Optimal risk and dividend strategies with transaction costs and terminal value

Economic Modelling 54 (2016) 522–536

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Optimal risk and dividend strategies with transaction costs and terminal value Gongpin Cheng a,⁎, Yongxia Zhao b a b

School of Statistics, Faculty of Economics and Management, East China Normal University, Shanghai 200241, China School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

a r t i c l e

i n f o

Article history: Accepted 14 January 2016 Available online xxxx Keywords: Dividend Refinancing Expensive reinsurance Transaction cost Terminal value Variance premium principle

a b s t r a c t This paper assumes that an insurance company can control the surplus by paying dividends, raising money and buying proportional reinsurance dynamically. The reinsurance premium is assumed to be calculated via the variance premium principle. Under the objective of maximizing the insurance company's value, we identify the optimal joint strategies and consider the effects of transaction costs and arbitrary terminal value at bankruptcy. From the results, we see that refinancing should be considered if and only if the terminal value and the transaction costs are not too high and the company is on the brink of bankruptcy, and the amount of each capital injection remains constant; the optimal ceded proportion of risk decreases with the current surplus and remains constant when the surplus exceeds some constant level; the optimal dividend distribution policy is of barrier type when the dividend rate is unrestricted or is of threshold type when the dividend rate is bounded, respectively. In particular, the insurance company should declare bankruptcy as soon as possible if the terminal value is high enough. © 2016 Elsevier B.V. All rights reserved.

1. Introduction From the viewpoint of corporate finance, a company's value can be measured by the expected discounted sum of dividend payments until the time of bankruptcy. Determining the optimal dividend strategy for maximizing the company's value is a long standing problem in mathematical insurance. Its origin can be traced as early as the work of De Finetti (1957). Since then much research on this topic has been carried out in varieties of risk models. The question of when to declare dividends and how much of them should be distributed is tricky. Research has shown that, when the dividend rate is unrestricted, the barrier strategy is often optimal. That is, no dividend is paid while the surplus is below a barrier b ≥ 0 and the overflow with respect to the barrier b is paid out as dividends immediately. Barrier strategy is practical and thus has been widely studied by Asmussen and Taksar (1997), Hϕgaard and Taksar (2004), Lϕkka and Zervos (2008), Kulenko and Schmidli (2008), Belhaj (2010), Yao et al. (2011), Hunting and Paulsen (2013), among others. However, when the considered dividend strategies are restricted to those of bounded dividend rates, the threshold strategy is often optimal. That is, dividends are paid at the maximum admissible rate as soon as the surplus exceeds a certain threshold u ≥ 0. Some literature on the threshold dividend strategy includes Asmussen and Taksar (1997), Gerber and Shiu (2006), Yao et al. (2014), Zhu (2015) and so on.

⁎ Corresponding author. E-mail addresses: [email protected] (G. Cheng), [email protected] (Y. Zhao).

http://dx.doi.org/10.1016/j.econmod.2016.01.009 0264-9993/© 2016 Elsevier B.V. All rights reserved.

It is well known that reinsurance plays an important role in both the theory and practice of insurance risk modeling, by which an insurer can transfer the risks to a second insurance carrier, namely, a reinsurer. A reinsurance contract is said to be “cheap” if the cedent pays the same fraction of the premium as the reinsured. While it is said to be “expensive” if the cedent pays a larger fraction of the premium than the fraction to be reinsured. The excess can be viewed as the transaction cost for a reinsurance contract. One of the typical reinsurance contracts is proportional reinsurance, under which the reinsurer takes a stated percentage share of each policy that an insurer issues. That is to say, the reinsurer will receive that stated percentage of the premiums and accordingly pay the stated percentage of claims. In addition, when the proportional reinsurance is taken as a risk control, the expectation premium principle is commonly used as the reinsurance premium principle due to its simplicity and popularity in practice. Although the variance principle is another important premium principle, few papers consider using it for risk control in a dynamic setting. Generally speaking, the expectation premium principle is commonly used in life insurance which has the stable and smooth claim frequency and claim sizes, while the variance premium principle is extensively used in property insurance. Both dividend and reinsurance are important issues in modeling insurance risk. To maximize the company's value, the insurer needs to balance risk control and dividend payout in terms of reinsurance and dividend distribution policies. Recently, some attention has been paid to the combined optimal dividend and reinsurance problem. As an extension of the classical dividend problem, it assumes that an insurer can control the dividend stream and risk exposure in terms of reinsurance. In the recent past, there have been many articles

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

published on this problem under the expectation premium principle, for example, Taksar and Zhou (1998), Hϕgaard and Taksar (1999), Choulli et al. (2003), Cadenillas et al. (2006), Meng and Siu (2011) and so on. Only a few papers investigated this problem under the variance premium principle, see Zhou and Yuen (2012), Yao et al. (2014). Notably, although there are fruitful research results on optimal dividend and reinsurance strategies, very little work considers the problem under a terminal value at bankruptcy, say P. The terminal value can be viewed as the salvage value for P ≥ 0 and the penalty amount for P b 0. This problem was firstly brought out in Taksar (2000), in which the company's value was defined as the expected discounted total dividends until the time of bankruptcy and the expected discounted terminal value at bankruptcy. Under the objective of maximizing the company's value, they obtained optimal dividend and reinsurance strategies by using some techniques in stochastic control theory. But the “expensive” reinsurance and the negative terminal value were not taken into account there. Liang and Young (2012) extended this problem by assuming the reinsurance was “expensive” and allowing for an arbitrary terminal value. They obtained the explicit solutions for the optimal dividend and reinsurance strategies by employing the Legendre transform. Other literature on this issue includes Taksar and Hunderup (2007), Xu and Zhou (2012) and Yao et al. (2014). In addition, the literature mentioned above did not consider the possibility of refinancing. As we know, when an insurance company encounters financial difficulty, it can continue the business by injecting capital. Of course, it requires financing costs, such as the proportional and fixed transaction costs generated by the advisory, consulting and issuance of securities, etc. Up to now, the combined dividend, refinancing and reinsurance problem with transaction costs has been studied extensively. The company's value was usually measured by the expected discounted total dividends minus the expected discounted costs of refinancing until the time of bankruptcy. To maximize the company's value, the insurance company must seek optimal dividend, refinancing and reinsurance strategies. For example, He and Liang (2009) and Barth and Moreno-Bromberg (2014) solved the optimal problem under the expectation premium principle. They also considered the effects of the fixed and proportional transaction costs in refinancing process. Peng et al. (2012) and Guan and Liang (2014) continued to investigate this problem under the assumption of “expensive” reinsurance. However, Zhou and Yuen (2012) solved the problem under the assumption of variance premium principle. The proportional cost and the “cheap” reinsurance were considered in the risk model. Yao et al. (2014) further extended the problem by allowing for the non-negative terminal value and the fixed transaction cost. They first focused on the combined optimization problem of dividend, refinancing and reinsurance with non-negative terminal value. They redefined the company's value as the expected sum of the discounted terminal value and the discounted dividends less the expected discounted costs of refinancing until the time of bankruptcy. Under the assumption of “cheap” proportional reinsurance, they obtained the explicit solutions of optimal strategies in both cases with unrestricted and restricted dividend rates and thus analyzed the effects of proportional and fixed transaction costs. As far as we know, with the exception of Yao et al. (2014), very little work has considered the combined optimal dividend, reinsurance and refinancing strategies with non-zero liquidation value. From the discussed literature, we can see that either the barrier dividend strategy or the threshold dividend strategy is often optimal, depending on whether there exist restrictions on dividend rates; the insurer would buy less reinsurance when the surplus increases; he may refinance when and only when the company is on the brink of bankruptcy and the size of each capital injection keeps constant. The decision to refinance or not depends on the relationships among the model's parameters. It is worthwhile to note that sometimes the transaction cost for reinsurance contract and negative terminal value at bankruptcy are unavoidable. So we need further research on the optimization problem in the case of “expensive” reinsurance and arbitrary terminal value.

523

Inspired by the above references, we extend the risk model in Yao et al. (2014) by including “expensive” reinsurance and an arbitrary terminal value in this paper. To maximize the insurance company's value, we seek the optimal dividend, refinancing and proportional reinsurance strategies. We solve the problem by using some techniques beyond the Legendre transform in Liang and Young (2012). The explicit solutions are given in both cases with unrestricted and restricted dividend rates, and the effects of transaction costs and terminal value P ϵ ℝ are analyzed. The rest of this paper is organized as follows. In Section 2, we use a diffusion approximation of the Cramér–Lundberg model with reinsurance to formulate the optimization problem for a controlled diffusion model with dividend, refinancing and “expensive” proportional reinsurance policies. In Section 3, we first consider two suboptimal problems when the dividend rate is unrestricted. Then we identify the value function and the optimal strategy with the corresponding solution in either category of suboptimal problems, depending on the relationships among the coefficients. In Section 4, we solve the problem when the dividend rate is bounded in a similar way. Finally, we conclude the study in Section 5. 2. Model formulation and the optimal control problem Let (Ω, F , { F t}t ≥ 0, P) be a probability space, on which all stochastic quantities in this paper are well defined. Here { F t}t ≥ 0 is a filtration, which satisfies the usual conditions. In mathematical insurance the surplus of an insurance portfolio is usually described in terms of the Cramér–Lunberg model process {Zt}t≥0 satisfying Nt X Z t ¼ x þ ct− Y n; n¼1

where Z0 = x is the initial surplus, c N 0 is the premium rate, {Nt}t≥0 is a Poisson process with constant intensity λ, random variables Yn ’ s are positive i.i.d. claims with common finite mean μ1 N 0 and finite second moment μ 22 N 0. Under the variance premium principle, it has c¼E

N1 X

! Yn

þ θ1 D

n¼1

N1 X

! Yn

  ¼ λ μ 1 þ θ1 μ 22 ;

ð2:1Þ

n¼1

where θ1 N 0 is a loading associated with the variance. E and D stand for expectation and variance, respectively. Suppose the insurer purchases a proportional reinsurance contract with ceded proportion a ∈ [0, 1]. That is to say, for each claim of size Yi, the insurer covers (1 − a)Yi and the reinsurer covers the rest aYi. Then the total ceded risks up to time t N

t aY n and the aggregate reinsurance premium under are given by ∑n¼1 the variance principle is

ca t :¼ E

Nt X

! aY n

þ θ2 D

n¼1

Nt X

! aY n

  ¼ λ aμ 1 þ θ2 a2 μ 22 t;

ð2:2Þ

n¼1

where ca is the rate of premiums and θ2 ∈ (θ1, ∞) is a loading associated with the variance of ceded risks. Here, the reinsurance is “expensive” due to the condition θ2 N θ1. Then the surplus process in the presence of “expensive” proportional reinsurance can be written as Z at ¼ x þ ðc−ca Þt−

Nt X ð1−aÞY n ;

ð2:3Þ

n¼1

with Z a0 = x. We approximate Eq. (2.3) by a pure diffusion model {X at , t ≥ 0} with the same drift and volatility. Specifically, Xat satisfies the following stochastic integral equation Z X at ¼ x þ

t 0

with X a0 = x.

