optimal reinsurance and dividend strategies under the markov-modulated insurance risk model

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This article was downloaded by: [University of Saskatchewan Library] On: 12 September 2012, At: 14:37 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Stochastic Analysis and Applications Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsaa20 Optimal Reinsurance and Dividend Strategies Under the Markov-Modulated Insurance Risk Model Jiaqin Wei a , Hailiang Yang b & Rongming Wang a a School of Finance and Statistics, East China Normal University, Shanghai, P. R. China b Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, P. R. China Version of record first published: 29 Oct 2010. To cite this article: Jiaqin Wei, Hailiang Yang & Rongming Wang (2010): Optimal Reinsurance and Dividend Strategies Under the Markov-Modulated Insurance Risk Model, Stochastic Analysis and Applications, 28:6, 1078-1105 To link to this article: http://dx.doi.org/10.1080/07362994.2010.515488 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

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Page 1: Optimal Reinsurance and Dividend Strategies Under the Markov-Modulated Insurance Risk Model

This article was downloaded by: [University of Saskatchewan Library]On: 12 September 2012, At: 14:37Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Stochastic Analysis and ApplicationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsaa20

Optimal Reinsurance and Dividend Strategies Under theMarkov-Modulated Insurance Risk ModelJiaqin Wei a , Hailiang Yang b & Rongming Wang aa School of Finance and Statistics, East China Normal University, Shanghai, P. R. Chinab Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, P.R. China

Version of record first published: 29 Oct 2010.

To cite this article: Jiaqin Wei, Hailiang Yang & Rongming Wang (2010): Optimal Reinsurance and Dividend Strategies Underthe Markov-Modulated Insurance Risk Model, Stochastic Analysis and Applications, 28:6, 1078-1105

To link to this article: http://dx.doi.org/10.1080/07362994.2010.515488

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.

Page 2: Optimal Reinsurance and Dividend Strategies Under the Markov-Modulated Insurance Risk Model

Stochastic Analysis and Applications, 28: 1078–1105, 2010Copyright © Taylor & Francis Group, LLCISSN 0736-2994 print/1532-9356 onlineDOI: 10.1080/07362994.2010.515488

Optimal Reinsurance and Dividend Strategies UndertheMarkov-Modulated Insurance RiskModel

JIAQIN WEI,1 HAILIANG YANG,2

AND RONGMING WANG1

1School of Finance and Statistics, East China Normal University,Shanghai, P. R. China2Department of Statistics and Actuarial Science, The Universityof Hong Kong, Hong Kong, P. R. China

In this article, we consider the optimal reinsurance and dividend strategy for aninsurer. We model the surplus process of the insurer by the classical compoundPoisson risk model modulated by an observable continuous-time Markov chain. Theobject of the insurer is to select the reinsurance and dividend strategy that maximizesthe expected total discounted dividend payments until ruin. We give the definition ofviscosity solution in the presence of regime switching. The optimal value functionis characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation and a verification theorem is also obtained.

Keywords Compound Poisson model; Dividend strategy; HJB equation; Regimeswitching; Reinsurance; Viscosity solution.

Mathematics Subject Classification Primary 93E20; Secondary 91B70, 60H30.

1. Introduction

With the application of control theory, the optimal dividend problem has beeninvestigated by many authors in the literature. Under the diffusion model, thisproblem was studyed by [2, 12] in the case of no reinsurance, by [14] in the

Received April 4, 2009; Accepted June 7, 2010The authors would like to thank the referee for careful reading the paper and the

helpful comments and suggestions. J. W. would like to acknowledge the PhD ProgramScholarship Fund of ECNU (No. 2010050). H. Y. would like to acknowledge the ResearchGrants Council of the Hong Kong Special Administrative Region, China (project No. HKU754008H); R. W. would like to acknowledge the National Natural Science Foundation ofChina (10971068), National Basic Research Program of China (973 Program) under grantnumber 2007CB814904, Program for New Century Excellent Talents in University (NCET-09-0356), and the Fundamental Research Funds for the Central Universities.

Address correspondence to Hailiang Yang, Department of Statistics and ActuarialScience, The University of Hong Kong, Pokfulam Road, Hong Kong, P. R. China; E-mail:[email protected]

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Optimal Reinsurance and Dividend Strategies 1079

case of proportional reinsurance, and by [3] and [8] in the case of excess-of-lossreinsurance. Under the classical compound Poisson model, it was discussed by [4]via the viscosity approach, by [13, 16, 22].

Recently, the Markov-modulated insurance risk model becomes popular. Thismodel was proposed by [1] in which the claim inter-arrivers and claim sizes areinfluenced by an external environment process �J�t��t≥0. This model can capturethe feature that insurance policies may need to change if economical or politicalenvironment changes. Under the regime-switching compound Poisson model, [17]considered the constant barrier strategy, but they did not address the optimaldividend strategy. Sotomayor and Cadenillas [25] and Jiang and Pistorius [15]discussed the optimal dividend strategy under the regime-switching diffusion model.In this article, we study the optimal reinsurance and dividend strategy underthe regime-switching compound Poisson model via the viscosity approach, andgeneralize some results of [4].

The notion of viscosity solution was introduced by [9, 18]. Nowadays, it is astandard tool for studying the Hamilton–Jacobi–Bellman equations (see, e.g., [5, 11,20, 21, 23, 24].

Suppose that �J�t��t≥0 is a homogenous continuous-time Markov chain takingvalues in a finite set � = �1� 2� � � � � N� with generator Q = �qij�N×N where −qii = qi,i = 1� 2� � � � � N . We further assume that �J�t��≥0 is irreducible.

At time t, given J�t� = i, the premium rate is ci, claims arrive according toPoisson process with rate �i, and the size of the claim which arrives at time tfollows the distribution Fi with density fi and mean �i. Furthermore, we assume thepremium rate ci is calculated using the expected value principle with relative safetyloading i > 0; that is, ci = �1+ i��i�i.

In the case of no control, suppose the initial surplus is u ≥ 0, the correspondingsurplus process �U�t��t≥0 of the insurer is given by

U�t� = u+∫ t

0cJ�s�ds −

∫ t

0

∫�0���

xNJ�s��ds × dx�� t ≥ 0�

where Ni�dt × dx� is a Poisson random measure with intensity measure �ifi�x�dt dx.Now, we assume that the insurer has the possibility of reinsurance in which each

individual claim of X is dividend between the insurer and the reinsurer according toa reinsurance strategy R�·� �0�� → �0�� such that 0 ≤ R�X� ≤ X: the insurer paysR�X�, the reinsurer pays X − R�X�. For this, the insurer pays a reinsurance premiumto the reinsurer. We restrict our analysis to the case of cheap reinsurance, whichmeans that the relative safety loading of the reinsurer is the same with the insurer’s.Then given the state i, the distribution of the claim sizes paid by the insurer is

Fi�R� x� =∫ �

01�R�y�≤x�dFi�y��

where 1�·� is the indicator function. We define �i�R� as its expectation. Then thepremium rate received by the insurer is

ci�R� = �1+ i��i�i�R��

Since R�X� ≤ X, we get that �i�R� ≤ �i and ci�R� ≤ ci.

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1080 Wei et al.

Considering a family � of reinsurance policies, we are going to find a dynamicchoice of both the reinsurance police R ∈ � and the dividend strategy whichmaximizes the cumulative expected discounted dividend payments.

Similar to [4], we consider the following families of reinsurance policies:

• �0 = �RI�, where RI�X� = X;• �P = �R R�X� = rX for some r ∈ �0� 1�� the set of proportional reinsurance

policies;• �XL = �R R�X� = min�r� X� for some r ∈ �0���� the set of excess-of-lossreinsurance policies;

• �A the set of all reinsurance policies.

Suppose that the aforementioned random variables and stochastic processes aredefined on the filtered probability space � �� �����, where � = ��t� t ≥ 0� is thefiltration generated by �U�t��t≥0 and �J�t��t≥0. We also assume that � satisfies theusual condition. Denote by �u�i the measure � conditioned on �U�0� = u� J�0� = i�,and write �u�i for the corresponding expectations.

