# The Optimal Dividend Payout Model with Terminal Values and Its Application

Authors: Xiankang Luo; Peimin Chen; Jiangming Ma
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

For some firms with large nonliquid assets, preferred shareholders can still get back a little bit of money when the firms finish disbursement of loans at the status of bankruptcy. For such a situation, to investigate the optimal dividend policy, a stochastic dynamic dividend model with nonzero terminal bankruptcy values is put forward in this paper. Moreover, an analytic solution for the optimal objective function of the discounted dividends is provided and verified. An important application of this result is that it can be employed to construct the solution for the optimal value function on the dividend problem with bailouts at bankruptcy. Further, the relationship for the solutions of these two different problems is demonstrated. In the end, some numerical examples are provided to support our theoretical results and the corresponding economic interpretations are illustrated.


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## Body

## 1. Introduction

In the past decades, optimal dividend problems have been an important issue in financial and actuarial sciences. Its origin can be traced as early as the work of de Finetti [1], in which a discrete time model for optimal dividend was introduced. Recently, diffusion models for firms with controllable risk exposure and dividends payout have drawn increasing attention of researchers. We refer the readers to Jeanblanc-Picqué and Shiryaev [2], Radner and Shepp [3], Taksar and Zhou [4], Højgaard and Taksar [5, 6], Hubalek and Schachermayer [7], Cadenillas et al. [8], Paulsen [9, 10], Paulsen and Gjessing [11], Avanzi and Wong [12], Hunting and Paulsen [13], Chen et al. [14], Eisenberg [15], Vierkötter and Schmidli [16], and so forth. In all of those works, the terminal value of a company is assumed to be equal to zero when there is a status of bankruptcy, where the bankruptcy is defined as the time when the liquid assets of the company vanish. But in the real world, sometimes preferred shareholders can get some money back at bankruptcy if the value of the nonliquid assets (such as real estate or the rights to conduct business or the trade name) is large enough to pay for loans. This means that for this case the value function is not zero at the time of ruin. For such firms, how to manage the liquid assets and how to distribute dividends become a necessary problem for managers to consider. Unfortunately, there are very few results concerned with the terminal values (see, e.g., Taksar and Hunderup [17], Taksar [18], Xu and Zhou [19], and Chen and Li [20]). In addition, for firms at status of bankruptcy, sometimes they can get bailouts from governments or other firms with abundant cash flows. In such a situation, the optimal dividend policy is also an urgent problem for managers to analyze. In this paper, we put forward a dividend model with nonzero terminal values and then provide optimal dividend strategies for these two problems.In the real financial market, capital injections are an important approach for insurance company to maintain the business when cash flow is insufficient. Recently, the optimal dividend problem with capital injections becomes hot issue in the research field of insurance. In works of Kulenko and Schimidli [21], Løkka and Zervos [22], He and Liang [23, 24], Yao et al. [25], Li and Liu [26], and so on, they assume that firms can raise capitals by issuing new shares or bonds. That implies the firms are healthy in management and financial situation and they can attract much more capitals from externals to scale up their business. In this case, the event and amounts of capital injections can be controlled by firms, so the capital injections can be a controllable variable in the objective function of optimal dividends (see, e.g., Løkka and Zervos [22] and He and Liang [23, 24]). However, nearby bankruptcy investors are in panic for prospects and firms have lots of difficulties to persuade shareholders to pay for new shares so that shareholders seldom inject capitals. In addition, at bankruptcy bailouts are mainly composed of loans or purchasing agreements from governments. For instance, in the financial crisis of 2008, the bailouts are mainly from the purchasing programs of US governments (see https://projects.propublica.org/bailout/list). Also, bailouts from governments are determined by a complicated political process, which is beyond firms’ control. Therefore, in this situation it is unrealistic to treat bailouts as a controllable and internal variable for shareholders in the value function although bailouts need to be refunded in the future. In all, it is reasonable for us to view bailouts as an exogenous variable, which does not appear in objective function.Following the classical model, we posit that the manager maximizes the expected discounted value of cumulative dividends payout. The value function is defined from shareholder’s perspective; meanwhile, a residual terminal value is permitted when the firm goes bankrupt. Very closely related to the above is the problem with nonterminal bankruptcy, where a company reaches the bankruptcy state, and it does not go out of business but rather stays in this state a random amount of time and then resumes “business as usual.” As in Sethi and Taksar [27], a diffusion limit description of such behavior would be via Brownian motion with delayed reflection at zero. With this setup, we then transform the optimal cumulative dividend problem to be an equivalent Hamilton-Jacobi-Bellman equation with mixed Dirichlet-Neumann boundary condition. Further, a closed form solution and optimal dividend policy for the original problem are obtained.The main contribution of this paper is to develop a new dividend model with terminal values or bailouts, in which bankruptcy is not terminal. A novel result obtained is the concise necessary and sufficient condition for immediate dividend events after bailouts. This result presents a guild way for regulators. To deter dividend distributions in embarrassing situation, a policy maker can set the rate of capital injection lower than the maximum dividend rate. Numerical examples are provided to support the theoretical results of the model.The rest of this paper is organized as follows. In Section2, we propose a mathematical model with nonzero terminal bankruptcy values for the optimal dividend problem. In Section 3, the HJB equation corresponding to this problem is derived and the detailed structure of candidate solutions is given under different situations. In Section 4, a nonterminal bankruptcy model with bailouts is presented. Then based on the results in Section 3, its smooth solution and the optimal control strategies are obtained. Meanwhile, some numerical examples are provided to verify the theoretical model and the corresponding economic interpretations are illustrated. Finally, in Section 5, we summarize our main findings and suggest a direction for future research.