  λμ 22 θ1 −θ2 a2 ds þ

Z t pffiffiffi λμ 2 ð1−aÞdBs ; 0

ð2:4Þ

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G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

Suppose that the ceded proportion a ∈ [0, 1] can be adjusted dynamically, we use the process {at}t ≥ 0 to describe a proportional reinsurance strategy. Furthermore, the insurer can control the surplus by paying dividends to shareholders and raising money from market. Let Lt denote the total amount of dividends paid from time 0 to t. ∞ Let Gt ¼ ∑n¼1 Ifτn ≤ tg ηn denote the total amount of capital raised by issuing equities from time 0 to t, where {τi, i = 1, 2, ⋯} denote the time points when the equity is issued and {ηi, i = 1, 2, ⋯} denote the amounts of equity issued. Then the surplus process controlled by the strategy π = (aπ,Lπ, G π) follows that X πt ¼ x þ þ

Z

t 0

∞ X

Z t pffiffiffiffi h  2 i   ds þ λμ 22 θ1 −θ2 aπs λμ 2 1−aπs dBs −Lπt 0

I fτπn ≤ tg ηπn ;

ð2:5Þ

n¼1

For some reasons such as safety, the dividend strategy is usually restricted in practice. Similar to Yao et al. (2014), we study a classic case in mathematical insurance when the dividend rate lt at time t is bounded by some dividend ceiling M N 0. Thereafter, the cumulated t

dividends up to time t can be written as Lt ¼ ∫ 0 ls ds. We thus need to redefine the admissible strategies. Definition 2.2. A strategy π ¼ ðaπ ; Lπ ; Gπ Þ is said to be admissible if (i) The ceded proportion aπ ¼ faπt gt ≥ 0 is an F Bt -adapted process with 0 ≤aπt ≤1 for all t ≥ 0. (ii) Lπ ¼ fLπt gt ≥ 0 is an increasing and F Bt -adapted càdlàg process and t π satisfies that Lπ0− ¼ 0 and Lπt ¼ ∫ 0 ls ds. π (iii) fτn g is a sequence of stopping times w.r.t. F Bt and 0 ≤ τ π1 b ⋯ b τπn b ⋯, a.s. (iv) ηπn ≥0; n ¼ 1; 2; ⋯ is measurable w.r.t. F Bτπ . n

X π0 = x.

π

π

π

with The insurer selects a strategy π = (a , L , G ) at any time t based on information available up to and including time t, say F Bt . Then we give the following definition of admissible policy which the company can select. Definition 2.1. A strategy π= (aπ, Lπ, Gπ) is said to be admissible if (i) The ceded proportion aπ = {aπt }t≥0 is an F Bt -adapted process with 0 ≤aπt ≤1 for all t ≥ 0. (ii) Lπ = {Lπt }t≥0 is an increasing and F Bt -adapted càdlàg process with L π0− = 0 and satisfies that ΔLπt =Lπt − Lπt− ≤X πt− for all t ≥ 0. (iii) {τπn} is a sequence of stopping times w.r.t. F Bt and 0 ≤ τπ1 b ⋯ b τπn b ⋯, a.s. (iv) ηπn ≥ 0 ,n = 1 , 2 , ⋯ is measurable w.r.t. F τBnπ. (v) Pð lim τ πn b tÞ ¼ 0; ∀t N 0:

(v) Pð lim τπn b tÞ ¼ 0; ∀t N 0: n→∞

The class of admissible strategies is denoted by Π. For each strategy π ∈ Π, the bankruptcy time is defined as T π ¼ infft : X πt b 0g, which is the first time that the surplus X πt becomes negative. Problem 2.2. When a ceiling M N 0 is imposed on the dividend rate, we measure the value of the company associated with π ∈ Π by the following performance function: Z V ðx; πÞ ¼ Ex β1

Tπ 0

! ∞   X π π π e−δs ls dsþPe−δT − e−δτn β2 ηπn þ K I τπ ≤ T π  : n¼1

ð2:8Þ

n→∞

Condition (ii) implies the total amount of dividends is less than the surplus available at that time. Condition (v) means that the issuance of equities may not occur infinitely in a finite time interval. We write Π for the space of these admissible strategies. For each strategy π ϵ Π, the time of bankruptcy is defined as T π = inf {t : Xπt b 0}, which is the first time that the surplus X πt becomes negative. Once bankruptcy occurs, the insurer has to claim a terminal value P ϵ ℝ. Then we are interested in the following problem. Problem 2.1. We measure the company's value associated with strategy π ϵ Π using the following performance function: Z U ðx; πÞ ¼ Ex β1

T

π

0

π

∞ X

e−δs dLπs þ Pe−δT −

n¼1

!

 π e−δτn β2 ηπn þ K I fτπ ≤ T π g ; n

ð2:6Þ which is the expected sum of the discounted terminal value and the discounted dividends less the expected discounted costs of refinancing until the time of bankruptcy. Ex denotes the expectation conditional on X π0 = x and δ N 0 is the discount rate. In the dividend distribution process, the quantity 1 − β1 ∈ (0, 1) denotes the tax rate at which the dividends are taxed. Thus, if the amount l of liquid surplus is paid out as dividends, the net amount of money that the shareholders receive is β1l. For each capital injection, β2 N 1 denotes the proportional cost factor and K N 0 is a fixed set-up cost, which is independent of the amount of refinancing. So if the amount of refinancing is η N 0, then the total amount of refinancing plus transaction costs is β2η + K. Then the objective of the company is to find the value function U ðxÞ ¼ max U ðx; πÞ π∈Π

and associated optimal strategy π⁎ ∈ Π, such that U(x) = U(x;π⁎).

ð2:7Þ

n

Correspondingly, we aim at finding the value function V ðxÞ ¼ max V ðx; πÞ;

ð2:9Þ

π∈Π





and the associated optimal strategy π ∈ Π such that VðxÞ ¼ Vðx; π Þ. Note that V(x) is bounded by β1δM. 3. Unrestricted dividends In this section, we focus on Problem 2.1, when there is no dividend rate ceiling. Suppose that u(x) : [0 , ∞ ) ↦ ℝ is a candidate function for the value function U(x). Let M denote the refinancing operator defined by M u(x) = supy ≥ 0{u(x + y) − β2y − K}, which represents the value of the strategy that consists in choosing the best immediate capital injection. Another notation used in this paper is the differential operator A a , denoted by A a uðxÞ ¼ 12 ð1−aÞ2 λμ 22 u00 ðxÞþ ðθ1 −θ2 a2 Þλμ 22 u0 ðxÞ−δuðxÞ: Suppose that u(x) is sufficiently smooth. Then, if the surplus process starts at x ≥ 0 and follows an optimal strategy, the associated performance function is u(x). On the other hand, if the surplus process starts at x, selects the best immediate capital injection and then follows an optimal strategy, then the performance function associated with this second strategy is M u(x). Because of the optimality of the value function, we have u(x) ≥ M u(x), with equality when it is optimal to inject capital. In fact, because of the time value of money, the insurer should postpone capital injection as long as possible, i.e., equity issuances may happen at the moments when and only when the surplus process reaches 0. Therefore u(x) N M u(x) holds strictly for all x N 0. In the continuation region, that is, when the insurer does not take any actions, we must have max0 ≤ a ≤ 1 f A a uðxÞg ¼ 0. In the dividend region, u′(x) = β1 holds. As the insurance company is on the brink of bankruptcy, the optimal strategy should either allow the surplus process to hit (−∞,0) by injecting no capital, which corresponds to the boundary condition

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

u(0) = P and M u(0)≤ u(0), or keep the surplus process in the interval [0 , ∞ ) by injecting capital, which corresponds to the boundary conditions u(0) ≥ P and M u(0) = u(0). Using stochastic control theory, see Fleming and Soner (1993), we characterize the value function by the following HJB (Hamilton–Jacobi–Bellman) equations:

max max f A a uðxÞg; β 1 −u0 ðxÞ; M uðxÞ−uðxÞ ¼ 0;

ð3:1Þ

maxf M uð0Þ−uð0Þ; P−uð0Þg ¼ 0:

ð3:2Þ

0≤a≤1

525

The function χ(z) : = 2θ2(α − z)(z − ρ) is decreasing on [0,a] with χ(0) = δ and χ(α) = 0. Assuming that a(0) : = a0 ∈ (0, α), we define a new function of x as Z

a0

Q ðxÞ ¼

1−z dz; 0 ≤ x ≤ a0 : 2θ2 ða−zÞðz−pÞ

x

Since the integrand of Eq. (3.10) is positive, Q(x) is strictly decreasing on [0,a0], which implies that the inverse Q−1(x) of the function Q(x) exists on [0,a0]. Then, together with Q(a0) = 0, we get aðxÞ ¼ Q −1 ðxÞ ∈ ½0; a0 ;

Theorem 3.1. Let u(x) be a twice continuously differentiable, increasing and concave solution of Eqs. (3.1) and (3.2), and u′(x) is bounded, then u(x) ≥ U(x; π) holds for each admissible strategy π ϵ Π. So u(x) = U(x) holds. Moreover, if there exists some strategy π⁎ = (aπ⁎, Lπ⁎, Gπ⁎) ∈ Π such that u(x) =U(x; π⁎), then u(x) = U(x) and π⁎ is optimal. Proof. See the proof in Appendix A. Theorem 3.1 drives us to find a twice continuously differentiable, increasing and concave solution of Eqs. (3.1) and (3.2) and then construct associated optimal strategy π⁎. The solutions will be discussed in two cases as follows depending on whether refinancing is profitable.