A control strategy � is a two-dimensional stochastic process �R��t�� L��t��t≥0,where R��t� ∈ � is the reinsurance policy and L��t� is the cumulative amount ofdividends paid out up to time t. Given such a control strategy �, the surplus of theinsurer is given by

U��t� = u+∫ t

0cJ�s��R

��s��ds −∫ t

0

∫�0���

xNJ�s��R��s�� ds × dx�− L��t�� (1.1)

where Ni�R��s�� dt × dx� is a Poisson random measure with intensity measure

�idtdFi�R��t�� x�.

We say that a control strategy � is admissible if

• the process R��t� is predictable;• the process L��t� is predictable, nondecreasing, no-negative and càglàd;• for any t ≥ 0, the process L��t� satisfies

L��t+�− L��t� ≤ U��t�� (1.2)

We denote by � the set of all admissible strategies.

Let �� = inf�t ≥ 0 U��t� < 0� be the corresponding ruin time of the insurer,and

V��u� i� = �u�i

[∫ ��

0e−�tdL��t�

](1.3)

be the expected present value of all dividend until ruin given initial surplus u andinitial state i, where � is the force of interest. Note that, from the definition of theadmissible strategy, the ruin can occur only by the arrival of a claim.

The value function of the insurer is

V�u� i� = sup�∈�

V��u� i�� (1.4)

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Optimal Reinsurance and Dividend Strategies 1081

The problem for the insurer is to identify a control strategy �∗ ∈ � that satisfiesV�u� i� = V�∗�u� i� for all u ≥ 0 and i ∈ �. However, this article discusses the viscosityproperty of the value function instead of construction of the optimal strategy.

The rest of the article is organized as follows. Section 2 gives some properties ofthe value function. In Section 3, we prove the Dynamic Programming Priciple andthe Hamilton–Jacobi–Bellman equation. In Section 4, we give the definition of theviscosity solution in the case of regime swithcing; and show that the value functionis an unique vicosity solution of the associated Hamilton–Jacobi–Bellman equation.In Section 5, we give a characterization of the value funtion and prove a verificationtheorem. The techniques used in Sections 4 and 5 are similar to [4]. In Section 6, thespecial case with exponential claim density is discussed.

2. Some Properties of the Value Function

In this section, we are going to show some properties of V�u� i� for u ≥ 0 and i ∈ �.The techniques are standard [4, 22]. To simplify the notation, we denote

Ai = 1+ �

�i� Bi = 1+ i and Ci =

qi�i�

Proposition 2.1. For u ≥ 0 and i ∈ �, the value function V�u� i� satisfies

(i) V�u� i� ≤ u+ Bi0�i0

/�Ai0− 1�, where i0 ∈ � such that ci0 = maxi∈��ci�;

(ii) V�u� i� ≥ u+ Bi�i/�Ai + Ci�.

Proof. (i) For any admissible strategy � = �R��t�� L��t��, L��t� cannot be greaterthan the sum of initial surplus and the total incoming premium without reinsuranceup to time t (see also (1.1) and (1.2)). Then we have

L��t� ≤ u+ ci0 t�

consequently,

V��u� i� ≤ �u�i

[∫ �

0e−�sd�u+ ci0s�

]= u+ Bi0

�i0

Ai0− 1

So V�u� i� = sup�∈� V��u� i� satisfies inequality (i).

(ii) Let T1 and S1 be the time at which the first transition of the environmentprocess �J�t��t≥0 occurs and the first claim comes, respectively. Given the initialsurplus u and initial state i, consider the admissible strategy �0 that pays the wholereserve u and pays the incoming premium without reinsurance ci = �1+ i��i�i asdividend until T1 ∧ S1. Then we have

V�0�u� i� ≥ u+ �1+ i��i�i�u�i

[∫ T1∧S1

0e−�sds

]= u+ Bi�i

Ai + Ci

Then by the definition (1.4), we get the result. �

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1082 Wei et al.

Proposition 2.2. For all i ∈ �, 0 ≤ u1 < u2, the value function V�u� i� satisfies

(i) V�u2� i�− V�u1� i� ≥ u2 − u1;(ii) V�u2� i�− V�u1� i� ≤

(e�Ai+Ci��u2−u1�/�Bi�i� − 1

)V�u1� i�.

Proof. (i) For each � > 0, take an admissible strategy � such that V��u1� i� ≥V�u1� i�− �. For 0 ≤ u1 < u2, we define a new strategy �0 as follows: pay u2 − u1 asdividends immediately, and then follow the strategy �. Then we get for each � > 0 that

V�u2� i� ≥ V�0�u2� i� = V��u1� i�+ �u2 − u1� ≥ V�u1� i�− �+ �u2 − u1��

The result follows from the arbitrariness of �.

(ii) Given the initial surplus u1 and initial state i, for each � > 0, consideran admissible strategy � such that V��u2� i� ≥ V�u2� i�− � for any u2 > u1. Takenow the strategy �0 that starting with initial surplus u1 and initial state i, paysno dividends and take no reinsurance if U�0�t� < u2 and follow strategy � afterthe current surplus reaches u2. Given the event of no claims and no transition ofthe environment process �J�t��t≥0, the surplus U�0�t� reaches u2 at time t0 = �u2 −u1�/�Bi�i�i�. Thus, we get

V�u1� i� ≥ V�0�u1� i� ≥ V��u2� i�e

−�t0� ��T1 ∧ S1� > t0�

≥ �V�u2� i�− ��e−�Ai+Ci��u2−u1�/�Bi�i�

and then we obtain the result. �

Remark 2.1. From the above propositions, we know that the value function V isincreasing and locally Lipschitz in �0���, and so for each i ∈ �, V ′�·� i� exists almosteverywhere in �0��� and satisfies 1 ≤ V ′�u� i� ≤ �Ai+Ci�

Bi�iV�u� i�.

3. The Hamilton–Jacobi–Bellman Equation

First, we will show the Dynamic Programming Principle for the value function V .

Theorem 3.1. Given u ≥ 0, i ∈ � and any stopping time �, we have

V�u� i� = sup�∈�

�u�i

[∫ �∧��

0e−�sdL��s�+ e−���∧���V �U��� ∧ ���� J�� ∧ ����

]� (3.1)

Proof. First, we prove (3.1) for the case � is a fixed time T > 0. Let

v�u� i� T� = sup�∈�

�u�i

[∫ T∧��

0e−�sdL��s�+ e−��T∧���V �U��T ∧ ���� J�T ∧ ����

]�

Next, we will show V�u� i� ≤ v�u� i� T�. For any admissible strategy � = �R��t��L��t��, we have

V��u� i� = �u�i

[1���≤T�

∫ ��

0e−�sdL��s�

]

+�u�i

[1���>T�

(∫ T

0e−�sdL��s�+ e−�T

∫ ��−T

0e−�sdL��s + T�

)]

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Optimal Reinsurance and Dividend Strategies 1083

= �u�i

[∫ T∧��

0e−�sdL��s�

]

+ e−�T�u�i

[1���>T��

(∫ ��−T

0e−�sdL��s + T�

∣∣∣U��T�� J�T�

)]

≤ �u�i

[∫ T∧��

0e−�sdL��s�

]+ e−�T�u�i

[1���>T�V�U

��T�� J�T��]

≤ �u�i

[∫ T∧��

0e−�sdL��s�

]+�u�i

[e−��T∧���V�U��T ∧ ���� J�T ∧ ����

]≤ v�u� i� T��

From (1.4) we get the result.Now, we are in the position to prove V�u� i� ≥ v�u� i� T�. For any � ∈ �, denote

u0 = U��T ∧ ��� and i0 = J�T ∧ ���. For each � > 0, we find an admissible strategy�0 such that V�0

