplus is

~�i :¼ � þ ð1þ �Þ�i:

Incorporating this impulse dividend strategy �, thesurplus process of the asset can be expressed as

X�ðtÞ ¼ xþ SðtÞ � ct�X1i¼1

If�i5tg~�i

¼ XðtÞ �X1i¼1

If�i5tg½� þ ð1þ �Þ�i�, ð2:4Þ

or, equivalently, written in the form of a stochasticdifferential equation,

dX�ðtÞ ¼

Z 10

zNðdt, dzÞ � cdt, �j 5 t � �jþ1,

X�ð�jþÞ ¼ X�ð�j�Þ þ DNSð�j Þ � ~�j, j ¼ 0, 1, 2, . . . ,

with X�(0)¼ x, �0¼ 0, and DNS(�j)¼S(�j)�S(�j�)denotes the size of the jump caused by the income processat time �j.

In practice, the company is strictly prohibited to paydividends under deficit. Hence, in this paper, it isreasonable to constrain every dividend payment so thatit will not lead to a negative surplus, i.e.

� � ~�i � X�ð�iÞ, or, equivalently,

0 � �i �1

1þ �½X�ð�iÞ � ��: ð2:5Þ

In particular, when the current surplus is less than thefixed cost �, it is not allowed to pay dividends because thecurrent surplus is not sufficient to cover the fixed cost.

We call the impulse dividend strategy � defined in (2.3)together with (2.5) an admissible control policy and denote� as the set of all admissible control policies. For eachadmissible control �2�, the time of ruin is defined as

�� ¼ infft � 0 : X�ðtÞ ¼ 0g,

and the corresponding performance functional J isdefined as

Jðx;�Þ ¼ ExX1i¼1

e��i If�i���g�i

" #,

where 40 is the discount factor. The goal of this paper isto find the value function defined as

VðxÞ ¼ sup�2�

Jðx;�Þ, for x � 0, ð2:6Þ

and the optimal dividend strategy �� ¼ ð��i , i ¼ 1, 2, . . . ;

��i , i ¼ 1, 2, . . .Þ 2 � such that V(x)¼ J(x;��).

3. The value function and the quasi-variational

inequalities

In this section, we establish the quasi-variational inequal-ities (QVI) associated with this stochastic impulse controlproblem and prove the verification theorem for a solutionto the QVI. Before doing so, we first establish a propertyof the value function.

Proposition 3.1: The value function V(x) defined by (2.6)is bounded by a linear function. More exactly, forevery x� 0,

ðx� �Þþ

1þ �� VðxÞ �

1

1þ �xþj��� cj

� �: ð3:1Þ

Proof: Apparently, the lower bound follows, because theentire surplus can be paid out instantaneously if x4� andno dividend is paid out if x� �. In the following weestablish the upper bound. For any admissible policy�2�, by the monotone convergence theorem, we have,from (2.4),

Jðx,�Þ¼ limt!þ1

ExX1i¼1

e��i If�i�t^��g�i

" #

� limt!þ1

1

1þ�Ex

Z t^��

0

e�sdXðsÞ�

Z t^��þ

0

e�sdX�ðsÞ

� �:

For the first integral, noting that t6 �� is a boundedstopping time, we have

Ex

Z t^��

0

e�sdXðsÞ

� �¼E

x

Z t^��

0

e�sdSðtÞ�c

Z t^��

0

e�sdt

� �

¼ ð��� cÞEx

Z t^��

0

e�sds

� �

� j��� cj

Z 10

e�sds

¼j���cj

: ð3:2Þ

Optimal dividend strategy with transaction costs 3

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For the second integral, from the Ito formula it follows

that

0 � e�ðt^��ÞX�ðt ^ ��þÞ ¼ x�

Z t^��þ

0

e�sX�ðsÞds

þ

Z t^��þ

0

e�sdX�ðsÞ:

Since X�(s)� 0 for s� t6 ��, it follows thatZ t^��þ

0

e�sdX�ðsÞ � �x: ð3:3Þ

Together with (3.2) and (3.3), we have

Jðx,�Þ �1

1þ �xþj��� cj

� �,

and (3.1) follows by taking the supremum over all

�2�. œ

For this impulse control problem, we start with the

definition of the intervention operatorM.