ð3:11Þ

0 ≤ x ≤ x0 ;

with x0 = Q(0). For 0 ≤ x ≤ x0, it follows from Eq. (3.7) that Z

0

u ðxÞ ¼ ke

x0 x

2θ2 aðzÞ dz; 1−aðzÞ

ð3:12Þ

where k N 0 is unknown coefficient. Obviously, we can see u″ ðx0 Þ ¼ −2k aðx0 Þ θ2 1−aðx ¼ 0. Also, Eq. (3.4) and smooth condition lead to u″(b) = 0. So 0Þ

we conclude that there exists a common switch level for optimal reinsurance and dividend policies under the variance premium principle, i.e., b = x0 = Q(0), which leads to k = β1. Using Eqs. (3.4), (3.5) and (3.12), we obtain

3.1. The case without refinancing Let's consider the first case when the strategy without refinancing is optimal. Then the boundary conditions correspond to u(0) = P and M u(0) ≤ u(0). In this case Eqs. (3.1) and (3.2) lead to the following equations,

ð3:10Þ

uðxÞ ¼

8 > < > :

Z β1

x

Z

e

b y

2θ2 aðzÞ dz 1−aðzÞ

dy þ P;

0 ≤ x ≤ b;

0

β1 ðx−bÞ þ uðbÞ;

ð3:13Þ

x ≥ b:

max f A a uðxÞg ¼ 0; 0 b x b b;

ð3:3Þ

Next, we come to identify the value of a0. Letting x = 0 in Eq. (3.8) results in

u0 ðxÞ−β1 ¼ 0;

ð3:4Þ

β1 λμ 22 ðθ1 −θ2 a0 Þe

0≤a≤1

Z

x ≥ b;

uð0Þ ¼ P;

ð3:5Þ

M uð0Þ−uð0Þ≤0;

ð3:6Þ

with some level b N 0. In fact, Eqs. (3.3) and (3.4) imply that the continuation region is (0, b) and the dividend region is [b , ∞ ), respectively. Differentiating Eq. (3.3) with respect to a and setting the derivative to zero yield u00 ðxÞ 2θ2 aðxÞ ¼− : u0 ðxÞ 1−aðxÞ

¼ δuðxÞ:

2θ2 ðα−aðxÞÞðaðxÞ−ρÞ ; 1−aðxÞ

φða0 Þ ¼ P;

ð3:15Þ

where the function φ(x) is of the form φðxÞ ¼

β1 λμ 22 ðθ1 −θ2 xÞ exp δ

Z

x 0

y dy ; ða−yÞðy−ρÞ

0 ≤ x b α:

x→α

ð3:8Þ

ð3:9Þ

with  

ð3:14Þ

After some simple calculations, we find that φ(0) = β1λθ1μ22/δ, lim φðxÞ ¼ −∞ and φ(x) is strictly decreasing on [0 , α) since

Taking derivative with respect to x on both sides of Eq. (3.8) and using Eq. (3.7) again, we obtain

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 α :¼ þ 2λθ1 θ2 μ 22 −δ þ 8λδθ22 μ 22 ∈ ðθ1 =θ2 ; 1Þ; 2 2 4λθ2 μ 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 1 2 2λθ θ μ −δ − 2λθ1 θ2 μ 22 −δ þ 8λδθ22 μ 22 ∈ ð−∞; 0Þ: ρ :¼ 1 2 2 4λθ22 μ 22 1

¼ δP:

Similar to Xu and Zhou (2012), let's do variable change of y = a(z) and use Eq. (3.9) and a(b) = 0, then we get

φ0 ðxÞ ¼ −

a0 ðxÞ ¼ −

2θ2 aðzÞ dz 1−aðzÞ

ð3:7Þ

Putting Eq. (3.7) into Eq. (3.3) yields λμ 22 ðθ1 −θ2 aðxÞÞu0 ðxÞ

b 0



2λθ1 θ2 μ 22 −δ

β1 exp 2θ2

Z

x 0

y δð1−xÞ dy b 0; ða−yÞðy−ρÞ ðα−xÞðx−ρÞ

0 b x b α:

Therefore, Eq. (3.15) has a unique root a0 ∈ (0, α) if the condition P b β1λθ1μ 22/δ holds. If P ≥ β1λθ1μ 22/δ, there is no solution to solve Eq. (3.15). In this case, we set b = 0, which means that the insurance company should pay all of the current surplus as dividend and claim the terminal value P at once. The solution is of the form u(x) = β1x + P. Then we shall check Eq. (3.6) in the following cases. b

∫0

2θ2 aðzÞ dz 1−aðzÞ

(1) P b β1λθ1μ22/δ and β2 ≥u0 ð0Þ ¼ β1 e . We know u′(x) ≤ β2 holds for any x N 0 since u′(x) is decreasing on (0,∞). Therefore M u(0) − u(0) = maxy ≥ 0{u(y) − β2y − K} − u(0) = − K b 0, Eq. (3.6) follows. Fig. 1(a) is a graph of u′(x) in this case.

526

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

Fig. 1. The graphs of the derivative U′(x).

b

β1λθ1μ22/δ

∫0

0

2θ2 aðzÞ dz 1−aðzÞ

(2) P b and β2 b u ð0Þ ¼ β1 e . Note that u′(x) is strictly decreasing from u′(0) to u′(b) = β1. Then there exists a unique number ξ⁎ ∈ (0, b) such that u′(ξ⁎) = β2. Define the integral Z IðξÞ :¼

ξ 0

ðu0 ðxÞÞ−β2 dx ¼ uð0Þ−β2 ξ:

ð3:16Þ

Obviously, Eq. (3.6) holds if and only when   K ≥I ξ :

ð3:17Þ

Fig. 1(b) provides a graph of u(x) in this case. (3) P ≥ β1λθ1μ 22/δ. It has u′(x) ≡β1 b β2. We can also conclude that M u(0) −u(0)= − Kb 0. Fig. 1(c) is a graph of u′(x) in this case.

Correspondingly, the candidate solution u1(x) for Eqs. (3.1) and (3.2) should satisfy max f A a u1 ðxÞg ¼ 0;

β1 −u01 ðxÞ ¼ 0;

ð3:21Þ

M u1 ð0Þ−u1 ð0Þ ¼ 0;

ð3:22Þ

with some level b1 ≥ 0, which is the switch level for dividend policy. Similar to Taksar (2000), we give the following solution with some parameter p⁎ N 0 8 > < > :

3.2. The case with forced refinancing

P b β1 λθ1 μ 22 =δ;

β2 b u0 ð0Þ ¼ β1 e

b 0

2θ2 aðzÞ dz 1−aðzÞ and K b I ξ :

ð3:18Þ

The analysis above shows that Mu(0) −u(0) ≤ 0 does not hold in the case of Eq. (3.18), so u(x) cannot solve the HJB equations now. In other words, it is no longer optimal to withdraw from the market when the surplus is null. The company should raise an appropriate amount of capital from the market to continue business. Then the associated boundary conditions become u(0) ≥ P and M u(0) − u(0) = 0.

ð3:20Þ

u1 ð0Þ≥P;

u1 ðxÞ ¼ uðx þ p Þ ¼

Z

ð3:19Þ

x ≥ b1 ;



Then, it remains to discuss the solution of HJB equations when

0 b x b b1 ;

0≤a≤1

Z β1

Z

xþp

e

b y

2θ2 að2Þ dz 1−aðzÞ

dy þ P;

0

β1 ðx−b1 Þ þ u1 ðb1 Þ;

0 ≤ x ≤ b1 ;

ð3:23Þ

x ≥ b1 ;

where u(x) is given by Eq. (3.13) and b1 = b − p⁎. Correspondingly, we define a reinsurance policy by 



aπ1 ðxÞ ¼ aπ ðx þ p Þ ¼



Q −1 ðx þ p Þ; 0 ≤ x ≤ b−p ; 0; x ≥ b−p :

ð3:24Þ

Apparently, u1(x) and aπ1⁎(x) satisfy Eqs. (3.19)–(3.21) automatically. Now we need to determine p⁎ N 0 using Eq. (3.22). Define a function of p as      ψðpÞ :¼ u ξ −uðpÞ−β 2 ξ −p −K;



0≤p≤ξ ;

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

where u(x) is shown in Eq. (3.13). It follows from Eq. (3.18) that      ψð0Þ ¼ u ξ −uð0Þ−β 2 ξ −K ¼ I ξ −K N 0:

ð3:25Þ

527

The optimal strategy is to pay all of the surplus as dividend and, thereafter, claim the terminal value P immediately, i.e., Lπt ⁎ ≡ x , Gπt ⁎ ≡ 0 and aπ⁎(x) can be arbitrary value in [0,1]. 2θ a b

2

∫0

In addition, we calculate that   ψ ξ ¼ −K b 0;

ð3:26Þ

ψ0 ðdÞ ¼ β2 −u0 ðdÞ b 0:

ð3:27Þ 

Consequently, there exists a unique p⁎ ∈ (0, ξ⁎), such that ψ(p⁎) =

dz

1−aðzÞ Case 4. If P b β1 λθ1 μ 22 =δ; β2 b u0 ð0Þ ¼ β1 e and K b I(ξ⁎), then the value function U(x) equals to u1(x) given by Eq. (3.23). The optimal dividend strategy Lπ⁎ is of barrier style with level b1 N 0, i.e.,

Lπt ¼ ðx−b1 Þþ þ

Z

t 0



IfX π ¼b1 g dLπs :

ð3:35Þ

t

0, i.e.,      u ξ −uðp Þ−β 2 ξ −p −K ¼ 0;

ð3:28Þ

It is profitable to refinance when and only when the surplus is null, the surplus immediately jumps to ξ1⁎ once it reaches 0 by issuing equities. Thus, Gπ⁎ is characterized by