�u0� i0� ≥ V�u0� i0�− �. We define an admissible strategy �̃ as

�̃ ={� when t ≤ T ∧ ���

�0 when t > T ∧ ���

Then we have

V�u� i� ≥ V�̃�u� i�

= �u�i

[∫ T∧��

0e−�sdL��s�

]+�u�i

[∫ ��̃

T∧��e−�sdL�0�s�

]

= �u�i

[∫ T∧��

0e−�sdL��s�

]+�u�i

[e−��T∧���

∫ ��̃−�T∧���

0e−�sdL�0�s + T ∧ ���

]

= �u�i

[∫ T∧��

0e−�sdL��s�

]

+�u�i

[e−��T∧����

(∫ ��̃−�T∧���

0e−�sdL�0�s + T ∧ ���

∣∣∣U��T ∧ ���� J�T ∧ ���

)]

= �u�i

[∫ T∧��

0e−�sdL��s�

]+�u�i

[e−��T∧���V�0

�U��T ∧ ���� J�T ∧ ����]

≥ �u�i

[∫ T∧��

0e−�sdL��s�

]+�u�i

[e−��T∧��� �V�U��T ∧ ���� J�T ∧ ����− ��

]�

Then the result follows from the arbitrariness of �. Thus, we have V�u� i� = v�u� i� T�.In the following, we are going to show (3.1) holds for any bounded stopping

time � by the standard methods (see [26]). With the assumption that 0 ≤ � ≤ T ,where T > 0 is a fixed time, we can define a sequence of stopping times

�n =k

2nT� if

k− 12n

T ≤ � ≤ k

2nT�

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1084 Wei et al.

It is obvious that �n → �, a�s. For each �n = tk = k2n T , the Equation (3.1) holds,

that is,

V�u� i� = sup�∈�

�u�i

[∫ tk∧��

0e−�sdL��s�+ e−��tk∧���V �U��tk ∧ ���� J�tk ∧ ����

]�

Multiplying by ���n = tk� on both sides of the above equation and summing themup over all 1 ≤ k ≤ 2n, then we get

V�u� i� =2n∑k=1

sup�∈�

�u�i

[∫ tk∧��

0e−�sdL��s�+ e−��tk∧���V �U��tk ∧ ���� J�tk ∧ ����

]���n = tk�

=2n∑k=1

sup�∈�

�u�i

[�( ∫ tk∧��

0e−�sdL��s�

+ e−��tk∧���V(U��tk ∧ ���� J�tk ∧ ���

) ∣∣∣ �n = tk

)]���n = tk��

which implies that

V�u� i� = sup�∈�

�u�i

[∫ �n∧��

0e−�sdL��s�+ e−���n∧���V �U���n ∧ ���� J��n ∧ ����

]�

Then by the continuity of the value function, letting n → � yields theEquation (3.1). For any stopping time �, we consider the sequence of boundedstopping times �̂n = � ∧ n. Then the result follows from the Dominated ConvergenceTheorem. �

To give the Hamilton–Jacobi–Bellman Equation, we show a lemma first.

Lemma 3.1. Given i ∈ �, let f�t� x� i� be a function which is continuously differentiablewith respect to t and x. Then f�t� U��t�� J�t�� is a finite variation process and

f�t� U��t�� J�t��− f�0� U��0�� J�0��

=∫ t

0

∑i∈�

qJ�s�if�s� U��s�� i�ds +

∫ t

0

∑i∈�

f�s� U��s−�� i�dm�s� i�

+∫ t

0

[ft�s� U

��s�� J�s��+ fx�s� U��s�� J�s��cJ�s��R

��s��]ds

−∫ t

0fx�s� U

��s�� J�s��dL�c�s�−∫ t

0

∫ L��s+�−L��s�

0fx�s� U

��s�− y� J�s��dy ds

+∫ t

0

∫ �

0�f�s� U��s−�− x� J�s��− f�s� U��s−�� J�s��� NJ�s��R

��s�� ds × dx��

(3.2)

where m�·� i� is a martingale.

Proof. Obviously,

f�t� R��t�� J�t�� =∑i∈�

f�t� R��t�� i���t� i� (3.3)

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Optimal Reinsurance and Dividend Strategies 1085

where ��t� i� = 1�J�t�=i�. It is shown in [19] that

��t� i� = ��0� i�+∫ t

0

∑j∈�

qji��s� j�ds +m�t� i�� i ∈ �� (3.4)

where m�t� i� is a square integrable martingale with right-continuous paths. FromItô formula and Equation (1.1), we know that f�t� U��t�� i� is a finite variationprocess and

f�t� U��t�� i�− f�0� U��0�� i�

=∫ t

0ft�s� U

��s�� i�ds +∫ t

0fx�s� U

��s�� i�cJ�s��R��s��ds −

∫ t

0fx�s� U

��s�� i�dL�c�s�

+∫ t

0

∫ �

0�f�s� U��s−�− x� i�− f�s� U��s−�� i�� NJ�s��R

��s�� ds × dx�

−∫ t

0

∫ L��s+�−L��s�

0fx�s� U

��s�− y� i�dy ds� (3.5)

where L�c�t� is the continuous part of L��t�, that is, L��t� = L�c�t�+∑0≤s≤t �L

��s+�− L��s��.Since ��·� i� and f�t� U��t�� i� are a finite variation processes, then from

integration by parts, we get

f�t� U��t�� i���t� i� = f�0� U��0�� i���0� i�+∫ t

0f�s� U��s−�� i�d��s� i�

+∫ t

0��s� i�df�s� U��s�� i�� (3.6)

The following equation is derived from Equations (3.4), (3.5), and (3.6).

f�t� U��t�� i���t� i�− f�0� U��0�� i���0� i�

=∫ t

0

∑j∈�

qji��s� j�f�s� U��s�� i�ds +

∫ t

0f�s� U��s−�� i�dm�s� i�

+∫ t

0��s� i�

[ft�s� U

��s�� i�+ fx�s� U��s�� i�cJ�s��R

��s��]ds

−∫ t

0��s� i�fx�s� U

��s�� i�dL�c�s�−∫ t

0��s� i�

∫ L��s+�−L��s�

0fx�s� U

��s�− y� i�dy ds

+∫ t

0��s� i�

∫ �

0�f�s� U��s−�− x� i�− f�s� U��s−�� i�� NJ�s��R

��s�� ds × dx��

(3.7)

Then Equation (3.2) follows from Equations (3.3) and (3.7). �

For all i ∈ � and R ∈ �, given a continuous vector function v�·� =�v�·� 1�� v�·� 2�� � � � � v�·� N��T , we define the operator

�i�R� v��u� = v′�u� i�ci�R�+ �i

∫ u

0v�u− x� i�dFi�R� x�

− ��i + ��v�u� i�+∑k∈�

qikv�u� k��

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Theorem 3.2. For all i ∈ �, assume the value function V�u� i� defined byEquation (1.4) is continuously differentiable with respect to u ∈ �0���. Then V�u� i�

satisfies the Hamilton–Jacobi–Bellman equation

max{1− V ′�u� i�� sup

R∈��i�R� V��u�

}= 0� (3.8)

Proof. From Equation (3.2), we have

e−���∧���V �U��� ∧ ���� J�� ∧ ����− V �U��0�� J�0��

=∫ �∧��

0e−�s

∑k∈�

qJ�s�kV�U��s�� k�ds +

∫ �∧��

0e−�s

∑k∈�

V�U��s−�� k�dm�s� k�

+∫ �∧��

0e−�s

[−�V�U��s�� J�s��+ V ′�U��s�� J�s��cJ�s��R��s��

]ds

−∫ �∧��

0e−�sV ′�U��s�� J�s��dL�c�s�−

∫ �∧��

0

∫ L��s+�−L��s�

0e−�sV ′�U��s�− y� J�s��dyds

+∫ �∧��

0

∫ �

0e−�s �V�U��s−�− x� J�s��− V�U��s−�� J�s��� NJ�s��R

��s�� ds × dx��

Applying conditional expectation to the above equation yields

�u�i

[e−���∧���V �U��� ∧ ���� J�� ∧ ����

]− V�u� i�

= �u�i

[∫ �∧��

0e−�s

∑i∈�

V�U��s−�� i�dm�s� i�

]