Definition 3.2: For each continuous function h

supported on [0, 1), the intervention operator M is

defined by

MhðxÞ

¼sup� hðx���ð1þ�Þ�Þþ �;0� �� x��

1þ�

n o, � � x51,

0, 0� x5�:

(

ð3:4Þ

The intervention operator can be intuitively interpreted

as follows. Let us view h(x) as the value function and

assume the current surplus is x. At the current time, there

are two possible choices for the decision maker: pay

dividends or not. If the decision maker decides to pay a

lump sum dividend �, it is reasonable to pay the amount

by which the supremum is attained in (3.4), otherwise the

dividend strategy will not be optimal. Therefore, if the

decision is right for the optimal strategy, we have

h(x)¼Mh(x). Otherwise, h(x)4Mh(x) follows from the

definition of the value function. The set Dh, defined by

Dh ¼ fx � 0; hðxÞ ¼ MhðxÞg,

is called the intervention region of h. Thus, for the value

function V(x) the inequality V(x)�MV(x) must hold.

This is the key step in finding the intervention region of

the value function.

Definition 3.3: A continuous function g(x) supported on

[0, 1) is said to satisfy the quasi-variational inequalities

of the impulse control problem if

gð0Þ ¼ 0, ð3:5Þ

gðxÞ �MgðxÞ � 0, x � 0, ð3:6Þ

AgðxÞ � gðxÞ � 0, x � 0, ð3:7Þ

½AgðxÞ � gðxÞ�½ gðxÞ �MgðxÞ� ¼ 0, x � 0: ð3:8Þ

We define the optimal impulse control policy associatedwith this solution in the following.

Definition 3.4: Given a solution (x) to QVI, an optimalimpulse control policy,

� ¼ ð��1 , ��2 , . . . ; ��1, �

�2, . . .Þ,

is defined inductively as follows:

��1 ¼ infft � 0 : XðtÞ 2 Dg, ��1 ¼ argMðXð��1 ÞÞ;

for j� 2,

��jþ1 ¼ infft4 ��j : X�jðtÞ 2Dg, ��jþ1 ¼ argMðX�jð��jþ1ÞÞ:

As a result, the control strategy �j is defined by

�j ¼ ð��1 , ��2 , . . . , ��j ; ��1, �

�2, . . . , ��j Þ:

The optimal impulse control policy suggests thefollowing facts. Given a candidate (x) for the valuefunction, the intervention takes place once the controlledprocess falls into the intervention region of , and theamount of dividend payment is determined by taking themaximum of the intervention operatorM(x).

In the following theorem, we show that a solution toQVI can be a candidate for the value function of thisstochastic impulse control problem. If the optimalimpulse control policy associated with this candidatesolution is taken, then the candidate coincides with thevalue function.

Theorem 3.5: Let (x) be a continuous function supportedon [0, 1) which is differentiable except for x¼ b� � 0. If(x) is a solution to the quasi-variational inequalitiesdefined in definition 3.3, such that

0 � ðxÞ � Kð1þ xÞ, for some constant K4 0, ð3:9Þ

then

ðxÞ � VðxÞ:

Furthermore, by taking the optimal impulse control policy� associated with , we have

ðxÞ ¼ VðxÞ ¼ Jðx;�Þ:

Proof: Choose any admissible strategy �¼ (�i, i¼ 1, 2,. . .; �i, i¼ 1, 2, . . .)2� and set �0¼ 0. By employing the Itoformula, we have, a.s. P

x (that is, excluding the null set{X�(s�)¼ b� for at most countable s� 0}), that

e�ðt^��ÞðX�ðt ^ ��þÞÞ � ðxÞ

¼

Z t^��þ

0

e�s½A � �ðX�ðs�ÞÞds

þX�i�t^��

e��i ½ðX�ð�iþÞÞ � ðX�ð�iÞÞ�:

Taking expectations on both sides of the above equality,together with (3.6) and (3.7), we obtain

Ex½e�ðt^�

�ÞðX�ðt ^ ��þÞÞ � ðxÞ � Ex

X�i�t^��

e��i�i

" #:

ð3:10Þ

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The above inequality becomes an equality when � is

replaced by the optimal impulse policy � associated with

(x). Because X� ðs�Þ 62 D for all s� 0, then

½A � �ðX� ðs�ÞÞ ¼ 0 for all s� 0. In addition, note

that X� ð�iÞ 2 D, then it follows that

ðX� ð��i þÞÞ � ðX� ð��i ÞÞ ¼ ��

�i :

Taking t!þ1 on both sides of (3.10) and by the

monotone convergence theorem, we have

limt!þ1

Ex½e�ðt^�

�ÞðX�ðt ^ ��þÞÞ� � ðxÞ � ExX�i���

e��i�i

" #:

On the other hand, together with (0)¼ 0 we have

Ex½e�ðt^�

�ÞðX�ðt ^ ��þÞÞ� ¼ Ex½e�tðX�ðtþÞÞIf��4tg�:

Since (x) is bounded by a linear function (see

equation (3.9)), we have

0 � Ex½e�ðt^�

�ÞðX�ðt ^ ��þÞÞ� � KEx½e�tð1þ X�ðtþÞÞ�

� KEx½e�tð1þ XðtÞÞ�:

Because

limt!þ1

Ex½e�tjXðtÞj� � lim

t!þ1e�t½xþ ðcþ ��Þt� ¼ 0,

we can conclude that

limt!þ1

Ex½e�ðt^�

�ÞðX�ðt ^ ��þÞÞ� ¼ 0,

and for every admissible strategy �2�,

ðxÞ � ExX�i���

e��i�i

" #¼ Jðx;�Þ:

The above inequality becomes an equality by taking

optimal impulse policy � associated with (x). Thus it

follows that

ðxÞ ¼ VðxÞ ¼ Jðx;�Þ:

Then the proof is completed. œ

4. Closed-form solutions to the quasi-variational

inequalities

From the last section, we know that a solution to QVI will

be a candidate for the value function. In this section, we

construct the closed-form solution to the value function

and optimal impulse dividend policy when the income size

follows an exponential distribution, i.e.

dFð yÞ ¼1

�e�ð1=�Þ ydy, y � 0:

In addition, we assume that the expected increase of the

surplus per unit time is positive, i.e.

E½Sð1Þ� � c ¼ ��� c4 0, ð4:1Þ

which is the so-called positive safe loading condition.

To construct a classical solution, we guess that the

intervention region of the value function is of the form

[b�,1). Thus, we first solve A(x)� (x)¼ 0, or,

equivalently,

c0ðxÞ þ ðxÞ � �

Z 10

½ðxþ yÞ � ðxÞ�dFð yÞ ¼ 0,

for x 2 ð0, b�Þ, ð4:2Þ

with boundary condition (0)¼ 0. It is worth noting that

this integro-differential equation (4.2) involves the form

of from x to1. It is impossible to give a solution to this

equation without knowing the expression of (x) for

x4b�. Fortunately, in the case of exponential incomes,

the integro-differential equation (4.2) can be reduced to

an ordinary differential equation by applying the operator

[(d/dx)� (1/�)]. In fact, if dF(y)¼ (1/�) e�(1/�)y dy, we

can derive that

d

dx

Z 10

ðxþ yÞdFð yÞ ¼1

�

Z 10

ðxþ yÞdFð yÞ � ðxÞ

� �,

by employing integration by parts, i.e.

d

dx�

1

�

� �Z 10

ðxþ yÞdFð yÞ ¼ �1

�ðxÞ:

Therefore, applying the operator [(d/dx)� (1/�)] to (4.2),

we have

c00ðxÞ þ þ ��c

�

� �0ðxÞ �

�ðxÞ ¼ 0,

which implies that the solution is of the form erx for some

constant r2R. Thus r must satisfy the equation

cr2 þ þ ��c

�

� �r�

�¼ 0: ð4:3Þ

This equation determines two roots, r1 and r2, such that

r2505r151/�. Combining with the positive loading

condition ��4c, we know that jr2j4r1. With this choice

of r1 and r2 determined by (4.3), we have ðxÞ ¼ A1er1xþ

A2er2x. Combining with the boundary condition (0)¼ 0,

(x) can be rewritten as

ðxÞ ¼A

1þ �ðer1x � er2xÞ, for x 2 ½0, b�Þ:

Motivated by the diffusion case studied by Jeanblanc-

Picque and Shiryaev (1995) and Cadenillas et al. (2007),

we try a two-barrier dividend strategy with 0� a� � b� as

the optimal dividend strategy. If this conjecture is right,

then the value function will be of the form

ðxÞ ¼

A1þ� ðe

r1x � er2xÞ, 0 � x5 b�,

ða�Þ þ x�a���1þ� , b� � x51,

(ð4:4Þ

where the three constants A, a� and b� are yet to be

determined. In addition, the value function (x) is

continuous at b�, which means

ðb��Þ ¼b� � a� � �

1þ �þ ða�Þ: ð4:5Þ

Optimal dividend strategy with transaction costs 5

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If D¼ [b�, 1), for any x� b�, we have

ðxÞ ¼ sup ðx� � � ð1þ �Þ�Þ þ �; 0 � � �x� �

1þ �

� �:

Thus the maximum, denoted by �ðxÞ, satisfies

x� � � ð1þ �Þ�ðxÞ ¼ a�, 0ða�Þ ¼1

1þ �and 00ða�Þ5 0:

From (4.4), we know that

0ða�Þ ¼A

1þ �ðr1e

r1a�

� r2er2a�

Þ ¼1

1þ �,

which implies

A ¼1

r1er1a�� r2er2a

� : ð4:6Þ

On the other hand, substituting (4.4) into (4.2) yields

c0ðxÞ þ ðþ �ÞðxÞ � �

Z b�

x

ð yÞdFð y� xÞ

��

1þ �

Z 10

ydFð yþ b� � xÞ � �ðb�Þ �Fðb� � xÞ ¼ 0,

05 x5 b�:

Letting x! b�� on both sides of the above equation

yields

c0ðb��Þ þ ðb�Þ ��

1þ �� ¼ 0:

Combining with (4.5), we have

A ¼��

ðcr1 þ Þer1b�� ðcr2 þ Þer2b

� : ð4:7Þ

From (4.6) and (4.7), if A is given, a� and b� can be solved

from these two equations simultaneously. Noting that

the solution to each equation might not be unique

(see figure 1), we view A as a function of a� and b�.

According to (4.6) and (4.7), we define two new functions,

Aa(x) and Ab(x), as follows:

AaðxÞ ¼1

r1er1x � r2er2x, ð4:8Þ

AbðxÞ ¼��

ðcr1 þ Þer1x � ðcr2 þ Þer2x: ð4:9Þ

Thus we need to find a�, b� and A such that

A¼Aa(a�)¼Ab(b

�). In what follows, we analyse the

property of the two function Aa(x) and Ab(x).First, at x¼ 0, we have

05Aað0Þ ¼1

r1 � r25

��=c

r1 � r2¼ Abð0Þ, ð4:10Þ

where the inequality follows from the positive loading

condition (4.1).Taking the derivative with respect to x on both sides

and after some algebra, we have

A0aðxÞ ¼ �eðr1þr2ÞxqaðxÞ

ðr1er1x � r2er2xÞ2, ð4:11Þ

A0bðxÞ ¼ ���eðr1þr2ÞxqbðxÞ

½ðcr1 þ Þer1x � ðcr2 þ Þer2x�2, ð4:12Þ

where

qaðxÞ ¼ r21e�r2x � r22e

�r1x, ð4:13Þ

qbðxÞ ¼ ðcr1 þ Þr1e�r2x � ðcr2 þ Þr2e

�r1x: ð4:14Þ

Since r2505r1 and jr2j4r1, it follows that qað0Þ ¼ ð1=cÞqbð0Þ ¼ r21 � r22 5 0 and q0aðxÞ ¼ r1r2ðr2e

�r1x� r1e�r2xÞ4 0.

As for qb(x), we have q0bðxÞ ¼ r1r2½ðcr2 þ Þe�r1x�

ðcr1 þ Þr2e�r2x�. From (4.3) and (4.1), it follows that

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8Aa(x )Ab(x )A(x )

Figure 1. Shapes of the functions Aa(x), Ab(x) and A(x).