ð3:29Þ

8Z ∞  > > I ft:X π N 0g dGπt ¼ 0; > > t > 0 n o > < π  τ 1 ¼ inf t ≥0 : X πt− ¼ 0 ; n o >   > π > τ n ¼ inf t N τπn−1 : X πt− ¼ 0 ; n ¼ 2; 3;   ; > > > : π   ηn ≡ ξ1 ¼ ξ −p ; n ¼ 1; 2;   :

or, equivalently,    u1 ξ1 −u1 ð0Þ−β2 ξ1 −K ¼ 0 with ξ1⁎ : = ξ⁎ − p⁎. Note that u′1(ξ1⁎) = u ′(ξ⁎) =β2, then    M u1 ð0Þ ¼ max fu1 ðξÞ−β2 ξ−K g ¼ u1 ξ1 −β2 ξ1 −K ¼ u1 ð0Þ ξ≥0

ð3:30Þ

is true, i.e., Eq. (3.22) holds. 3.3. The solution of problem 2.1 Based on the analysis above, we can identify the explicit solution to the value function and construct the associated optimal strategy in this section. Theorem 3.2. When there is no restriction on the dividend rate, the value function U(x) and associated optimal strategy π⁎ = (aπ⁎, Lπ⁎, Gπ⁎) can be identified in the following 4 cases, which exhaust all of the possibilities. b 2θ2 aðzÞ ∫0 dz 2 0 Case 1. If P b β1λθ1μ 2/δ and β2 ≥u ð0Þ ¼ β1 e 1−aðzÞ , then the value function U(x) coincides with u(x) in Eq. (3.13). The optimal reinsurance strategy is determined by the ceded proportion 

aπ ðxÞ ¼



Q −1 ðxÞ; 0;

0 ≤ x ≤ b; x ≥ b;



Lπt ¼ ðx−bÞþ þ

Z

t 0



IfX π ¼bg dLπs :

ð3:32Þ

s

It is unprofitable to refinance all the time, so Gπ⁎ t = 0. The controlled surplus process associated with π⁎ = (aπ⁎, Lπ⁎, Gπ⁎) satisfies 8 < :



X πt ¼ x þ π

X t ≤ b:

Z t Z th     2    ipffiffiffiffi   λμ 2 dBs −Lπt ; λμ 22 ds þ θ1 −θ2 aπ X πs 1−aπ X πs 0

0

ð3:33Þ b

∫0

2θ2 aðzÞ dz 1−aðzÞ

Case 2. If P b β 1 λθ1 μ 22 =δ; β2 b u0 ð0Þ ¼ β1 e and K ≥ I(ξ⁎), the value function and optimal strategies take the same forms as Case 1 and the controlled surplus process follows Eq. (3.33). Case 3. If P ≥ β1λθ1μ 22/δ, then U ðxÞ ¼ β 1 x þ P:

The optimal reinsurance strategy is determined by the ceded proportion 

aπ1 ðxÞ ¼



Q −1 ðx þ p Þ; 0 ≤ x ≤ b1 ; 0; x ≥ b1 ;

ð3:37Þ

which is also a decreasing function. The controlled surplus process π⁎ π⁎ associated with π ⁎ = (aπ⁎ 1 , L , G ) follows 8 Z t Z th     2    ipffiffiffiffi  > π > λμ 22 ds þ θ1 −θ2 aπ1 X πs 1−aπ1 X πs λμ 2 dBs > < Xt ¼ x þ 0 0 X∞ π π þ n¼1 Ifτπ ≤ tg ηn −Lt ; > > n > :  0 ≤X πt ≤b1 :

ð3:38Þ

ð3:31Þ

with b = Q(0). Obviously, less reinsurance is bought when the surplus increases since aπ⁎(x) is decreasing with respect to x, but reinsurance is always necessary before the time of bankruptcy. The barrier dividend strategy Lπ⁎ with level b N 0 is optimal, which can be described by

Proof. We can prove that the function U(x) given in 4 cases above is indeed a twice continuously differentiable, increasing and concave solution to the HJB Eqs. (3.1) and (3.2). By applying Theorem 3.1, we can establish that U(x) and π ⁎ are solutions to Problem 2.1. We only provide the detailed proof of Case 4 in Appendix B as an example and the method is also applicable to other cases. Remark 3.1. As shown in Theorem 3.2, the optimal strategies depend on the relationships among the parameters. The strategy with refinancing is optimal if and only if the condition (3.18) holds, b

2θ2 aðzÞ

dz ∫0  1−aðzÞ and K b Iðξ Þ. Economicali.e., P b β 1 λθ1 μ 22 =δ; β2 b u0 ð0Þ ¼ β1 e 2 ly, β1λθ1μ 2/δ is the present after-tax value of a perpetuity with discount rate δ and rate of income β1λθ1μ 22, which is the expected after-tax rate of

profit under the strategy without reinsurance. When the terminal value is smaller than the present after-tax value of this perpetuity, the insurer had better continue the business. Or else, it is better to declare bankruptcy and claim the terminal value immediately. When the company is on the brink of bankruptcy, the insurer would like to raise money if and only if both the proportional cost factor β2 and fixed cost factor K b

ð3:34Þ

ð3:36Þ

are not too high. Precisely, supremums, respectively.

u0 ð0Þ ¼ β1 e

∫0

2θ2 aðzÞ dz 1−aðzÞ

and I(ξ⁎) are

528

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

3.4. Numerical example In this section, we shall make some numerical examples to reflect the effects of β1, β2, K and P on the value functions and optimal strategies. Assume that the claim sizes are exponentially distributed with parameter 1, i.e., Yi ~ exp(1), then μ1 =1,μ 22 =2. Without loss of generality, we further set λ ¼ 0:56; θ1 ¼ 0:5; θ2 ¼ 0:8; δ ¼ 0:05 . The figures and calculations in this paper are all made with the help of Matlab. Example 3.1. In this example, we set β 2 ¼ 1:2; λ ¼ 0:56; θ1 ¼ 0:5; θ2 ¼ 0:8; μ 1 ¼ 1; μ 22 ¼ 2; δ ¼ 0:05; P ¼ 6; K ¼ 0:3 and let β1 vary. The results are shown in Table 1 and Fig. 2. It can be seen from Table 1 that Eq. (3.18) holds when β1 = 0.80, 0.85, 0.9, and 0.95, then the strategy with refinancing (denoted by “r”) is optimal and U(x) equals to u1(x) in Eq. (3.23). As the factor β1 increases, the ceded proportion ⁎ aπ1 (0) and dividend barrier b1 will decrease, but the amount of refinancing ξ1⁎ will increase. Namely, when the tax rate 1 − β1 decreases, the insurer would buy less reinsurance and leave less funds with the company to protect against financial risk and increase the size of refinancing. Meanwhile, both u′(0) and I(ξ⁎) increase, which means that lower tax rate makes the insurer more willing to refinance. In the case of β1 = 0.70 or 0.75, the condition (3.18) fails due to the fact K N I(ξ⁎), then the strategy without refinancing (denoted by “n”) is optimal and U(x) coincides with u(x) in Eq. (3.13), i.e., the insurer will not choose to refinance in a high tax environment. In addition, lower tax rate 1 − β1 leads to higher ceded proportion aπ ⁎(0) and dividend barrier b. It is also clear from Fig. 2 that lower tax rate results in larger value function. Example 3.2. In this example, we set β1 = 0.8, λ = 0.56 , θ1 = 0.5 , θ2 = 0.8 , μ1 = 1 , μ 22 = 2 , δ = 0.05 , P = 6 , K = 0.3 and let β2 vary. The results are shown in Table 2 and Fig. 3(a). We can see from Table 2 that the condition (3.18) holds when β2 = 1.05 , 1.10 , 1.15 , 1.20 , 1.25, thus the strategy with refinancing (denoted by “r”) is optimal and U(x) equals to u1(x) in Eq. (3.23). Moreover, it is clear that larger β2 results in larger aπ1 ⁎(0) and b1, the insurer should buy more reinsurance and pay dividends lately. Meanwhile, ξ1⁎ and I(ξ⁎) become smaller, so the insurer will decrease the size of each capital injection and would like accept a smaller fixed cost for refinancing. Moreover, the condition (3.18) fails when β2 = 1.30, so the strategy without refinancing (denoted by “n”) is optimal and U(x) coincides with u(x) in Eq. (3.13). Thus the insurer will not refinance if the proportional costs of equity issuances are too high, which justifies our intuition. Fig. 3(a) shows that larger β2 results in smaller u1(x). Also, the value of u(x) is independent of β2. Example 3.3. In this example, we assign β1 = 0.8 , β2 = 1.2 , λ = 0.56 , θ1 = 0.5 , θ2 = 0.8 , μ1 = 1 , μ 22 = 2 , δ = 0.05 , P = 6 and let K vary. The results are shown in Table 3 and Fig. 3(b). We can see from Table 3 that Eq. (3.18) always holds when K varies from 0.05 to 0.30, so the strategy with refinancing (denoted by “r”) is optimal and U(x) equals to u1(x) in Eq. (3.23). Meanwhile, the values of aπ1 ⁎(0) , b1 and ξ1⁎ all become larger. In other words, the insurer will purchase more reinsurance and reserve more money to protect against risks and increase the size of capital injection to decrease the frequency of equity issuance.