−�u�i

[∫ �∧��

0e−�sV ′�U��s�� J�s��dL�c�s�

]

−�u�i

[∫ �∧��

0

∫ L��s+�−L��s�

0e−�sV ′�U��s�− y� J�s��dyds

]

+�u�i

[∫ �∧��

0e−�s�J�s��R

��s�� V��U��s��ds

](3.9)

Since V�·� i� is bounded and m�t� i� is a martingale, it is easy to show

�u�i

[∫ �∧��

0e−�s

∑k∈�

V�U��s−�� k�dm�s� k�

]= 0�

From Remark 2.1, we know that V ′�u� i� ≥ 1. Then

−�u�i

[∫ �∧��

0e−�sV ′�U��s�� J�s��dL�c�s�

]

−�u�i

[∫ �∧��

0

∫ L��s+�−L��s�

0e−�sV ′�U��s�− y� J�s��dyds

]

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Optimal Reinsurance and Dividend Strategies 1087

≤ −�u�i

[∫ �∧��

0e−�sdL�c�s�

]−�u�i

[∫ �∧��

0

∫ L��s+�−L��s�

0e−�sdy ds

]

= −�u�i

[∫ �∧��

0e−�sdL��s�

]�

For any admissible strategy � and h > 0, let ���h� = h ∧ inf�t U��t� ∈ �/�u−h� u+ h��. Then ���h� < � a.s. and ���h� → 0 a.s. Furthermore, if we choose h < u,then ���h� ≤ ��.

Thus, from the Dynamic Programming Principle (3.1) and Equation (3.9), forany � > 0, we can choose an admissible strategy � and h < u such that

V�u� i�− � ≤ �u�i

[∫ ���h�

0e−�sdL��s�+ e−�����h��V �U�����h��� J����h���

]

≤ V�u� i�+�u�i

[∫ ���h�

0e−�s�J�s��R

��s�� V��U��s��ds

]�

Because of the arbitrariness of �, we have

�u�i

[∫ ���h�

0e−�s�J�s��R

��s�� V��U��s��ds

]≥ 0�

Dividing the above equation by �u�i����h�� and then letting h → 0, we get

supR∈�

�i�R� V��u� ≥ 0�

Now, we consider an admissible strategy �0 such that R�0�t� can be anyelement R ∈ �, L�0�t� = 0 for t < ��0�h� and arbitrary for t ≥ ��0�h�. Then from theDynamic Programming Principle (3.1) and Equation (3.9), we have

V�u� i� ≥ �u�i

[e−����0 �h��V

(U�0��0�h��� J���0�h��

)]= V�u� i�+�u�i

[∫ ��0 �h�

0e−�s�J�s��R

�0�s�� V��U�0�s��ds

]�

Similarly, we have

supR∈�

�i�R� V��u� ≤ 0�

Thus, we get the Hamilton–Jacobi–Bellman equation

max{1− V ′�u� i�� sup

R∈��i�R� V��u�

}= 0�

In the above proof, we assume that the value function V is a sufficiently soothfunction which is not the case in most of the examples. Thus, in the following wewill study the viscosity solutions of the Hamilton–Jacobi–Bellman equation.

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4. Viscosity Solution

Now we are going to study the viscosity property of the value function. Sometechniques used in this section and the next one are similar to [4].

First we write the Hamilton–Jacobi–Bellman Equation (3.8) in the matrix form.

max{1− V′�u�� sup

R∈���R�V��u�

}= 0� (4.1)

where

V�u� = �V�u� 1�� V�u� 2�� � � � � V�u� N��T

and

��R�V��u� = ��1�R� V��u���1�R� V��u�� � � � ��N �R� V��u��T �

Next we give the definition of viscosity solution. Note that our equation isderivative constrained and V makes sense only in �0���.

Definition 4.1. (i) We say that a continuous vector function (each elementis continuous) v�·� = �v�·� 1�� v�·� 2�� � � � � v�·� N��T �0��� → �N is a viscositysubsolution of (4.1) at u ∈ �0���, if any continuously differentiable vector function(each element is continuously differentiable) ��·� = ���·� 1�� ��·� 2�� � � � � ��·� N��T

�0��� → �N with v�u� = ��u� such that for each i ∈ �, v�·� i�− ��·� i� reaches themaximum at u satisfies

max{1− �′�u� i�� sup

R∈��i�R� ���u�

}≥ 0� ∀i ∈ ��

(ii) We say that a continuous vector function v�·� =�v�·� 1�� v�·� 2�� � � � � v�·� N��T �0��� → �N is a viscosity supersolution of(4.1) at u ∈ �0���, if any continuously differentiable vector function ��·� =���·� 1�� ��·� 2�� � � � � ��·� N��T �0��� → �N with v�u� = ��u� such that for eachi ∈ �, v�·� i�− ��·� i� reaches the minimum at u satisfies

max{1− �′�u� i�� sup

R∈��i�R� ���u�

}≤ 0� ∀i ∈ ��

(iii) We say that a continuous vector function v�·� =�v�·� 1�� v�·� 2�� � � � � v�·� N��T �0��� → �N is a viscosity solution of (4.1) if it isboth viscosity subsolution and viscosity supersolution at any u ∈ �0���.

Before showing that the value function V is the unique viscosity solution of(4.1), we give an equivalent definition of the viscosity solution. When � containsonly one element, the next lemma is standard in the context of viscosity theory; seefor example, [4, 10]; and the proof can be found in [6, 20, 24].

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Optimal Reinsurance and Dividend Strategies 1089

For all i ∈ �, R ∈ �, given a continuous vector function v�·� =�v�·� 1�� v�·� 2�� � � � � v�·� N��T and a continuously differentiable vector functionf�·� = �f�·� 1�� f�·� 2�� � � � � f�·� N��T , we define the operator

�i�R� v� f��u� = f ′�u� i�ci�R�+ �i

∫ u

0v�u− x� i�dFi�R� x�

− ��i + ��v�u� i�+∑k∈�

qikv�u� k��

Lemma 4.1.

(i) A continuous vector function v�·� = �v�·� 1�� v�·� 2�� � � � � v�·� N��T �0��� → �N

is a viscosity subsolution of (4.1) at u ∈ �0���, if and only if any continuouslydifferentiable vector function ��·� = ���·� 1�� ��·� 2�� � � � � ��·� N��T �0��� → �N

such that for each i ∈ �, v�·� i�− ��·� i� reaches the maximum at u satisfies

max{1− �′�u� i�� sup

R∈��i�R� v� ���u�

}≥ 0� ∀i ∈ ��

(ii) A continuous vector function v�·� = �v�·� 1�� v�·� 2�� � � � � v�·� N��T �0��� → �N isa viscosity supersolution of (4.1) at u ∈ �0���, if and only if any continuouslydifferentiable vector function ��·� = ���·� 1�� ��·� 2�� � � � � ��·� N��T �0��� → �N

such that for each i ∈ �, v�·� i�− ��·� i� reaches the minimum at u satisfies

max{1− �′�u� i�� sup

R∈��i�R� v� ���u�

}≤ 0� ∀i ∈ ��

Proof. We only proof the statement (i), and (ii) can be proven similarly.First, we prove the sufficiency. For any continuously differentiable vector

function ��·� with v�u� = ��u� such that for each i ∈ �, v�·� i�− ��·� i� reaches themaximum at u, we have

max{1− �′�u� i�� sup

R∈��i�R� v� ���u�

}≥ 0� ∀i ∈ ��

Since ��y� i� ≥ v�y� i�, we have supR∈� �i�R� ���u� ≥ supR∈� �i�R� v� ���u�. Thus weget

max{1− �′�u� i�� sup

R∈��i�R� ���u�

}≥ 0� ∀i ∈ ��

Next, we are going to show the necessity. For any continuously differentiablevector function ��·� such that for each i ∈ �, v�·� i�− ��·� i� reaches the maximumat u, we define �n�y� i� for each i ∈ � as follows

�n�y� i� = �n�y� ���y� i�+ v�u� i�− ��u� i��+ �1− �n�y��v�y� i� for y ∈ �0����

where �n�·� is a smooth function satisfying

0 ≤ �n�·� ≤ 1�

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1090 Wei et al.