6 M. Zhou and K.F.C. Yiu

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cr2þ ¼ (1/�) [(/r2)þ (c� ��)]50. Thus we also have

q0bðxÞ4 0. Both qa(x) and qb(x) are strictly increasing

functions. Let �xa and �xb satisfy qað �xaÞ ¼ 0 and qbð �xbÞ ¼ 0,

respectively, i.e.

�xa ¼1

r1 � r2lnr22r21, ð4:15Þ

�xb ¼1

r1 � r2lnðcr2 þ Þr2ðcr1 þ Þr1

: ð4:16Þ

With the above analysis, we can conclude that the

function Aa(x) is strictly increasing on ½0, �xa� and strictly

decreasing to zero on ½ �xa,1Þ, and the function Ab(x) is

strictly increasing on ½0, �xb� and strictly decreasing to zero

on ½ �xb,1Þ. In addition, from (4.3), it follows that

�xb ¼1

r1 � r2ln� ð��� cÞr2� ð��� cÞr1

¼1

r1 � r2lnr22ðð1=�Þ � r1Þ

r21ðð1=�Þ � r2Þ

51

r1 � r2lnr22r21¼ �xa:

In addition,

AaðxÞ�AbðxÞ ¼½ðcr1þ Þ���r1�e

r1x�½ðcr2þ Þ���r2�er2x

ðr1er1x� r2er2xÞ½ðcr1þ Þer1x�ðcr2þ Þer2x�

¼�eðr1þr2ÞxqbðxÞ

ðr1er1x� r2er2xÞ½ðcr1þ Þer1x�ðcr2þ Þer2x�,

which means Aa(x)5Ab(x) on ½0, �xbÞ, Aa(x)4Ab(x) on

½ �xb,1Þ and Aað �xbÞ ¼ Abð �xbÞ.Combining with Aa(x) and Ab(x), we define a new

function A(x)¼min{Aa(x), Ab(x)}, or, equivalently,

AðxÞ ¼AaðxÞ, 0 � x � �xb,AbðxÞ, x � �xb:

�ð4:17Þ

Thus, the function A(x) is strictly increasing on ½0, �xb� and

strictly decreasing to zero on ½ �xb,1Þ. The shapes of the

functions A(x), Aa(x) and Ab(x) are shown in figure 1.In what follows we will give the solutions of a�, b� and

A in terms of the function A(x). Define �b4 �xb such that

Að0Þ ¼ Að �bÞ. From figure 1 we know that, for any

b 2 ½ �xb, �b�, there exists a unique a 2 ½0, �xb� such that

A(a)¼A(b). Consider a as a function of b and denote it by

a(b). For any b 2 ½ �xb, �b�, we have A(a(b))¼A(b), að �xbÞ ¼�xb and að �bÞ ¼ 0. Thus, the function a(b) is strictly

decreasing from �xb to zero on ½ �xb, �b�. For b4 �b, we

define a(b)� 0.Define a new function of the form

ðb, xÞ ¼ AðbÞðer1x � er2xÞ, b � 0 and 0 � x � b,

where the function A(x) is determined by (4.17).

Combined with the fact that (x) is continuous at b�,

this implies that b� must solve the equation

ðbÞ ¼ ðaðbÞÞ þb� aðbÞ � �

1þ �,

or, equivalently,Z b

aðbÞ

1�@

@x ðb, xÞ

� �dx ¼ �:

Lemma 4.1: Define a function

gðbÞ ¼

Z b

aðbÞ

1�@

@x ðb, xÞ

� �dx, b 2 ½ �xb,1Þ,

then g(b) is increasing from zero to þ1 on ½ �xb,1Þ.