On the contrary, the condition (3.18) is not true if either K N I(ξ⁎) = 0.3647 or β2 N u ′ (0) = 2.5788 holds, then the strategy without refinancing becomes optimal and U(x) coincides with u(x) in Eq. (3.13). It is also clear from Fig. 3(b) that larger K results in smaller u1(x). Also, the value of u(x) is independent of K. Example 3.4. In this example, we set β1 = 0.8, β2 = 1.2 , λ = 0.56 , θ1 = 0.5 , θ2 = 0.8 , μ1 = 1 , μ 22 = 2 , δ = 0.05 , K = 0.3 and let P vary. The results are shown in Table 4 and Fig. 4. We can see from Table 4 that Eq. (3.18) holds for P = − 4 , − 2 , 0 , 2 , 4 , 6, so the strategy with refinancing (denoted by “r”) is optimal and U(x) equals to u1(x) in Eq. (3.23). Instead, the condition (3.18) is not true if either K N I(ξ⁎) = 0.3647 or β2 N u ′ (0) = 2.5788 holds, then the strategy without capital injection becomes optimal and U(x) coincides with u(x) in Eq. (3.13). With the increasing of the terminal value P, both aπ⁎(0) and b become smaller, then the insurer will buy less reinsurance and reserve less money to guard against the risks. Moreover, the values of u ′ (0) and I(ξ⁎) decreases, which means that the insurer's appetite for refinancing is diminishing and the bankruptcy has become gradually acceptable. It ⁎ is interesting to see that the values of aπ⁎ 1 (0), b1 and ξ1 and the figure of u1(x) remains unchanged for different P. In other words, if the strategy with refinancing is optimal, then it is independent of the value P. In Table 5, we summarize the effects of the parameters on the crucial levels which characterize the optimal strategies. In the table, “↑” denotes the level is increasing, “↓” denotes the level is decreasing and “−” means the level is independent of some parameter. The results in the first three rows are obtained when P N 0. 4. Restricted dividend In this section, we consider Problem 2.2, when a ceiling M is imposed on the dividend rate. Suppose the candidate solution v(x) for V(x) is smooth enough. We conjecture that the interval [0, ∞ ) can be divided into three parts: In the continuation region (0, d) with some d N 0, that is, when the insurer does not take any actions, we must have max0 ≤ a ≤ 1;0 ≤ l ≤ M f A a vðxÞ þ lðβ1 −v0 ðxÞÞg ¼ max0 ≤ a ≤ 1 f A a vðxÞg ¼ 0 ; In the dividend region [d , ∞ ), the dividends are distributed at the maximum rate M, that is, max0 ≤ a ≤ 1;0 ≤ l ≤ M f A a vðxÞ þ lðβ1 −v0 ðxÞÞg ¼ max0 ≤ a ≤ 1 f A a vðxÞ þ Mðβ1 −v0 ðxÞÞg ¼ 0 holds; On the set {0}, the insurance company is on the brink of bankruptcy, then the insurer either allows the surplus process to hit (−∞, 0) by injecting no capital, which corresponds to the boundary condition v(0) = P and M v(0) ≤ v(0), or keeps the surplus process in the interval [0 , ∞ ) by issuing equities, which corresponds to the boundary condition v(0) ≥ P and M v(0) = v(0). Now, we can use optimal control theory to get the following HJB equations corresponding to Problem 2.2: max

max

0 ≤ a ≤ 1;0 ≤ l ≤ M

f A a vðxÞ þ lðβ 1 −v0 ðxÞÞg; M vðxÞ−vðxÞ ¼ 0;

maxf M vð0Þ−vð0Þ; P−vð0Þg ¼ 0:

ð4:1Þ

ð4:2Þ

Table 1 The effect of β1 on optimal strategy. β2 ¼ 1:2; π⁎

λ ¼ 0:56;

θ1 ¼ 0:5;

θ2 ¼ 0:8;

μ 1 ¼ 1;

β1 ↑

a (0)↑

b↑

⁎ aπ1 (0)↓

0.70 0.75 0.80 0.85 0.90 0.95

0.3690 0.4469 0.4952 0.5256 0.5456 0.5595

2.2104 2.4756 2.6518 2.7753 2.8663 2.9362

– – 0.4818 0.4657 0.4484 0.4298

μ 22 ¼ 2;

δ ¼ 0:05;

P ¼ 6;

K ¼ 0:3

b1 ↓

ξ⁎1↑

u ′ (0)↑

I(ξ⁎)↑

strategy

– – 2.6013 2.5423 2.4809 2.4164

– – 0.6706 0.7112 0.7580 0.8133

1.3078 1.8795 2.5788 3.3678 4.2186 5.1124

0.0051 0.1278 0.3647 0.6720 1.0256 1.4122

n n r r r r

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

529

Fig. 2. The effects of β1 on u(x) and u1(x)

Theorem 4.1. Let v(x) be an increasing, concave and twice continuously differentiable solution of the HJB Eqs. (4.1) and (4.2), and v′(x) is bounded, then vðxÞ ≥ Vðx; πÞ for each admissible strategy π ∈ Π , such that v(x) ≥ V(x) holds. Moreover, if there exists some strategy π ¼     π π π ða ; L ; G Þ ∈ Π, such that v(x) = V(x; π⁎), then v(x) = V(x) and π is optimal. Proof. The proof can be done by repeating a similar procedure to Appendix A, so it is omitted here.

Similar to Eq. (3.9), the maximizer a(x) satisfies the following differential equation a0 ðxÞ ¼ −

2θ2 ðα−aðxÞÞðaðxÞ−ρÞ : 1−aðxÞ

ð4:7Þ

Let's define a function of x as Z

According to Theorem 4.1, we should find an increasing, concave and twice continuously differentiable solution to the HJB equations  and then construct the associated optimal strategy π . In what follows, we will do that according to different conditions, such that Eq. (4.2) holds.

where a⁎ b a0 b α will be determined later. Then we can express a(x) as

4.1. The case without capital injections

aðxÞ ¼ S−1 ðxÞ ∈ ½a ; a0 ; 0 ≤ x ≤ d;

Suppose that it is optimal to get out of the business whenever the surplus reaches 0, then the corresponding boundary conditions are v(0) = 0 and M v(0) − v(0) ≤ 0. Following the approach in control theory, the candidate solution v(x) should satisfy that

which is a decreasing function with a(d) : = a⁎ N 0 and a(0) : = a0 N a⁎. Repeating a similar process as in Subsection 3.1, we provide the candidate solution to Eq. (4.6) with

max

0 ≤ a ≤ 1;0 ≤ l ≤ M

f A a vðxÞ þ lðβ 1 −v0 ðxÞÞg ¼ 0;

x ≥ 0;

M vð0Þ−vð0Þ ≤ 0:

ð4:5Þ

Suppose that there exists a switch point d N 0 such that v′(d) = β1. Then we have v′(x) ≥ β1 for 0 ≤ x ≤ d, so max

f A a vðxÞ þ lðβ 1 −vðxÞÞg ¼ max f A a vðxÞg:

ð4:6Þ

0≤a≤1

1−z dz; 2θ2 ðα−zÞðz−ρÞ

Z

Z vðxÞ ¼ β1

ð4:4Þ

a0

x

ð4:3Þ

P−vð0Þ ¼ 0;

0 ≤ a ≤ 1;0 ≤ l ≤ M

SðxÞ ¼

d

x

e

y

2θ2 aðzÞ dz 1−aðzÞ dy þ P;

0

a ≤ x ≤ a0 ;

ð4:8Þ

ð4:9Þ

0 ≤ x ≤ d:

ð4:10Þ

For x N d, it has v ' (x) b β1. We conjecture that it is optimal to pay dividends at the maximum rate M and cede the risks with a constant proportion a⁎. Accordingly, max



0 ≤ a ≤ 1;0 ≤ l ≤ M

f A a vðxÞ þ lðβ 1 −v0 ðxÞÞg ¼ A a vðxÞ þ Mðβ1 −v0 ðxÞÞ ¼ 0; ð4:11Þ

Table 2 The effect of β2 on optimal strategy. β1 = 0.8 , λ = 0.56 ,θ1 = 0.5 , θ2 = 0.8 , μ1 = 1 ,μ22 = 2 , δ = 0.05 , P = 6 ,K = 0.3 β2 ↑

aπ (0)

b

aπ1 (0)↑

b1 ↑

ξ⁎1↓

u ′ (0)

I(ξ⁎)↓

strategy

1.05 1.10 1.15 1.20 1.25 1.30

0.4952 0.4952 0.4952 0.4952 0.4952 0.4952

2.6518 2.6518 2.6518 2.6518 2.6518 2.6518

0.4542 0.4646 0.4738 0.4818 0.4890 –

2.5014 2.5386 2.5716 2.6013 2.6283 –

0.8114 0.7555 0.7094 0.6706 0.6372 –

2.5788 2.5788 2.5788 2.5788 2.5788 2.5788

0.4895 0.4438 0.4024 0.3647 0.3302 0.2985

r r r r r n

⁎

⁎

530

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

Fig. 3. The effects of the refinancing costs on u1(x).

or equivalently,    1 ð1−a Þ2 λμ 22 v″ ðxÞ þ θ1 −θ2 a2 λμ 22 −M v0 ðxÞ−δvðxÞ þ β1 M ¼ 0: 2 ð4:12Þ

Finally, we need to determine the value of a0. Similar to Eq. (3.8), we have λμ 22 ðθ1 −θ2 aðxÞÞv0 ðxÞ ¼ δvðxÞ:

ð4:18Þ

Therefore, combining with (4.10), we arrive at Note that v(x) is bounded and v′(d) =β1, then we have β M β vðxÞ ¼ 1 þ 1 eγðx−dÞ ; γ δ

x ≥ d;

Z

ð4:13Þ

β1 λμ 22 ðθ1 −θ2 a0 Þe

d 0

2θ2 aðzÞ dz 1−aðzÞ

¼ δP:

ð4:19Þ

Doing variable change of y = a(z) and combining with Eqs. (4.7) and (4.17), we can obtain that

where γ is the negative root of the equation    1 ð1−a Þ2 λμ 22 γ 2 þ θ1 −θ2 a2 λμ 22 −M γ−δ ¼ 0: 2

ð4:14Þ

ϕða0 Þ ¼ P;

ð4:20Þ

where ϕ(x) is of the form In addition, to match the continuity condition 00

ϕðxÞ :¼

00

v ðd−Þ v ðdþÞ ¼ 0 ; v0 ðd−Þ v ðdþÞ

β1 λμ 22 ðθ1 −θ2 xÞ exp δ

Z

x a

y dy ; a ≤a0 b α: ðα−yÞðy−ρÞ

The function ϕ(x) is strictly decreasing on [a⁎ ,α) since

we have 2θ2 a ¼ γ: − 1−a

ð4:15Þ

Then, by substituting (4.15) into (4.14), we see that a⁎ ∈ (0, α) is the unique root of the following equation   2λμ 22 θ22 ða Þ2 þ 2Mθ2 þ δ−2θ1 θ2 λμ 22 a −δ ¼ 0:

ð4:16Þ

ϕ0 ðxÞ ¼ −

Z x

β1 y ð1−xÞ exp dy b 0: 2θ2 ðα−xÞðx−ρÞ a ðα−yÞðy−ρÞ

Moreover, we derive that lim ϕðxÞ ¼ −∞ and ϕ(a⁎) = β1λμ22(θ1 − x→α

θ2a⁎)/δ. Therefore, we conclude that Eq. (4.20) has a unique solution a0 ∈ (a⁎, α) if and only if the condition P b β1λμ 22(θ1 − θ2a⁎)/δ holds. When P ≥ β1λμ22(θ1 − θ2a⁎)/δ, Eq. (4.20) has no solution. In this case, we conjecture that v′(x) ≤ β1 for all x ≥0. Then