�n�y� = 1 for y ∈(u− 1

n� u+ 1

n

)�

�n�y� = 0 for y ∈ � \(u− 2

n� u+ 2

n

)�

Observe that �n�·� i� is continuously differentiable with �n�u� i� = v�u� i�� �′n�u� i� =

�′�u� i� and

v�y� i�− �n�y� i� = �n�y� �v�y� i�− ��y� i�− v�u� i�+ ��u� i��

≤ 0

= v�u� i�− �n�u� i��

Since v is a viscosity subsolution, we have

max{1− �′�u� i�� sup

R∈��i�R� �n��u�

}≥ 0� ∀i ∈ ��

Letting n → � and using the Dominated Convergence Theorem, we conclude that

max{1− �′�u� i�� sup

R∈��i�R� v� ���u�

}≥ 0� ∀i ∈ ��

Theorem 4.1. V�·� is a viscosity solution of (4.1).

Proof. Let us prove first that V is a viscosity supersolution. Let h > 0, T1 and S1be the time at which the first transition of the environment process �J�t��t≥0 occursand the first claim comes, respectively.

We consider the following admissible strategy �0: between the interval �0� T1 ∧S1 ∧ h�, the reinsurance police R�0 is any element in �, dividends are paid at ratel�0 ≥ 0, that is, L�0�t� = l�0 t, and thereafter, an optimal strategy is applied.

Given the initial surplus u and initial state i, denote ��t� = �ci�R�0�− l�0� t. By

total probability formula, we see that the expectation of the present value of alldividends until ruin is

I1 + I2 + I3 = e−��i+qi+��hV�u+ ��h�� i�

+∫ h

0�ie

−��i+qi+��s∫ u+��s�

0V�u+ ��s�− y� i�dFi�R

�0� y�ds

+∫ h

0qie

−��i+qi+��t∑k =i

qikqi

V�u+ ��t�� k�dt +��h�

≤ V�u� i��

where

I1 =∫ �

hqie

−qit∫ �

h�ie

−�is

[∫ h

0l�0e−�xdx + e−�hV�u+ ��h�� i�

]ds dt�

I2 =(∫ �

h�ie

−�is∫ h

0qie

−qit +∫ h

0�ie

−�is∫ s

0qie

−qit

)

×[∫ t

0l�0e−�xdx + e−�t

∑k =i

qikqi

V�u+ ��t�� k�

]dt ds�

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Optimal Reinsurance and Dividend Strategies 1091

I3 =(∫ �

hqie

−qit∫ h

0�ie

−�is +∫ h

0qie

−qit∫ t

0�ie

−�is

)

×[∫ s

0l�0e−�xdx + e−�s

∫ u+��s�

0V�u+ ��s�− y� i�dFi�R

�0� y�

]ds dt�

I�h� = l�0

��1− e−�h�e−��i+qi�h +

∫ h

0

l�0

��1− e−�s��ie

−��i+qi�sds

+∫ h

0

l�0

��1− e−�t�qie

−��i+qi�tdt�

Then we have

V�u+ ��h�� i�− V�u� i�

h− 1− e−��i+qi+��h

hV�u+ ��h�� i�

+ 1h

∫ h

0�ie

−��i+qi+��s∫ u+��s�

0V�u+ ��s�− y� i�dFi�R

�0� y�ds

+ 1h

∫ h

0qie

−��i+qi+��t∑k =i

qikqi

V�u+ ��t�� k�dt + 1hI�h� ≤ 0� (4.2)

Let ��·� = ���·� 1�� ��·� 2�� � � � � ��·� N��T �0��� → �N be a continuouslydifferentiable vector function such that for each i ∈ �, V�·� i�− ��·� i� reaches theminimum at u. From (4.2) we get

��u+ ��h�� i�− ��u� i�

h− 1− e−��i+qi+��h

hV�u+ ��h�� i�

+ 1h

∫ h

0�ie

−��i+qi+��s∫ u+��s�

0V�u+ ��s�− y� i�dFi�R

�0� y�ds

+ 1h

∫ h

0qie

−��i+qi+��t∑k =i

qikqi

V�u+ ��t�� k�dt + 1hI�h� ≤ 0�

Letting h → 0 in the above inequality gives

0 ≥ l�0�1− �′�u� i��+ ci�R�0��′�u� i�+ �i

∫ u

0V�u− y� i�dFi�R

�0� y�

− ��i + ��V�u� i�+∑k∈�

qikV�u� k��

Since this inequality holds for all l�0 ≥ 0, we have that �′�u� i� ≥ 1, and taking l�0 =0 we get �i�R

�0� V� ���u� ≤ 0, so

max{1− �′�u� i�� sup

R∈��i�R� V� ���u�

}≤ 0

and we have the result.It remains to prove that V is a viscosity subsolution at any u > 0. Arguing by

contradiction, we assume that V is not a viscosity subsolution of (4.1). Using thesame method of [4], we can find � > 0, h ∈ �0� u/2� and a continuously differentiable

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1092 Wei et al.

vector function ��·� = ���·� 1�� ��·� 2�� � � � � ��·� N��T with V�u� = ��u� such that forall i ∈ �

1− �′�y� i� ≤ 0 (4.3)

when y ∈ �0� u+ h�,

supR∈�

�i�R� ���y� ≤ −2�� (4.4)

when y ∈ �u− h� u+ h� and also

V�y� i� ≤ ��y� i�− 3� (4.5)

when y ∈ �0� u− h� ∪ �u+ h�.Since � is continuously differentiable we can find a positive constant C such

that for all i ∈ �

supR∈�

�i�R� ���y� ≤ C (4.6)

when y ∈ �0� u+ h�. Since the value function V is continuous, we can find � > 0small enough such that for all i ∈ �

� <

(�

2C∧ 1

4�

)(4.7)

and

V�y + ci0�� i� ≤ V�y� i�+ �/2� ��y + ci0�� i� ≥ ��y� i�− �/2 (4.8)

for all y ∈ �u− h− ci0�� u− h�.Let us take any admissible strategy � = �R��t�� L��t��, and define

� = inf�t > 0 U��t� ≥ u+ h�� � = inf�t > 0 U��t� ≤ u− h��

and �̂ = � ∧ ��+ ��. If ruin occurs between ��� �+ ��, we define U��t� = 0 for t > ��.It is obvious that U���� = u+ h, U���� ∈ �0� u− h+ ci0�� and U��t� ∈ �0� u+ h�

for t ∈ �0� �̂�. Thus, when U���̂� ∈ �0� u− h� ∪ �u+ h�, (4.5) yields

V�U���̂�� J��̂�� ≤ ��U���̂�� J��̂��− 2�� (4.9)

When U���̂� ∈ �u− h� u− h+ ci0��, we can write

U���̂� = y0 + ci0��

where y0 ∈ �u− h− ci0�� u− h�. Then from (4.5) and (4.8), inequality (4.9) stillholds.