Proof: Note that A(x)¼Aa(x) on ½0, �xb� and A(x)5Aa(x) on ð �xb,1Þ. Given b 2 ½ �xb,1Þ and a(b)� x� b, we

have A(b)�Aa(x) and

@

@x ðb,xÞ ¼AðbÞðr1e

r1x� r2er2xÞ �AaðxÞðr1e

r1x� r2er2xÞ ¼ 1,

i.e. 1� (@/@x) (b, x)� 0 for x2 [a(b), b]. Also note that

a(b) is decreasing on ½ �xb, þ1Þ. Thus, for �xb � b1 5b2 51, we have [a(b1), b1]� [a(b2), b2], and

gðb1Þ ¼

Z b1

aðb1Þ

1�@

@x ðb1,xÞ

� �dx

5Z b1

aðb1Þ

1�@

@x ðb2, xÞ

� �dx

�

Z b2

aðb2Þ

1�@

@x ðb2, xÞ

� �dx ¼ gðb2Þ,

which implies that g(b) is strictly decreasing on ½ �xb,1Þ.

Since að �xbÞ ¼ �xb, we know that gð �xbÞ ¼ 0 and

gðþ1Þ ¼ limb!þ1

Z b

aðbÞ

1�@

@x ðb, xÞ

� �dx

¼ limb!1

b� AbðbÞðer1b � er2bÞ ¼ þ1:

This completes the proof. œ

According to the above lemma, we know that there

exists a unique constant b�, such that

gðb�Þ ¼

Z b�

aðb�Þ

1�@

@x ðb�, xÞ

� �dx ¼ �, b� 2 ½ �xb,1Þ:

ð4:18Þ

In addition, since A(b�)40, we know that

@

@x ðb�, xÞ ¼ Aðb�Þðr1e

r1x � r2er2xÞ4 0, ð4:19Þ

which implies that b� � a(b�)� � from (4.18). The gap

between the upper barrier and the lower barrier is always

greater than the fixed cost.Based on the above analysis, we conclude the solution

to the value function in the following theorem.

Theorem 4.2: The value function is of the form

ð1þ �ÞVðxÞ

¼Aðb�Þðer1x � er2xÞ, 0 � x5 b�,

Aðb�Þðer1a�

� er2a�

Þ þ ðx� a� � �Þ, b� � x,

�ð4:20Þ

where b� 2 ½ �xb,1Þ is determined by (4.18) and a� ¼ a(b�),

such that 0 � a� � �xb � b�51. In addition, the value

function is not differentiable at x¼ b� except for the case

of �¼ 0.

Optimal dividend strategy with transaction costs 7

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Proof: According to the above analysis, we know thatV(x) with the form (4.20) is continuous on (0,1) anddifferentiable on (0, b�)[ (b�, 1). This function is notdifferentiable at x¼ b� except for the case where �¼ 0.This is because

ð1þ �ÞV0ðb��Þ ¼@

@x ðb�, xÞjx¼b�

¼ Aðb�Þðr1er1b�

� r2er2b�

Þ

� Aaðb�Þðr1e

r1b�

� r2er2b�

Þ

¼ 1, ð4:21Þ

where the equality follows only when �¼ 0 and thenb� ¼ �xb. So we have V0(b� � )51/(1þ �) when b�4 �xb,and V0(b��)¼ 1/(1þ �) when b� ¼ �xb. Note thatV0(b�þ)¼ 1/(1þ �). Hence V0(b��)� 1/(1þ �)¼V0(b�þ).The equality follows only when b� ¼ �xb, which is the casewhere the fixed cost �¼ 0 in (4.18).

In order to show that V(x) is a solution to QVI, weneed the following two steps.

Step 1: We verify DV¼ [b�, 1). According to thedefinition of the intervention operator, we have

MVðxÞ ¼

Vða�Þ þ x�a���1þ� , x � a� þ �,

Vðx� �Þ, �5 x5 a� þ �,

0, 0 � x � �:

8><>:

Without loss of generality, we assume that 05� �a�5a� þ �5b�.