## 2. The Mathematical Model

Let(Ω,F,P) be a probability space endowed with a filtration {Ft}t≥0, and let {Wt}t≥0 be a standard Brownian motion adapted to that filtration. To understand motivation for our diffusion control problem, one can start with the classical Cramér-Lundberg model of an insurance company. Assume that claims arrive according to a Poisson process Nt with rate λ and the size of ith claim is Ui, where {Ui} are i.i.d. with mean μ^ and variance s2. The risk process representing the liquid assets of the company, also called reserve or surplus, is governed by (1)Xt=x+pt-∑i=1NtUi,where p is the amount of premium per unit time received by the insurance company and the initial reserve X0=x is supposed to be F0-measurable. As in Taksar [28], this process can be approximated by a diffusion process with a constant drift μ=p-λμ^ and diffusion coefficient σ=s2+μ^2. Thus, in the absence of control the reserve process Xt can be modelled as (2)dXt=μdt+σdWt,X0=x.For an insurance company, it often considers to do reinsurance to reduce its risk and to pay out dividends to show its bright prospect. Proportional reinsurance, which we will consider, consists of paying a certain fraction of the premiums to the reinsurance company in exchange for an obligation from the latter to pick up the same fraction of each claim. If1-a is the fraction of each claim picked up by the reinsurance company, then we call a the risk exposure of the cedent. When a is fixed, the reserve of the insurance company evolves as follows: (3)Xt=x+apt-∑i=1NtaUi.A diffusion approximation for the above process yields a Brownian motion with drift aμ and diffusion coefficient aσ, where μ and σ are the same as before. We describe a control policy π by a two-dimensional stochastic process {aπ(t),lπ(t)}, where 0≤aπ(t)≤1 corresponds to the risk exposure and 0≤lπ(t)≤M(M>0) is the restricted dividend rate paid out to the shareholders at time t. Consequently, the dynamics of the reserve process under this policy are given by(4)dXtπ=μaπt-lπtdt+σaπtdWt,(5)X0π=x.The set of all admissible policies is denoted by Π. For a given admissible policy π∈Π, the corresponding value function VPπ(x) is defined as(6)VPπx≔Ex∫0τπe-δtlπtdt+e-δτπP,where τπ=inf{t≥0;Xtπ=0} is the time of bankruptcy, δ>0 denotes a discount rate, and P is the residual values left to shareholders when the company goes bankrupt. Generally, once bankruptcy occurs, shareholders have very few opportunities to get imbursements. However, in the case of some companies with large nonliquid assets, sometimes shareholders with preferred rights can still have a finger in the pie and obtain some returned money. So in this paper, we assume that P≥0.For givenx, the objective is to find(7)VPx=supπ∈ΠVPπx.That is, we wish to find an admissible policy so as to maximize the expected present value of the cumulative dividend payouts and bankruptcy amounts shareholders achieve. The function defined by (7) is called the optimal value function, and the policy π∗, which satisfies VP(x)=VPπ∗(x), is termed the optimal policy.The following gives a characterization of the value functionVP(x).Proposition 1.
The functionVP(x) defined by (7) is concave in [0,∞).Proof.
Letπx be an admissible policy for the initial reserve x and πy for the initial reserve y. Take 0<k<1 and define πz by (8)aπzt=kaπxt+1-kaπyt,(9)lπzt=klπxt+1-klπyt.Then, by linearity of (4), πz is an admissible policy for the initial reserve z=kx+(1-k)y and Xtπz=kXtπx+(1-k)Xtπy with τπz=τπx∨τπy. The linearity of (6) and (9) implies (10)VPπzz=kVPπxx+1-kVPπyy.For any ε>0, we can choose πx and πy such that (11)VPπxx≥VPx-ε,VPπyy≥VPy-ε.As πz is suboptimal, it follows that (12)VPz≥kVPx+1-kVPy-ε.From the arbitrariness of ε, we conclude that VP(x) is concave.