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Optimal Reinsurance and Dividend Strategies 1093

From Lemma 3.1 we have

e−��̂� �U���̂�� J��̂��− � �U��0�� J�0��

=∫ �̂

0e−�s

∑k∈�

qJ�s�k��U��s�� k�ds +

∫ �̂

0e−�s

∑k∈�

��U��s−�� k�dm�s� k�

+∫ �̂

0e−�s

[−���U��s�� J�s��+ �′�U��s�� J�s��cJ�s��R��s��

]ds

−∫ �̂

0e−�s�′�U��s�� J�s��dL�c�s�−

∫ �̂

0

∫ L��s+�−L��s�

0e−�s�′�U��s�− y� J�s��dy ds

+∫ �̂

0

∫ �

0e−�s ���U��s−�− x� J�s��− ��U��s−�� J�s��� NJ�s��R

��s�� ds × dx�

(4.10)

From (4.3) we get

−∫ �̂

0e−�s�′�U��s�� J�s��dL�c�s�−

∫ �̂

0

∫ L��s+�−L��s�

0e−�s�′�U��s�− y� J�s��dy ds

≤ −∫ �̂

0e−�sdL�c�s�−

∫ �̂

0

∫ L��s+�−L��s�

0e−�sdy ds

= −∫ �̂

0e−�sdL��s��

Thus,

e−��̂� �U���̂�� J��̂��− � �U��0�� J�0��

≤∫ �̂

0e−�s�J�s��R

��s�� ���U��s��ds +∫ �̂

0e−�s

∑k∈�

��U��s−�� k�dm�s� k�

−∫ �̂

0e−�sdL��s�+M��̂�� (4.11)

where

M�t� =∫ t

0

∫ �

0e−�s ���U��s−�− x� J�s��− ��U��s−�� J�s��� NJ�s��R

��s�� ds × dx�

−∫ t

0

∫ �

0e−�s ���U��s−�− x� J�s��− ��U��s−�� J�s��� �J�s�dFJ�s��R

��s�� dx�ds

is a martingale.Using the inequalities (4.4), (4.6), and (4.19), we get

∫ �̂

0e−�s�J�s��R

��s�� ���U��s��ds

=∫ �∧�

0e−�s�J�s��R

��s�� ���U��s��ds +∫ �̂

�∧�e−�s�J�s��R

��s�� ���U��s��ds

≤ −2��∫ �∧�

0e−�sds + �

2� (4.12)

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Also from (4.19), we have

2��∫ �∧�

0e−�sds = 2��

∫ �̂

0e−�sds − 2��

∫ �̂

�∧�e−�sds

≥ 2��∫ �̂

0e−�sds − �

2� (4.13)

From (4.9), using (4.11), (4.12), and (4.13), it follows that

e−��̂V �U���̂�� J��̂�� ≤ (� �U��0�� J�0��− e−��̂2�

)+ (

e−��̂� �U���̂�� J��̂��− � �U��0�� J�0��)

≤ (� �U��0�� J�0��− e−��̂2�

)− 2��∫ �̂

0e−�sds + �+M��̂�

+∫ �̂

0e−�s

∑k∈�

��U��s−�� k�dm�s� k�−∫ �̂

0e−�sdL��s�� (4.14)

For all i ∈ �, since ��y� i� is continuously differentiable, we have ��y� i� isbounded for y ∈ �0� u+ h�. Consequently,

�u�i

[∫ �̂

0e−�s

∑k∈�

��U��s−�� k�dm�s� k�

]= 0�

Furthermore,

�u�i

[1− e−��̂

] = ��u�i

[∫ �̂

0e−�sds

]�

From (4.14) and the Dynamic Programming Principle (3.1), we have

V�u� i� = sup�∈�

�u�i

[∫ �̂

0e−�sdL��s�+ e−��̂V �U���̂�� J��̂��

]≤ ��u� i�− �

and this contradicts the assumption that V�u� = ��u�. �

The following proposition gives us uniqueness of the viscosity solution withboundary condition v�0� among all the function v that satisfy the followingconditions.

(C1) ] For all i ∈ �, v�·� i� �0��� → � is locally Lipschitz.(C2) ] For all i ∈ � and 0 ≤ u1 < u2, v�u2� i�− v�u1� i� ≥ u2 − u1.(C3) ] For For all i ∈ � and u1 ≥ 0, there exists a constant k > 0 such that v�u� i� ≤

u+ k.

Proposition 4.1. Assume

maxi∈�

∑n∈�

�qin� < �� (4.15)

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Optimal Reinsurance and Dividend Strategies 1095

and for all u > 0, v�u� and v�u� are viscosity subsolution and supersolution of (4.1),respectively; and both satisfying conditions (C1), (C2), and (C3). If for all i ∈ �,v�0� i� ≤ v�0� i� then for all i ∈ �, v�·� i� ≤ v�·� i� in �0���.

Proof. We consider vs�u� = sv�u� with s > 1. It is easy to verify that vs�u� is alsoa supersolution with ∀i ∈ �, v�0� i� ≤ vs�0� i�. In fact, if ��u� is a continuouslydifferentiable function such that for i ∈ � the minimum of vs�u� i�− ��u� i� isattained at a point u0 > 0, then

1− �′�u0� i� ≤ 1− s < 0 for all i ∈ �� (4.16)

Arguing by contradiction, assume that

= �i ∈ � ∃ui ≥ 0 such that v�ui� i� > v�ui� i�� = ∅�

For i ∈ , we can choose s0 such that v�ui� i� > vs0�ui� i�. From conditions (C2) and(C3)we obtain for i ∈ �

v�u� i�− vs0�u� i� ≤ k+ �1− s0�u�

Hence, we have

v�u� i�− vs0�u� i� ≤ 0 for u ≥ b and i ∈ �� (4.17)

where

b = k

s0 − 1> 0�

For i ∈ � let

Mi = supu≥0

(v�u� i�− vs0�u� i�

)

and

M = Mi0= max

i∈��Mi��

Obviously, i0 ∈ , and from (4.17) we have

0 < v�ui0� i0�− vs0�ui0

� i0� ≤ M = supu∈�0�b�

(v�u� i0�− vs0�u� i0�

)� (4.18)

And let u∗ = arg supu∈�0�b�(v�u� i0�− vs0�u� i0�

).

Since v�u� i0� and vs0�u� i0� satisfy (C1), there exist a constant m > 0 such that

v�u1� i0�− v�u2� i0�

u1 − u2

≤ m�vs0�u1� i0�− vs0�u2� i0�

u1 − u2

≤ m (4.19)

for 0 ≤ u2 ≤ u1 ≤ b.

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1096 Wei et al.

Define the set

= ��x� y� 0 ≤ x ≤ y ≤ b�

and for all a > 0 the function

�a�x� y� i0� = v�x� i0�− vs0�y� i0�−a

2�x − y�2 − 2m

a2�y − x�+ a� (4.20)

Let Ma = max �a�x� y� i0� and �xa� ya� = argmax �a�x� y� i0�. Obviously, ya ≥ xaand

Ma ≥ �a�u∗� u∗� i0� = M − 2m

a�

and from (4.18) we get Ma > 0 for a ≥ 4m/M and

lim infa→� Ma ≥ M� (4.21)

By the same argument of [4], we can show that there exist a0 > 0 large enoughsuch that for any a ≥ a0 we have that

�xa� ya� � ��

Here we omit the details of the proof.Let

��x� i0� = vs0�ya� i0�+a

2�x − ya�

2 + 2ma2�ya − x�+ a

and

��y� i0� = v�xa� i0�−a

2�xa − y�2 − 2m

a2�y − xa�+ a�

It is obvious that �′�xa� i0� = �′�ya� i0� and both of them are continuouslydifferentiable. Furthermore, the maximum of v�x� i0�− ��x� i0� is attained at xaand the minimum of vs0�y� i0�− ��y� i0� is attained at ya. Therefore, we have thefollowing inequalities

max{1− �′�xa� i0�� sup

R∈��i0

�R� v� ���xa�

}≥ 0 (4.22)

and

max{1− �′�ya� i�� sup

R∈��i0

�R� vs0� ���ya�

}≤ 0� (4.23)

From (4.16) and �′�xa� = �′�ya�, we have from (4.22),

supR∈�

�i0�R� v� ���xa� ≥ 0� (4.24)

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Optimal Reinsurance and Dividend Strategies 1097

Therefore, from (4.23) and (4.24) we get

�i0 + �

�i0

[v�xa� i0�− vs0�ya� i0�

]

≤ supR∈�

{∫ xa

0v�xa − x� i0�dFi0

�R� x�−∫ ya

0vs0�ya − x� i0�dFi0

�R� x�

}

+ 1�i0

∑n∈�

qi0�n�v�xa� n�− vs0�ya� n��� (4.25)