In addition, the equality a(d) = S−1(d) = a⁎ yields

max

0 ≤ a ≤ 1;0 ≤ l ≤ M

d ¼ Sða Þ ≥ 0:

f A a vðxÞ þ lðβ 1 −v0 ðxÞÞg ¼ max f A a vðxÞ þ M ðβ1 −v0 ðxÞÞg; 0≤a≤1

ð4:17Þ

ð4:21Þ

Table 3 The effect of K on optimal strategy. β1 = 0.8 , β2 = 1.2 ,λ = 0.56 , θ1 = 0.5 ,θ2 = 0.8 ,μ1 = 1 , μ22 = 2 , δ = 0.05 , P = 6 ⁎

⁎

K↑

aπ (0)

b

aπ1 (0)↑

b1 ↑

ξ⁎1↑

u ′(0)

I(ξ⁎)

strategy

0.05 0.10 0.15 0.20 0.25 0.30

0.4952 0.4952 0.4952 0.4952 0.4952 0.4952

2.6518 2.6518 2.6518 2.6518 2.6518 2.6518

0.3807 0.4139 0.4370 0.4549 0.4695 0.4818

2.2500 2.3621 2.4414 2.5040 2.5563 2.6013

0.3193 0.4314 0.5107 0.5733 0.6256 0.6706

2.5788 2.5788 2.5788 2.5788 2.5788 2.5788

0.3647 0.3647 0.3647 0.3647 0.3647 0.3647

r r r r r r

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

531

Table 4 The effect of P on optimal strategy. β1 = 0.8 , β2 = 1.2 , λ = 0.56 , θ1 = 0.5 ,θ2 = 0.8 , μ1 = 1 ,μ22 = 2 , δ = 0.05 , K = 0.3 ⁎

⁎

P↑

aπ (0)↓

b↓

aπ1 (0)

b1

ξ⁎1

u′(0)↓

I(ξ⁎)↓

Strategy

−4 −2 0 2 4 6

0.6331 0.6300 0.6250 0.6153 0.5912 0.4952

3.6877 3.6071 3.5035 3.3603 3.1329 2.6518

0.4818 0.4818 0.4818 0.4818 0.4818 0.4818

2.6013 2.6013 2.6013 2.6013 2.6013 2.6013

0.6706 0.6706 0.6706 0.6706 0.6706 0.6706

27.6424 22.1545 16.7579 11.5197 6.6081 2.5788

9.1216 7.2184 5.3426 3.5146 1.7874 0.3647

r r r r r r

i.e., the company will pay dividends at the maximum rate M all the time. Specifically,

   1 ð1−aÞ2 λμ 22 v00 ðxÞ þ 1−a2 θλμ 22 −M v0 ðxÞ−δvðxÞ þ β1 M ¼ 0: 0≤a≤1 2 max

ð4:22Þ Taking derivative with respect to a yields v00 ðxÞ 2θ2 aðxÞ ¼− : v0 ðxÞ 1−aðxÞ

(1) In the case of P b β1λμ 22(θ1 − θ2a⁎)/δ and β2 ≥ v′(0), we have β2 ≥ v′(x) for all x N 0, as v′(x) is decreasing on (0,∞). Therefore M v(0) − v(0) = maxy ≥ 0{v(y) − β2y − K} − v(0) = − K b 0, Eq. (4.5) follows. Fig. 5(a) is a graph of v ′ (x) in this case. (2) In the case of P b β1λμ22(θ1 − θ2a⁎)/δ and β2 b v ′ (0), v ′ (x) is strictly decreasing from v ′ (0) to 0 and v ′ (d) = β1. Then, there exists a unique number ζ ⁎ ∈ (0, d), such that v ′ (ζ⁎) = β2. Define the integral

ð4:23Þ

Z J ðζ Þ :¼

We further expect the optimal ceded proportion a(x) to be some constant, which proves to be a⁎ ∈ (0, 1) later. Then the solution to Eq. (4.22) takes the form vðxÞ ¼ C þ Deγx

0

ðv0 ðxÞ−β2 Þdx ¼ vðζ Þ−vð0Þ−β 2 ζ :

ð4:27Þ

Apparently, Eq. (4.5) holds if and only if

ð4:24Þ

with

ς

K ≥ J ðζ  Þ:

2θ2 a : γ¼− 1−a

ð4:25Þ β1 M δ

By putting Eq. (4.24) into Eq. (4.22), we deduce that C ¼ and β1 M ⁎ a ∈ (0, α) satisfies Eq. (4.16). Then D ¼ P− δ can also be obtained by the boundary condition v(0) = P. We remain to verify the condition v ′ (x) ≤β1 for all x ≥0, equivalently,

Fig. 5(b) is a graph of v′(x) in this case. (3) In the case of P ≥ β1λμ 22(θ1 − θ2a⁎)/δ, it has v′(x) = β1 b β2 for all x ≥ 0. Then, for the same reason, Mu(0) − u(0) = − K b 0 is right. Fig. 5(c) provides a graph of v′(x) in this case. 4.2. The case with forced capital injections



β1 β1 M β ð1−a Þ β1 M þ þ ¼− 1 γ 2θ2 a δ δ 2  β1 λμ 2 ðθ1 −θ2 a Þ ; ¼ δ

v0 ð0Þ ¼ Dγ ≤β1 ⇔ P ≥

ð4:28Þ

Now, it is left to discuss the solution of HJB equations when ð4:26Þ

where the last equality comes from Eq. (4.16). Finally, we continue to check Eq. (4.5) in following cases.

Z

Pb

β1 λμ 22 ðθ1 −θ2 a Þ=δ; β2

Fig. 4. The effects of P on u(x) and u1(x).

0

b v ð0Þ ¼ β1 e

d 0

2θ2 aðzÞ dz 1−aðzÞ

and K b J ðζ  Þ: ð4:29Þ

532

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

Fig. 5. The graphs of the derivative V′(x).

Above analysis shows that M v(0) − v(0) ≤ 0 does not hold in the case of Eq. (4.29) and v(x) cannot solve the HJB equations now. We conjecture that the candidate solution v1(x) for v(x) satisfies max

0 ≤ a ≤ 1;0 ≤ l ≤ M

f A a v1 ðxÞ þ lðβ1 −v1 ðxÞÞg ¼ 0;

x ≥ 0;

ð4:30Þ

v1 ð0Þ≥P;

ð4:31Þ

M v1 ð0Þ−v1 ð0Þ ¼ 0:

ð4:32Þ

Eqs. (4.31) and (4.32) imply that the optimal strategy is to refinance whenever the surplus reaches 0. With the same argument as in Subsection 3.2, we give the solution as

v1 ðxÞ ¼ vðx þ q Þ ¼

8 > > Z > > <β

Z xþq

1

e

d y

2θ2 aðzÞ dz 1−aðzÞ

0 > > β1 M β1 γðx−d1 Þ > > e ; þ : γ δ

dy þ P;

0 ≤ x ≤ d1 ;

4.3. The solution of Problem 2.2 Based on the analysis above, we establish the explicit solution to Problem 2.2 in this subsection. In particular, the proofs resemble those in Section 3, so we present the main results in this section but omit most of interpretations. Theorem 4.2. When there exists a ceiling M on the dividend rate, the value function V(x) and associated optimal strategy π can be identified in the following 4 cases, which exhaust all of the possibilities. d

2θ2 aðzÞ dz 1−aðzÞ

¼ β1 e , the value Case 1. function is given by Eqs. (4.10) and (4.13). Lπ is a threshold dividend strategy with level d, which is depicted by the following dividend rate

ð4:33Þ l

x ≥ d1 ;

∫0

If P b β1λμ22(θ1 − θ2a⁎)/δ and β2 ≥v0 ð0Þ

π







¼ l ðxÞ ¼

M; x N d; 0; 0 ≤ x ≤ d:

ð4:37Þ 

where v(x) is given by Eqs. (4.10) and (4.13), d1 = d − q⁎ and q⁎ ∈ (0, ζ⁎) is the unique solution of the following equation

It is unprofitable to refinance and bankruptcy is allowed, so Gπt ≡ 0. The optimal reinsurance strategy is determined by the ceded proportion

vðζ  Þ−vðq Þ−β2 ðζ  −q Þ−K ¼ 0:

ð4:34Þ

aπ ðxÞ ¼

ð4:35Þ

Then the surplus process controlled by π ¼ ðaπ ; Lπ ; Gπ Þ satisfies that

We can rewrite Eq. (4.34) as   v1 ζ 1 −v1 ð0Þ−β2 ζ 1 −K ¼ 0

aπ1 ðxÞ ¼



−1



S ðx þ q Þ; 0 ≤ x ≤ d1 ; a ; x ≥ d1 :

Fig. 5(d) provides a graph of v1′(x).



S−1 ðxÞ; 0 b x ≤ d; a ; x N d:

ð4:38Þ 

with ζ1⁎ : = ζ ⁎ − q⁎. The associated optimal reinsurance strategy is described by 



ð4:36Þ



X πt ¼ x þ







Z t Z th     2    ipffiffiffi   λμ 2 dBs −Lπt : θ1 −θ2 aπ X πs 1−aπ X πs λμ 22 dsþ 0

0

d

β1λμ22(θ1 − θ2a⁎)/δ,

0

∫0

2θ2 aðzÞ ð4:39Þ dz 1−aðzÞ

Case 2. If P b β2 b v ð0Þ ¼ β1 e and  K ≥ J(ζ⁎), the value function V(x) and associated optimal strategy π take the same forms as those in Case 1.