## 3. The Hamilton-Jacobi-Bellman Equation and Its Solution

In this section, we firstly derive the Hamilton-Jacobi-Bellman equation satisfied byVP(x), and then discuss how to find its analytic solutions.For any functionφ(x), we define a differential operator La,l on it by (13)La,lφx≔12σ2a2φ′′x+μa-lφ′x-δφx,0≤a≤1,0≤l≤M.Proposition 2.
Assume that the functionVP(x) defined by (7) is twice continuous differential on [0,∞); then it satisfies the Hamilton-Jacobi-Bellman (HJB) equation(14)maxa∈0,1,l∈0,M12σ2a2VP′′x+μa-lVP′x-δVPx+l=0,with the boundary condition(15)VP0=P.Proof.
By the similar arguments of Højgaard and Taksar [29], the HJB equation (14) can be derived. To avoid tedious repetition, here we omit it. In addition, at x=0, it means that a firm is at the state of bankruptcy. Under no arbitrage theorem in markets, the summation of future dividends should be equal to the residual values of the firm at bankruptcy. Thus, it follows that VP(0)=P.


### 3.1. Constructing a Solution to the HJB Equation

In this section, we construct a solutionf(x) of (14) and (15), where for convenience VP(x) is replaced by f(x). Let x1≔inf{x:f′(x)=1}. Then, by concavity of f(x), we have (16)lx=0,x<x1,M,x>x1.Therefore, for all x<x1, the HJB equation (14) becomes(17)maxa∈0,112σ2a2f′′x+μaf′x-δfx=0.Let a(x) be the maximizer of the expression on the left-hand side of (17); then(18)ax=-μf′xσ2f′′x.Put (18) into (17); we obtain(19)-μ2f′x22σ2f′′x-δfx=0.A general solution of (19) with the boundary condition f(0)=P is given by(20)fx=C1x+PC11/γγ,where C1 is a free constant and γ is given by(21)γ=δδ+μ2/2σ2.This solution, however, is valid only when a(x)∈(0,1). From (18) and (20), it follows that(22)ax=μ1-γσ2x+PC11/γ.Since a(x) above is an increasing linear function of x, then a(x)≤1 if and only if x≤x0, where(23)x0=1-γσ2μ-PC11/γ.Comparingx0 and x1, there are two possible cases to be considered: x1≥x0 and x1<x0, whose necessary and sufficient conditions will be given later.