For 0� x5a� þ �, we have V0ðxÞ ¼ Aðb�Þðr1er1x�

r2er2xÞ4 0, which implies

VðxÞ4Vð0Þ ¼ 0 ¼MVðxÞ, for 0 � x5 �,

and

VðxÞ4Vðx� �Þ ¼ MVðxÞ, for � � x5 a� þ �:

For a� þ � � x5b� we have

V0ðxÞ ¼ Aðb�Þðr1er1x � r2e

r2xÞ5Aaðb�Þðr1e

r1x � r2er2xÞ

� AaðxÞðr1er1x � r2e

r2xÞ ¼1

1þ �,

which implies

VðxÞ4Vðb�Þ �b� � x

1þ �¼ Vða�Þ þ

b� � a� � �

1þ ��b� � x

1þ �

¼ Vða�Þ þx� a� � �

1þ �¼MVðxÞ:

For x� b�, it is clear that V(x)¼MV(x).

Step 2: We verify (3.7) of QVI. For 0� x� b�, we havethat AV(x)� V(x)¼ 0 and V(0)¼ 0. To see this,replacing (x) by V(x) (defined in (4.20)) in the equationbelow (4.6), and noting that dF(y)¼ (1/�) e�(1/�)ydy, wehave, for any 0� x� b�,

�AVðxÞ þ VðxÞ

¼ cV0ðxÞ þ ðþ �ÞVðxÞ � �

Z b�

x

Vð yÞdFð y� xÞ

��

1þ �

Z 10

ydFð yþ b� � xÞ � �Vðb�Þ �Fðb� � xÞ

¼Aðb�Þ

1þ �

�cr1 þ þ �þ

�

r1�� 1

� �er1x

� cr1 þ þ �þ�

r1�� 1

� �er2x

�

�

�Aðb�Þ

1þ �

��þ

�

r1�� 1

� �er1b

�

� �þ�

r2�� 1

� �er2b

�

�

þ��

1þ �

�e�ðb

��xÞ=�:

In addition, from (4.3), we know that

cri þ þ �þ�

ri�� 1¼ 0, i ¼ 1, 2, ð4:22Þ

and

�þ�

ri�� 1¼ �cri þ , i ¼ 1, 2: ð4:23Þ

Note that b� � �xb, thus

Aðb�Þ ¼ Abðb�Þ ¼

��

ðcr1 þ Þer1b�� ðcr2 þ Þer2b

� :

This, together with (4.22) and (4.23), implies thatAV(x)� V(x)¼ 0 for 0� x� b�.

For x4b�, noting that V0(x)¼ 1/(1þ �), we haveAV(x)¼ (��� c)/(1þ �), and therefore

AVðxÞ � VðxÞ ¼ AVðb�þÞ � VðxÞ

5AVðb�þÞ � Vðb�þÞ

� AVðb��Þ � Vðb��Þ ¼ 0,

where the last two steps follow from the continuity ofV(x) at x¼ b� and V0(b��)� 1/(1þ �)¼V0(b�þ). Thus,(3.7) of QVI follows.

With the above two steps, we know that V(x) definedin (4.20) is a solution to QVI (3.5–3.8). Then, according totheorem 3.5, this completes the proof. œ

Remark 1: According to the procedure described in thissection, the two barriers of the optimal strategy aredetermined by (4.17) and (4.18). Therefore, the barriersare only related to the fixed transaction cost and do notdepend on the proportional transaction cost.

Denote a critical value for the fixed cost by

�0 ¼ gð �bÞ,

where the function g is defined in lemma 4.1. If 0� �5�0,it follows that 05 a� � b�5 �b. Otherwise, if � � �0, thenb� � �b and a� ¼ 0. This observation implies that if thefixed cost is less than the critical value (low fixed cost),there exists a regular two-barrier dividend strategy. If thefixed cost is greater than the critical value (high fixedcost), then the optimal dividend strategy is reduced to astrategy of waiting for a chance to take the money andrun away. In particular, for the special case where thefixed cost is zero, i.e. �¼ 0,

a� ¼ b� ¼ �xb ¼1

r1 � r2lnðcr2 þ Þr2ðcr1 þ Þr1

,

8 M. Zhou and K.F.C. Yiu

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which was obtained by Avanzi et al. (2007), where theproportional cost � is also assumed to be zero.

5. Numerical examples

In theorem 4.2, we obtain the value function and theoptimal dividend strategy in the case of exponentiallydistributed income sizes. In this section, we perform somenumerical calculations and study the impact of a fixedtransaction cost on the value function and the optimaldividend strategy.