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

Case 3. If P ≥

β1λμ22(θ1 − θ2a⁎)/δ,

533

The optimal reinsurance strategy is determined by the ceded proportion

then

 β M β M γx V ðxÞ ¼ 1 þ P− 1 e : δ δ



ð4:40Þ

aπ1 ðxÞ ¼



S−1 ðx þ q Þ; a ; x ≥d1 :

0 ≤x≤d1 ;

ð4:47Þ 



The optimal strategy is to pay dividends at the maximum rate M until the time of bankruptcy, raise no money all the time and buy reinsurance with the ceded proportion a⁎. That is to say, π





l ðxÞ ¼ l ¼ M;

x ≥0;



X πt ¼ x þ



0

ð4:43Þ 







Then the surplus process controlled by π ¼ ðaπ ; Lπ ; Gπ Þ satisfies that Z

t 0

Z t pffiffiffi  ðθ1 −θ2 a Þλμ 22 dsþ ð1−a Þ λμ 2 dBs −Lπt :

ð4:48Þ

d

∫0

0

Proof. The proof can be done by repeating a similar procedure to Appendix B, so it is omitted here. Remark 4.1. Theorem 4.2 shows that the optimal strategies depend on the relationships among the model's parameters. The strategy with forced capital injections is optimal if and only if the condition d

ð4:44Þ

0

β1 λμ 22 ðθ1 −θ2 a Þ=δ; β2

2θ2 aðzÞ dz 1−aðzÞ

b v ð0Þ ¼ β1 e and Case 4. If P b K b J(ζ ⁎), the value function V(x) equals to v1(x) given by Eq. (4.33). Lπ is a threshold dividend strategy with level d1 N 0, which is depicted by the following dividend rate

∫0

π





¼ l ðxÞ ¼



M; 0;

x N d1 ; 0≤x ≤d1 :

ð4:45Þ

It is optimal to refinance when and only when the surplus is null, and the surplus immediately jumps to ζ1⁎ once it reaches 0 by issuing 

equities. Mathematically, Gπ is characterized by 8Z ∞  > > > I t:X π N 0 dGπt ¼ 0; > > t > 0 n o >  < π τ1 ¼ inf t ≥ 0 : X πt− ¼ 0 ; n o   > > π > τn ¼ inf t N τ πn−1 : X πt− ¼ 0 ; n ¼ 2; 3; ⋯; > > > > π : ηn ≡ ζ 1 ¼ ζ  −q ; n ¼ 1; 2; ⋯:

ð4:46Þ

2θ2 aðzÞ dz 1−aðzÞ

(4.29) holds, i.e., P b β1 λμ 22 ðθ1 −θ2 a Þ=δ; β2 b v0 ð0Þ ¼ β1 e and K b Jðζ  Þ. In fact β1λμ22(θ1 − θ2a⁎)/δ is the present after-tax value of a perpetuity with discount rate δ and rate of income β1λμ22(θ1 − θ2a⁎), which is the expected after-tax profit rate under the reinsurance with ceded proportion a⁎. When the company is on the brink of bankruptcy, the insurer would like to raise money if and only if both the proportional cost factor β2 and fixed cost factor K are not too high. Precisely, v0 ð0Þ ¼ d

∫0

l

n

ð4:42Þ

x≥0;





0

∞ X   þ Iτπ ≤ t  ηπn −Lπt

ð4:41Þ

Gπt ≡ 0:

X πt ¼ x þ



Z t Z th     2    ipffiffiffi  λμ 22 ds þ θ1 −θ2 aπ1 X πs 1−aπ1 X πs λμ 2 dBs

n¼1

aπ ðxÞ ≡ a ;



The surplus process controlled by π ¼ ðaπ1 ; Lπ ; Gπ Þ follows that

2θ2 aðzÞ dz 1−aðzÞ

and J(ζ⁎) are supremums, respectively. In particular, β1 e when the terminal value P is greater than β1λμ 22(θ1 − θ2a⁎)/δ, it is optimal for the insurer to pay dividends at the maximum rate M and declare bankruptcy as soon as possible. Remark 4.2. All results obtained in Section 3 are compatible with those in Section 4 when the dividend ceiling M goes to infinity. That is, the optimal control problem without dividend restriction can be seen as the limiting optimal control problem with bounded dividend rate, which will be illustrated by Example 4.1 and Fig. 6. Remark 4.3. Under the assumptions of variance premium principle and “cheap” proportional reinsurance, Zhou and Yuen (2012) studied an optimal dividend, refinancing and proportional reinsurance problem with proportional cost and zero terminal value. They proved that either the barrier dividend strategy or threshold dividend strategy is optimal,

M

M Fig. 6. The effects of M on v(x) and v1(x).

534

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

transform employed in Liang and Young (2012), we use some different techniques in the solving process.

Table 5 The effects of parameters on the crucial levels.

β1 ↑ β2 ↑ K↑ P↑

aπ⁎(0)

b

aπ1⁎(0)

b1

ξ⁎1

u ' (0)

I(ξ⁎)

↑ – – ↑

↑ – – ↑

↓ ↑ ↑ –

↓ ↑ ↑ –

↑ ↓ ↑ –

↑ – – ↑

↑ ↑ – ↑

depending on whether there exist restrictions on dividend policies, and that the optimal ceded proportion of risk exponentially decreases with the surplus and remains constant when the initial surplus exceeds the dividend threshold, and that new capital could be injected continuously if it is profitable. Yao et al. (2014) further studied this problem by allowing for a non-negative terminal value and the fixed cost in refinancing process. Similar results for dividend and reinsurance strategies were obtained but with different initial value of ceded proportion, and new capital could be injected discretely due to the existence of fixed transaction cost. This paper extends the risk model of Yao et al. (2014) by assuming that the proportion reinsurance is “expensive” and the terminal value is arbitrary, which makes the problem more practical and difficult to deal with. Similar results as in Yao et al. (2014) are obtained in a more sophisticated way. Correspondingly, all results in Yao et al. (2014) can be obtained by letting θ2 → θ1 and P ≥ 0; all results in Zhou and Yuen (2012) can also be given by letting P = K =0 and θ2 → θ1. Under the expectation premium principle and “expensive” proportional reinsurance, Liang and Young (2012) solved the optimal dividend and reinsurance problem with an arbitrary terminal value. They also proved that the insurance company should pay dividends according to the barrier strategy when the dividend rated was unrestricted and buy less reinsurance when the surplus increases. However, there are some important differences: First, this paper incorporates the refinancing strategy in the combined dividend and reinsurance problem. So that, besides declaring bankruptcy, refinancing may be selected by the insurer when the company runs into financial difficulty. Moreover, the transaction costs generated by refinancing are included. Second, this paper assumes that the premium is calculated via the variance principle instead of the expectation premium principle in Liang and Young (2012). Furthermore, we find that there exists a common switch level for the optimal dividend and reinsurance policies instead of two levels under the expectation premium principle. Accordingly, the reinsurance contract is always meaningful for the insurance company in this paper. By the way, we note that Chen et al. (2013) have shown that there was a common switch level for the optimal dividend and reinsurance policies under the exponential premium principle. Third, we also pay attention to the problem when the dividend policy is restricted, associated threshold dividend strategy is studied. Finally, beyond the Legendre

4.4. Numerical example Similar to Section 3.4, we can do some numerical examples to show the effects of each parameter on the value function and associated optimal strategy. To avoid repetition, we only observe the effect of the ceiling M in the following example. Example 4.1. Suppose that the claim sizes are exponentially distributed with parameter 1, then μ1 = 1 and μ22 =2. We further set β1 = 0.8 , β2 = 1.2 , λ = 0.56 , θ1 = 0.5 , θ2 = 0.8 , δ = 0.05 , P = 6 , K = 0.3 and let M vary. With the help of Matlab, we can make the numerical calculations for each set of parameters. The results are shown in Table 6 and Fig. 6. When M = 3 ; 5 ;10; 100 ; 1000, the condition (4.29) holds, the strategy with refinancing (denoted by “r”) is optimal and the value function V(x) is consistent with v1(x) in Eq. (4.33). When M = 1, the strategy without refinancing (denoted by “n”) is optimal since Eq. (4.29) does not hold. The changing trends of the crucial levels can be seen in Table 6. Obviously, the crucial levels characterizing the optimal strategies tend to those in Tables 1–4 when the dividend ceiling M goes to infinity, which supports the statement in Remark 4.2. As shown in Fig. 6, the differences among the crucial levels are so small when M varies that some figures overlap closely. (See Table 6.)

5. Conclusions In this paper, we consider a combined optimal dividend, refinancing and proportional reinsurance problem for an insurer. The controlled diffusion model is established as an approximation of the classical risk model with “expensive” proportional reinsurance under the variance premium principle. Then the effects of terminal value and transaction costs are considered in the model. And also, by using optimal control method, explicit solutions to the value function and optimal strategies are presented in two cases which depend on whether the dividend rates are bounded. We find that either barrier dividend strategy or threshold dividend strategy is optimal, depending on whether there exist restrictions on dividend policies; the insurer should buy less insurance when the surplus increases, but the ceded proportion of risk remains constant once the surplus exceeds some level; and refinancing is optimal if and only if the terminal value and the transaction costs are not too high and the company is on the brink of bankruptcy, and the amount of each capital injection remains constant. In particular, the insurance company should declare bankruptcy as soon as possible if the terminal value is high enough. Acknowledgments The authors are grateful to Prof. Sushanta Mallick and the three anonymous referees for their valuable insights and helpful suggestions.

Table 6 The effect of M on the optimal strategy. β1 = 0.8 , β2 = 1.2 , λ = 0.56 ,θ1 = 0.5 , θ2 = 0.8 ,μ1 = 1 , μ22 = 2 ,δ = 0.05 , P = 6 ,K = 0.3 ⁎

⁎

M↑

aπ (0)↑

a ⁎↓

d↑

aπ1 (0)↓

d1 ↑

ζ ⁎1↑

v ′ (0)↑

J(ζ ⁎)↑

Strategy

1 3 5 10 100 1000

0.4737 0.4939 0.4948 0.4951 0.4952 0.4952

0.0596 0.0126 0.0070 0.0033 0.0003 0.0000

1.8406 2.4436 2.5324 2.5940 2.6462 2.6513

– 0.4823 0.4820 0.4819 0.4818 0.4818

– 2.3996 2.4840 2.5440 2.5957 2.6008

– 0.5931 0.6693 0.6702 0.6705 0.6706

2.2133 2.5549 2.5710 2.5770 2.5787 2.5788

0.2345 0.3559 0.3618 0.3640 0.3647 0.3647

n r r r r r

G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

This work was sponsored by Program of Shanghai Subject Chief Scientist (14XD1401600), the 111 Project (B14019), and the National Natural Science Foundation of China (11571113, 11231005, 11501321).