#### 3.1.1. The Case:x1≥x0

From (23), we notice that x0 depends on the uncertain parameter C1, which would lead to two different cases for x0. So, in the following we consider two cases, x0>0 and x0≤0, and discuss them, respectively.Ifx0>0, then a(x)≥1 for x0≤x<x1. But as a(x)∈[0,1], we must have a(x)=1 for x∈[x0,x1). Consequently, (17) becomes(24)12σ2f′′x+μf′x-δfx=0.A general solution to (24) is given by (25)fx=C2eθ+x-x0+C3eθ-x-x0,x0≤x<x1,where θ+ and θ- are defined by(26)θ+=-μ+μ2+2δσ2σ2,θ-=-μ-μ2+2δσ2σ2,and C2, C3 are free constants.The continuity of the functionf(x) and its derivative f′(x) at x0 implies that (27)C2=C1a1,C3=C1a2,where(28)a1=βγ-1γ-θ-βθ+-θ-,a2=βγ-1θ+β-γθ+-θ-,and β=(1-γ)σ2/μ.Forx≥x1, we have a(x)=1 and l(x)=M. Thus, f(x) satisfies the following equation:(29)12σ2f′′x+μ-Mf′x-δfx+M=0.Concavity of f(x) implies that f(x)~ax+b when x→∞ with a,b>0. Thereby, a solution of (29) is given by (30)fx=Mδ+C4eθ^x-x1,x≥x1,where(31)θ^=-μ-M-μ-M2+2δσ2σ2.Summarizing the above, we get the structure of the solution to (14) and (15) for x0>0(32)fx=C1x+PC11/γγ,0≤x<x0,C1a1eθ+x-x0+C1a2eθ-x-x0,x0≤x<x1,Mδ+C4eθ^x-x1,x≥x1,where C1, C4, and x1 are unknown constants and x0 are determined by (23).By continuity off′(x) and f′′(x) at x1, it implies that (33)C4θ^=1,(34)C1a1θ+eθ+x1-x0+C1a2θ-eθ-x1-x0=1,(35)C1a1θ+2eθ+x1-x0+C1a2θ-2eθ-x1-x0=θ^,where we have used (33) in (35).In view of (33), we get(36)C4=1θ^.Dividing (35) by (34), we eliminate C1 and solve for x1, thereby obtaining(37)x1=x0+1θ+-θ-ln⁡a2θ-θ^-θ-a1θ+θ+-θ^.To simplify this expression, letα=μ2+2δσ2. Then θ+θ-=-2δ/σ2, γ=2δσ2/α2, and β=μσ2/α2. This yields (38)γ-θ-β=2δσ2-μσ2θ-α2=μ2+2δσ2+μαα2=α2+μαα2=1+μα,(39)a1=σ2βγ-11+μ/α2α=βγμ+α2μ=σ2βγ2μ-θ-.By the same argument as above, we have(40)a2=σ2βγ2μ-θ+.Substituting (39) and (40) into (37), we get(41)x1=x0+1θ+-θ-ln⁡θ^-θ-θ+-θ^.By (34), we easily obtain(42)C1=a1θ+eθ+x1-x0+a2θ-eθ-x1-x0-1=a1θ+θ^-θ-θ+-θ^θ+/θ+-θ-+a2θ-θ^-θ-θ+-θ^θ-/θ+-θ--1=μβ-γδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1.Consequently, all free constants have been determined. In addition, as in Lemma2.1 of Højgaard and Taksar [29], it shows that x1≥x0 if and only if M≥μ/2+δσ2/μ.Remark 3.
From (23), it easily shows that the assumption, x0>0, is equivalent to (43)P<C1βγ;that is,(44)P<μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1.On the other hand, for the case ofx0≤0, namely, (45)P≥μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1,we also need to discuss the following two cases by x1>0 and x1≤0, respectively.Ifx0≤0 and x1>0, it implies that a(x)≡1 for all x>0. Then, the HJB equation (14) becomes(46)12σ2f′′x+μf′x-δfx=0,0≤x<x1,12σ2f′′x+μ-Mf′x-δfx+M=0,x≥x1.Using the similar argument as above, we obtain a general solution of (46) as follows: (47)fx=C2eθ+x+C3eθ-x,0≤x<x1,Mδ+C4eθ^x-x1,x≥x1,where θ± is given by (26), θ^ by (31), and C2, C3, C4, and x1 need to be determined.The continuity off′(x) and f′′ at x1, together with f(0)=P, implies that(48)C2+C3=P,(49)C4θ^=1,(50)C2θ+eθ+x1+C3θ-eθ-x1=1,(51)C2θ+2eθ+x1+C3θ-2eθ-x1=θ^.By (49), we get(52)C4=1θ^.Combining (50) and (51), for given x1 we can solve for C2 and C3(53)C2=θ^-θ-θ+θ+-θ-e-θ+x1,C3=θ+-θ^θ-θ+-θ-e-θ-x1.Put (53) into (48); we obtain the following equation, where x1 must satisfy(54)θ^-θ-θ+θ+-θ-e-θ+x1+θ+-θ^θ-θ+-θ-e-θ-x1=P.By continuity of f(x) at x1, (52) and (53), it follows that(55)C2eθ+x1+C3eθ-x1=θ^-θ-θ+θ+-θ-+θ+-θ^θ-θ+-θ-=Mδ+1θ^.Remark 4.
Letgx=θ^-θ-/θ+θ+-θ-e-θ+x1+(θ+-θ^/θ-θ+-θ-)e-θ-x1. Under the condition of M≥μ/2+δσ2/μ, it is easily verified that g′(x)<0 on [0,∞). Thus, maxx∈[0,∞)g(x)=g(0)=(θ^-θ-)/θ+(θ+-θ-)+(θ+-θ^)/θ-(θ+-θ-)=M/δ+1/θ^ owing to (55). Suppose that P satisfies(56)μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1≤P<Mδ+1θ^;then by intermediate value theorem, it shows that (54) has a unique positive root x1. Thereby, the values of C2 and C3 can be determined certainly.Secondly, ifx0≤0 and x1≤0, that is,(57)P≥Mδ+1θ^,we get (29) and find as before a solution (58)fx=Mδ+C4eθ^x,x≥0,where C4=P-M/δ.Summarizing the above, in the case ofM≥μ/2+δσ2/μ, the solutions of (14) and (15) are presented by the following theorem.Theorem 5.
Assume thatM≥μ/2+δσ2/μ. Let θ+, θ-, θ^, and γ be defined as above.
If (44) holds, then the solution to (14) and (15) is given by(59)fx=C1x+PC11/γγ,0≤x<x0,C1a1eθ+x-x0+C1a2eθ-x-x0,x0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where C1 is given by (42), x0 by (23), and x1 by (41).
If (56) is true, then(60)fx=C2eθ+x+P-C2eθ-x,0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where C2 is given by (53) and x1 is determined by (54).
If (57) is true, then(61)fx=Mδ+P-Mδeθ^x,x≥0.Moreover, in any of the cases described above, f(x) is a concave solution of (14) and (15).Proof.
From the construction above, it is easily to verify that the expressions (59)–(61) are solutions to (14) and (15). In addition, the proof on concavity of f(x) is similar to that of Theorem 2.1 in Højgaard and Taksar [29], so we omit it here.