Using the inequality

�a�xa� xa� i0�+�a�ya� ya� i0� ≤ 2�a�xa� ya� i0��

we obtain that

a�xa − ya�2 ≤ v�xa� i0�− v�ya� i0�+ vs0�xa� i0�− vs0�ya� i0�+ 4m�ya − xa��

then from (4.19), we have

a�xa − ya�2 ≤ 6m�xa − ya�� (4.26)

We can find a sequence an → � such that �xan� yan� → �x∗� y∗� ∈ . From (4.14), weget that

� xan − yan �≤ 6m/an

and this implies x∗ = y∗. From (4.25) we have

�i0 + �

�i0

[v�x∗� i0�− vs0�x

∗� i0�]

≤ supR∈�

{∫ x∗

0v�x∗ − x� i0�dFi0

�R� x�−∫ x∗

0vs0�x

∗ − x� i0�dFi0�R� x�

}

+ 1�i0

∑n∈�

qi0�n�v�x∗� n�− vs0�x

∗� n��� (4.27)

Since

∑n∈�

qi0�n�v�x∗� n�− vs0�x

∗� n�� ≤ ∑n∈�

� qi0�n � �v�x∗� n�− vs0�x∗� n��

≤ M∑n∈�

� qi0�n ��

the inequality (4.27) yields

(�i0 + �

) [v�x∗� i0�− vs0�x

∗� i0�] ≤ M

(�i0 +

∑n∈�

�qi0�n�)� (4.28)

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From (4.26) we get that

limn→� an�xan − yan�

2 = 0�

thus from (4.15), (4.20), (4.21), and (4.28), we obtain

M ≤ lim infa→� Ma ≤ lim

n→�Man

= v�x∗� i0�− vs0�x∗� i0�

≤ �i0 +∑

n∈� �qi0�n��i0 + �

M < M

which implies a contradiction. �

Note that the value function V satisfies the conditions (C1), (C2), and (C3). Thefollowing corollary is a direct consequence.

Corollary 4.1. There is at most one viscosity solution of (4.1) with boundary conditionV�0� = (

v01� v02� � � � � v

0N

)Tamong all the functions that satisfy the conditions (C1), (C2),

and (C3).

5. Verification Theorem

In this section, we will prove a verification theorem; namely, if for some admissiblestrategy �, V� is a supersolution, then it is the value function V and � is the optimalstrategy. Using the same arguments of the proof of Lemma A.2 in [4], we can showthe next lemma.

Lemma 5.1. Let v be an supersolution of (4.1) with absolution continuous elementssatisfying the condition (C3). We can find a sequence of positive function vn such thatfor each i ∈ � and y ∈ �0���

(i) vn�y� i� is continuously differentiable.(ii) vn�y� i� satisfies the condition (C3).(iii) 1 ≤ v′n�y� i� ≤ 1

ci���i + ��vn�u� i�−

∑k∈� qikvn�u� k��.

(iv) vn�y� i� converges to v�y� i� uniformly on compact set and v′n�y� i� converges tov′�y� i� a.e.

Theorem 5.1. Let v�u� be an supersolution of (4.1) with absolution continuouselements satisfying the condition (C3), then for each i ∈ �, v�u� i� ≥ V�u� i�.

Proof. Consider an admissible strategy � = �R��t�� L��t��. Since the functionsvn�y� i� defined in Lemma 5.1 are continuously differentiable, by Lemma 3.1 we have

e−��t∧���vn �U��t ∧ ���� J�t ∧ ����− vn �U

��0�� J�0��

=∫ t∧��

0e−�s

∑k∈�

qJ�s�kvn�U��s�� k�ds +

∫ t∧��

0e−�s

∑k∈�

vn�U��s−�� k�dm�s� k�

+∫ t∧��

0e−�s

[−�vn�U��s�� J�s��+ v′n�U

��s�� J�s��cJ�s��R��s��

]ds

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Optimal Reinsurance and Dividend Strategies 1099

−∫ t∧��

0e−�sv′n�U

��s�� J�s��dL�c�s�−∫ t∧��

0

∫ L��s+�−L��s�

0e−�sv′n�U

��s�− y� J�s��dyds

+∫ t∧��

0

∫ �

0e−�s �vn�U

��s−�− x� J�s��− vn�U��s−�� J�s��� NJ�s��R

��s�� ds × dx��

From Lemma 5.1, v′n�y� i� ≥ 1, similar to (4.11) we have

e−��t∧���vn �U��t ∧ ���� J�t ∧ ����− vn �U

��0�� J�0��

≤∫ t∧��

0e−�s�J�s��R

��s�� vn��U��s��ds +

∫ t∧��

0e−�s

∑k∈�

vn�U��s−�� k�dm�s� k�

−∫ t∧��

0e−�sdL��s�+Mn�t ∧ ���� (5.1)

where

Mn�t� =∫ t

0

∫ �

0e−�s �vn�U

��s−�− x� J�s��− vn�U��s−�� J�s��� NJ�s��R

��s�� ds × dx�

−∫ t

0

∫ �

0e−�s �vn�U

��s−�− x� J�s��− vn�U��s−�� J�s��� �J�s�dFJ�s��R

��s�� dx�ds

is a martingale.Using the Monotone Convergence Theorem, we get

limt→��u�i

[∫ t∧��

0e−�sdL��s�

]= �u�i

[∫ ��

0e−�sdL��s�

]� (5.2)

From Lemma 5.1(iii), we have

−��J�s� + ��vn�U��s�� J�s�� ≤ �J�s��R

��s�� vn��U��s�� ≤ �J�s�vn�U

��s�� J�s���

From Lemma 5.1(ii) and U��s� ≤ u+ ci0s, we get that

vn�U��s�� J�s�� ≤ k0 + U��s� ≤ k0 + u+ ci0s (5.3)

for some k0. So, using the Bounded Convergence Theorem, we obtain

limt→��u�i

[∫ t∧��

0e−�s�J�s��R

��s�� vn��U��s��ds

]

= �u�i

[∫ ��

0e−�s�J�s��R

��s�� vn��U��s��ds

]� (5.4)

From (5.1), (5.2), and (5.4), we get

limt→��u�i

[e−��t∧���vn �U

��t ∧ ���� J�t ∧ ����]− vn�u� i�

≤ �u�i

[∫ ��

0e−�s�J�s��R

��s�� vn��U��s��ds

]− V��u� i�� (5.5)

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1100 Wei et al.

By Lemma 5.1(iv) and Dominated Convergence Theorem, we get

limn→��u�i

[∫ ��

0e−�s�J�s��R

��s�� vn��U��s��ds

]

= �u�i

[∫ ��

0e−�s�J�s��R

��s�� v��U��s��ds

]≤ 0� (5.6)

Since

limn→� lim

t→��u�i

[e−��t∧���vn �U

��t ∧ ���� J�t ∧ ����] ≥ 0�

from (5.5) and (5.6), we have

v�u� i� = limn→� vn�u� i� ≥ V��u� i�

and the result holds. �

The following characterization of value function V is directly from Corollary 4.1and Theorem 5.1.

Corollary 5.1. The value function V can be characterized as the unique viscositysolution of (4.1) satisfying conditions (C1), (C2), and (C3); and for each i ∈ �, theboundary condition

V�0� i� = inf �v�0� i� v is viscosity solution of (4.1) satisfying condition (C3)�

holds.

Theorem 5.2. Let � be an admissible strategy such that V� is an absolutely continuoussupersolution of (4.1), then V = V�.

Proof. Since for all i ∈ � and u ≥ 0, V��u� i� ≤ V�u� i�, and V satisfies the condition(C3), then V� also satisfies the condition (C3) and the result follows fromTheorem 5.1. �

Up to this point we do not know if there exists an admissible strategy � whichsatisfies the conditions of Theorem 5.2. In the next section, we are going to presentan example in which such a strategy exists.