535

Appendix B. The proof of Case 4 in Theorem 3.2

b

∫0

Appendix A. The proof of Theorem 3.1.

Proof. For each strategy π = (aπ, Lπ, Gπ) ∈ Π, let's define ΛπL = {s : Lπs− ≠ π Lsπ} , ΛπG = {s : Gπs − ≠ Gπs } = {τπ1, τπ2, ⋯ , τπn, ⋯}. Let ^Lt ¼ ∑ ðLπs −Lπs− Þ be s∈ΛπL ;s ≤ t

π π the discontinuous part of Lπt and ~Lt ¼ Lπt −^Lt be the continuous part π π ^ and G ~ stand for the discontinuous and continuous of Lπt . Similarly, G t

t

^ formula, we obtain parts of Gπt respectively. Then, according to It o that   π e−δðt∧T Þ u X πt∧T π −uðxÞ ¼

Z

Z

t∧T π

 π  e−δs A a u X πs dsþ

0

Z −

2θ2 aðzÞ dz 1−aðzÞ

Proof. Suppose that P ≤ β1 λθ1 μ 22 =δ; β2 b u0 ð0Þ ¼ β1 e and K b I(ξ⁎). We first verify that u1(x) and aπ⁎ 1 satisfies the HJB Eqs. (3.1) and (3.2).

• Step 1: To show max0 ≤ a ≤ 1 f A a u1 ðxÞg≤0 on [0, ∞ ). (i) If 0 ≤ x ≤ b1, by construction, u1(x) and aπ⁎ 1 (x) satisfy Eq. (3.19) with π b1. That is max0 ≤ a ≤ 1 f A a u1 ðxÞg ¼ A a1 u1 ðxÞ ¼ 0. (ii) If x N b1, then u1(x) ≥ u1(b1) , u′1(x) = u′1(b1) = β1 and u″1(x) = u″1(b1) = 0. So, for each a ∈ [0, 1], we derive that A a u1 ðxÞ ¼

t∧T π pffiffiffi 0

  λμ 2 1−aπs dBs

Z t∧T π   π   π ~ e−δs u″ X πs d~Ls þ e−δs u0 X πs dG s 0 0 X   π  π  −δs þ e u X s −u X s− : s∈ΛπL ∪ΛπG ;s ≤ t∧T π t∧T π

ðA:1Þ

  1 0 ð1−aÞ2 λμ 22 u″1 ðxÞ þ θ1 −θ2 a2 λμ 22 u1 ðxÞ−δu1 ðxÞ 2   1 0 ¼ ð1−aÞ2 λμ 22 u″1 ðb1 Þ þ θ1 −θ2 a2 λμ 22 u1 ðb1 Þ−δu1 ðxÞ 2   1 0 ≤ ð1−aÞ2 λμ 22 u″1 ðb1 Þ þ θ1 −θ2 a2 λμ 22 u1 ðb1 Þ−δu1 ðb1 Þ 2 a ¼ A u1 ðb1 Þ ≤0:

• Step 2: To show β1 − u′1(x) ≤ 0. It can be established directly from the expression of u1(x) in Eq. (3.23). • Step 3: To show M u1(x) ≤ u1(x). We have that

The last term on the right hand side can be written as X ∪ ¼

s∈ΛπL

M u1 ðxÞ−u1 ðxÞ ¼ max fu1 ðx þ yÞ−β 2 y−Kg−u1 ðxÞ

     e−δs u X πs −u X πs−

ΛπG ;s ≤ t∧T π

X

     e−δs u X πs −u X πs− þ

s∈ΛπL ;s ≤ t∧T π

X

≤−

y≥0



s∈ΛπL ;s ≤ t∧T π

e−δs β1 Lπs −Lπs−



X s∈ΛπG ;s ≤ t∧T π

∞ X  π e−δτn β2 ηπn þ K Ifτπ ≤ t∧T π g ; þ n

ðA:2Þ where the inequality holds since u(x) satisfies the HJB Eq. (3.1) with u ' (x) ≥ β1 and M u(x) ≤ u(x). Moreover, in view of Eq. (3.1), the first term on the right hand side of Eq. (A.1) is non-positive. So substituting Eq. (A.2) into Eq. (A.1), we obtain   π e−δðt∧T Þ u X πt∧T π ≤uðxÞ þ Z −β1

t∧T π

−δs

e 0

dLπs

t∧T 0

π

∞ X  π þ e−δτn β2 ηπn þ K I fτπ ≤ t∧T π g :

ðA:3Þ

n

n¼1

Since u(x) is an increasing function with u(0)≥ P, we have Z t∧T π pffiffiffi     π e−δs λμ 2 1−aπs u0 X πs dBs e−δðt∧T Þ P ≤uðxÞ þ 0 Z t∧T π ∞ X  π −β1 e−δs Lπs þ e−δτn β2 ηπn þ K Ifτπ ≤ t∧T π g : 0

ðA:4Þ

n

n¼1

The stochastic integral with respect to the Brownian motion in Eq. (A.4) is a uniformly integrable martingale, if u′(x) is bounded. Taking expectation and limit on both sides of Eq. (A.4) yields Z uðxÞ ≥Ex ðβ 1

(i) If 0 ≤ x ≤ ξ1⁎, then u'1(x)−β2 ≥0 if and only if 0 ≤ x ≤ ξ1⁎. Therefore,

Z M u1 ðxÞ−u1 ðxÞ ¼ max f y≥0

0

∞ X

e−δs dDπs −

n¼1

 π π e−δτn β 2 ηπn þ K I fτπ ≤ T π g þ Pe−δT Þ ¼ U ðx; πÞ: n

ðA:5Þ

Z M u1 ðxÞ−u1 ðxÞ ¼ max f

□

Z



ξ1 0

ðu0 1 ðsÞ−β2 Þds−K ¼ 0;

xþy  x

 u01 ðsÞ−β2 dsg−K ¼ −K b 0:

• Step 4: To show maxf max0 ≤ a ≤ 1 f A a u1 ðxÞg; β1 − u10 ðxÞ; M u1 ðxÞ− u1 ðxÞg ¼ 0 (i) If x =0, by Eq. (3.22), it has M u1(x) − u1(x)= 0. (ii) If 0 ≤ x ≤ b1, by Eq. (3.19), it has max0 ≤ a ≤ 1 f A a u1 ðxÞg ¼ 0. (iii) If x N b1, by Eq. (3.20), it has β1 − u′1(x) = 0. • Step 5: Clearly, u1(x) = u(x + p⁎) N P is true since u′1(x) ≥ β1 and u1(0) = u(p⁎) N u(0) = P. Therefore, u1(x) satisfies Eqs. (3.1) and (3.2). Obviously, u1(x) is indeed a twice continuously differentiable, increasing and concave function and its derivative satisfies β1 ≤ u′1(x) ≤ u′1(0) b ∞. So u1(x) ≥ U(x) holds according to Theorem 3.1. Finally, let's verify the optimality of ⁎ strategy π⁎ = (aπ1 , Lπ⁎, G π⁎) ∈ Π described by Eqs. (3.35)–(3.38). Since 

A a1 u1 ðX πt Þ ¼ 0 for 0 ≤ X πt ⁎ ≤ b1, we have Z

t∧T π



−δs

e 0

Consequently, u(x) ≥ U(x) follows.

x

ðu0 1 ðsÞ−β2 Þdsg−K ≤

(ii) If ξ1⁎ b x b ∞, the inequality u'1(x) − β2 b 0 is always true. Then

π

Tπ

xþy

the equality holds if and only if x =0 and y =ξ1⁎.

y≥0

pffiffiffi     e−δs λμ 2 1−aπs u0 X πs dBs

xþy

y≥0

     e−δs u X πs −u X πs−

n¼1

Z

¼ max f∫ x ðu01 ðsÞ−β 2 Þdsg−K:

A

aπ1



Z   u1 X πs ds ¼

t∧T π 0



  π e−δs A a1 u1 X πs I f0 ≤ X π ≤ b1 g ds ¼ 0: s

ðB:1Þ

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G. Cheng, Y. Zhao / Economic Modelling 54 (2016) 522–536

Furthermore, Eqs. (3.36) and (3.37) indicate that       e−δs u1 X πs −u1 X πs−

X

∪ΛπG ;s ≤ t∧T π       X ¼ e−δs u1 X πs −u1 X πs− IfX π ¼b1 g

 s∈ΛπL



s∈ΛπL ;s ≤ t∧T π

þ

s



X

e−δs



s∈ΛπG ;s ≤ t∧T π

¼−



X



s∈ΛπL ;s ≤ t∧T π





     u1 X πs −u1 X πs− I fX π ¼0g s−

∞    X   π   e−δs β1 Lπs −Lπs− þ e−δτn β2 ηπn þ K Ifτπ ≤ t∧T π g : n

n¼1

ðB:2Þ Therefore, by replacing π , T π , u with π⁎ , T π = ∞ , u1 in Ito formula (A.1) and taking expectations, we have ⁎

h   i u1 ðxÞ ¼ Ex e−δt u1 X πt Z t ∞ X   π   þ Ex ðβ1 e−δs dLπs − e−δτn β2 ηπn þ K Ifτπ ≤ t g Þ: 0

n

n¼1

ðB:3Þ

Letting t→ ∞, the first term on the right hand side vanishes, then we obtain Z u1 ðxÞ ¼ Ex ðβ1

∞ 0



e−δs dLπs −

∞ X π  e−δτn β2 ηπn þ KÞIfτπ b∞g Þ ¼ U ðx; π Þ; n¼1

n

ðB:4Þ which, together with u1(x) ≥ U(x), establishes that u1(x) = U(x) = U(x; π⁎). □ References Asmussen, S., Taksar, M., 1997. Controlled diffusion models for optimal dividend payout. Ins. Math. Econ. 20 (1), 1–15. Barth, A., Moreno-Bromberg, S., 2014. Optimal risk and liquidity management with costly refinancing opportunities. Ins. Math. Econ. 57 (1), 31–45. Belhaj, M., 2010. Optimal dividend payments when cash reserves follow a jump-diffusion process. Math. Financ. 20 (2), 313–325. Cadenillas, A., Choulli, T., Taksar, M., Zhang, L., 2006. Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm. Math. Financ. 16 (1), 181–202.

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