Example 5.1: Set the parameters as follows: ¼ 0.1,�¼ 1, c¼ 1 and the jump size follows a Expð12Þ distribu-tion, i.e. f(x)¼ e�(1/2)x, x� 0. Then �¼ 2. The two rootsof (4.3) are r1¼ 0.0742 and r2¼� 0.6742. Then we have�xb ¼ 4:5436, �b ¼ 29:0024 and �0¼ 17.5191.

Figure 2 displays the optimal two-barrier strategies as afunction of the fixed transaction cost. Without the fixedcost (�¼ 0), the optimal strategy is a common barrierpolicy with level barrier �xb ¼ 4:5436, as represented bythe thin horizontal line. In the presence of a fixed cost, theoptimal strategy is a two-barrier policy. The amount ofsurplus over the lower barrier a� is paid out as dividendsplus costs once the surplus reaches the upper barrier b�.Thus, the fixed transaction has a significant impact on theoptimal dividend strategy. In addition, it should also benoted that, as the fixed cost increases, the upper barrierincreases, but the lower barrier decreases, making thedividend payments larger but less frequent. When thefixed cost exceeds �0¼ 17.5191, the lower barrier will stayat zero. When the surplus reaches the upper barrier b�,the amount of the whole surplus is paid out and ruinoccurs immediately because of the dividend payments.We call this a chance-waiting strategy to take the moneyand run away.

Set the proportional transaction cost rate �¼ 0.01.Figure 3 displays the shapes of the value function for thefixed cost �¼ 0, 10 and 20. As we can see, the valuefunction is convex when the fixed cost is zero. In thepresence of a fixed cost, the value function is no longerconvex. Thus, the fixed cost has a significant impact onthe value function. With an increase in the fixed cost, thevalue function decreases. In addition, as proved in the lastsection, the value function is not differentiable at x¼ b� inthe presence of a fixed cost, while it is differentiable whenthe fixed cost � ¼ 0.

Table 1 shows the optimal two-barrier strategies underdifferent fixed costs, and the corresponding value func-tion at x¼ b� is calculated. Figure 4 displays the valuefunction V(x) at x¼ b� as a function of the fixed cost. Aswe can see, V(b�) increases sharply when the fixed cost islow. This suggests that even a rather low fixed cost has avery significant impact on the value function.

6. Concluding remarks

The main contribution of this paper has been to studymathematically the optimal dividend strategies in upward

jump models in the presence of transaction costs.

The closed-form solution to the value function and

hence the optimal dividend strategy have been established

in the case of exponentially distributed income sizes. We

have shown that a two-barrier strategy is optimal and

both the upper barrier and the lower barrier depend only

on the fixed cost, but not the proportional cost.

Furthermore, it has been shown that an increased fixed

cost results in larger but less frequent dividend payments.

0 5 10 15 20 250

5

10

15

20

25

30

35

40

a*

b*

Figure 2. The two-barrier strategies as a function of the fixedcost.

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

x

V(x

)

= 20

= 10

= 0

Figure 3. The value function with different fixed costs.

Table 1. Optimal two-barrier strategies and V(b�) for differentfixed costs (�0¼ 17.5191, �xb ¼ 4:5436, �b ¼ 29:0024).

�

0 5 10 15 20 25 30

a� 4.5436 1.9470 1.0280 0.3199 0 0 0b� 4.5436 15.1120 21.1229 26.4512 31.4833 36.4833 41.4833V(b�) 9.9010 11.3690 11.3696 11.3696 11.3696 11.3696 11.3696

Optimal dividend strategy with transaction costs 9

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As an extension of this work, it would certainly be ofinterest to study the optimal dividend strategy with amore general distributed income size.

Acknowledgements

The first author was supported by a grant from theNatural Science Foundation of China (10701082) and bythe Scientific Research Foundation for ReturnedOverseas Chinese Scholars, State Education Ministry.The second author was supported partially by RGC grantPolyU(5001/11P) during revision of the paper, TheAMSS-PolyU Joint Research Institute and the ResearchCommittee of Hong Kong Polytechnic University.

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0 2 4 6 8 109.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4V

(b* )

Figure 4. The value function at b� as a function of �.

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