6. Case Study

In this section we consider the special case with � = �1� 2�, no reinsurance andexponential claim density, that is, fi�x� = �ie

−�ix, i = 1� 2. We consider the followingstrategy � which is called modulated barrier strategy in [15]. For two constants bi >0, i = 1� 2, when the state is i and the surplus u < bi, pay out no dividend; when the

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Optimal Reinsurance and Dividend Strategies 1101

state is i and the surplus u ≥ bi, pay out everything above bi as dividends. By thestandard method (see [12]), for i = 1� 2, we have

0 = ciV

′��u� i�+ �i�i

∫ u

0 V��u− y� i�e−�iydy

− ��i + qi + ��V��u� i�+ qiV��u� 3− i�� if 0 ≤ u < bi�

V��u� i� = u− bi + V��bi� i�� if u ≥ bi�

(6.1)

where

V��bi� i� =ci

�i + qi + �+ �i

�i + qi + ��i

∫ bi

0V��u− y� i�e−�iydy

+ qi�i + qi + �

V��bi� 3− i�� (6.2)

In this section, we consider b1 ≤ b2; the case b1 > b2 may be treated similarly.Here we have to consider three cases: u ∈ �0� b1�, u ∈ �b1� b2� and u ∈ �b2���.

If u ∈ �0� b1�, system (6.1) gives the following system of integro-differentialequations:

−c1V′��u� 1� = �1�1e

−�1u∫ u

0V��y� 1�e

�1ydy

− ��1 + q1 + ��V��u� 1�+ q1V��u� 2��

−c2V′��u� 2� = �2�2e

−�2u∫ u

0V��y� 2�e

�2ydy

− ��2 + q2 + ��V��u� 2�+ q2V��u� 1��

(6.3)

The right-hand sides of the both equations are differentiable. Therefore,

c1V′′� �u� 1� = �1�

21e

−�1u∫ u

0V��y� 1�e

�1ydy − �1�1V��u� 1�

+ ��1 + q1 + ��V ′��u� 1�− q1V

′��u� 2��

c2V′′� �u� 2� = �2�

22e

−�2u∫ u

0V��y� 2�e

�2ydy − �2�2V��u� 2�

+ ��2 + q2 + ��V ′��u� 1�− q2V

′��u� 2��

Plugging in the original equations yields

0 = c1V′′� �u� 1�− ��1 + q1 + �− c1�1�V

′��u� 1�

−�1�q1 + ��V��u� 1�+ �1q1V��u� 2�+ q1V′��u� 2��

0 = c2V′′� �u� 2�− ��2 + q2 + �− c2�2�V

′��u� 2�

−�2�q2 + ��V��u� 2�+ �2q2V��u� 1�+ q2V′��u� 1��

(6.4)

We conjecture that the solution is given by{V��u� 1� = A11e

�1�u−b1� + A12e�2�u−b1� + A13e

�3�u−b1� + A14e�4�u−b1��

V��u� 2� = A21e�1�u−b1� + A22e

�2�u−b1� + A23e�3�u−b1� + A24e

�4�u−b1��(6.5)

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1102 Wei et al.

By plugging (6.5) in (6.4) and comparing the coefficients, we know that if we canshow the system of equations

{c1�

2 − ��1 + q1 + �− c1�1��− �1�q1 + ��+ �1q1� + q1�� = 0�

c2�2 − ��2 + q2 + �− c2�2��− �2�q2 + ��+ �2q2/� + q2�/� = 0

(6.6)

has four roots ��� ��, then we can determine ��j� A2j/A1j�, j = 1� 2� 3� 4.

Lemma 6.1. The system of equations (6.6) has four real roots ��j� �j�, j = 1� 2� 3� 4,and �1 < �2 < 0 < �3 < �4.

Proof. Let

gi��� = ci�2 − ��i + qi + �− ci�i��− �i�qi + ��� i = 1� 2

and

G��� = g1���g2���− q1q2��1 + ����2 + ���

Obviously, for i = 1� 2, gi��� = 0 has two roots i1 and i2, such that i1 < 0 < i2.Since G��� = G�−�� = � and G�0� > 0, the proof is then completed by showingthat there are 1 and 2, such that 1 < 0 < 2 and G�i� < 0, i = 1� 2. First, we have12 > 0 and G�12� < 0. Second, for i = 1� 2, gi�−�i� > 0, then −�i < i1 < 0. Thus,we have 1 = max�11� 21�. In fact, G�1� = −q1q2��1 + 1���2 + 1� < 0. �

Hence, we identify ��j� A2j/A1j�, j = 1� 2� 3� 4 as the four roots of (6.6). Hencefor j = 1� 2� 3� 4, we have

A2j = − g1��j�

�1q1 + q1�jA1j = −�2q2 + q2�j

g2��j�A1j � (6.7)

Moreover, plugging (6.5) in (6.3) yields

A11

�1 + �1e−�1b1 + A12

�2 + �1e−�2b1 + A13

�3 + �1e−�3b1 + A14

�4 + �1e−�4b1 = 0

A21

�1 + �2e−�1b1 + A22

�2 + �2e−�2b1 + A23

�3 + �2e−�3b1 + A24

�4 + �2e−�4b1 = 0

(6.8)

If u ∈ �b1� b2�, system (6.1) gives the following system of integro-differentialequations:

V��u� 1� = u− b1 + V��b1� 1��

−c2V′��u� 2� = �2�2e

−�2u∫ u

0V��y� 2�e

�2ydy − ��2 + q2 + ��V��u� 2�+ q2V��u� 1��

(6.9)

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Optimal Reinsurance and Dividend Strategies 1103

V��b1� 1� can be calculated by (6.5) and similar to (6.4), we have

0 = c2V′′� �u� 2�− ��2 + q2 + �− c2�2�V

′��u� 2�− �2�q2 + ��V��u� 2�

+ �2q2�u− b1 + V��b1� 1��+ q2� (6.10)

We conjecture that the solution of V��u� 2� is given by

V��u� 2� = B1e�̃1�u−b2� + B2e

�̃2�u−b2� + �1u+ �2� (6.11)

Plugging (6.11) in (6.10) and comparing the coefficients show that �̃i, i = 1� 2 are theroot of the equation

c2�̃2 − ��2 + q2 + �− c2�2��̃− �2�q2 + �� = 0� (6.12)

and{−�q2 + ���1 + �2q2 = 0�

−��2 + q2 + �− c2�2��1 − �2�q2 + ���2 + �2q2�−b1 + V��b1� 1��+ q2 = 0(6.13)

Moreover, plugging the second equation of (6.5) and (6.11) into (6.9) yields

4∑i=1

A2i

�i + �2�1− e��i−�2�b1 �−

2∑i=1

Bi

�̃i + �2e�̃i�b1−b2� − �1b1 + �2 − 1

�2= 0 (6.14)

Finally, if u ∈ �b2���, system (6.1) gives the following system of integro-differential equations:

{V��u� 1� = u− b1 + V��b1� 1��

V��u� 2� = u− b2 + V��b2� 2��(6.15)

Here, we want to find a continuous version of V��u� i�. Thus, by letting u →bi in (6.1) and noting that V ′′

� �u� 2� exists at b1, we have the following smooth-fitcondition:

V��b1−� 2� = V��b1+� 2��

V ′��b1−� i� = V ′

��b1+� i�� for i = 1� 2�

V ′��b2−� 2� = 1�

V ′′� �b1−� 2� = V ′′

� �b1+� 2��

(6.16)

The solution of the system of Equations (6.7), (6.8), (6.13), (6.14), and (6.16) willgive us the values for b1 and b2, and also the values for Aij� Bi� �i� i = 1� 2� andj = 1� 2� 3� 4.

We can verify that V��u� i� is a supersolution of (4.1) by the same method asin Section 4 in [25]. Although we get a modulated-barrier strategy in this specialcase, a modulated-barrier strategy maybe not the optimal strategy in general cases.Azcue and Muler [4] considers only one regime shows that the optimal strategy is aband strategy in general cases.

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1104 Wei et al.

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Optimal Reinsurance and Dividend Strategies 1105

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