#### 3.1.2. The Case:x1<x0

Now we turn to the casex1<x0, which is equivalent to M<μ/2+δσ2/μ. From Højgaard and Taksar [29], it shows that there exists only one “switching point” x1 because of x0=∞. Therefore, the solution of (14) and (15) depends on whether x1>0 or x1≤0.Ifx1>0, for x<x1, we obtain (17) and find as before a solution as follows: (62)fx=C1x+PC11/γγ,where γ is given by (21) and C1 is a free constant.Forx≥x1, we have l(x)=M. Thus, the HJB equation (14) becomes(63)maxa∈0,112σ2a2f′′x+μa-Mf′x-δfx+M=0.Let f∗(x) be a solution of the following homogeneous equation:(64)maxa∈0,112σ2a2f′′x+μa-Mf′x-δfx=0.Then, a solution of (63) can be expressed by(65)fx=f∗x+Mδ.Differentiating (64) with respect to a yields the maximizer(66)ax=-μf′xσ2f′′x.Substituting (66) into (64), we obtain(67)-μ2f′x22σ2f′′x-Mf′x-δfx=0.A general solution of (67) can be given by(68)f∗x=C2eρx-x1,x≥x1,where C2(C2≠0) and ρ are free constants.Input (68) into (67); we obtain (69)C2-μ22σ2-Mρ-δeρx-x1=0.Then, it is easy to derive that(70)ρ=-δMγ,where γ is defined by (21).Thus, combining (65), (68), and (70), a solution of (63) is given by (71)fx=C2e-δ/Mγx-x1+Mδ,x≥x1.Consequently, for the case ofx1>0, the following solution to (14) and (15) is suggested (72)fx=C1x+PC11/γγ,0≤x<x1,C2e-δ/Mγx-x1+Mδ,x≥x1,where C1, C2, and x1 need to be specified.Making “smooth fit” atx1, we have the following equations:(73)-C2δMγ=1,(74)C1γx1+PC11/γγ-1=1,(75)C1γγ-1x1+PC11/γγ-2=-δMγ,where we have used (73) in (75).From (73), we obtain (76)C2=-Mγδ.Dividing (75) by (74) yields that(77)x1+PC11/γ=Mγ1-γδ.Put (77) into (74); we get(78)C1=1γMγ1-γδ1-γ.Combining (77) with (78), it follows that(79)x1=Mγ1-γδ1-δPM1-γ1/γ.In addition, by (79), it shows that x1>0 is equivalent to(80)P<M1-γδ.On the other hand, ifx1≤0, that is,(81)P≥M1-γδ,we get (63) and find as above a solution as follows: (82)fx=C2e-δ/Mγx+Mδ,x≥0.By the boundary condition f(0)=P, we have(83)fx=P-Mδe-δ/Mγx+Mδ,x≥0.Sincea(x)<1 for all x∈[0,∞), the following theorem follows easily from construction.Theorem 6.
Assume thatM<μ/2+δσ2/μ. Let γ be defined by (21).
If (80) is true, then the solution to (14) and (15) is given by(84)fx=C1x+PC11/γγ,0≤x<x1,Mδ1-γe-δ/Mγx-x1,x≥x1,where C1 is given by (78) and x1 by (79).
On the other hand, if (81) holds, then(85)fx=P-Mδe-δ/Mγx+Mδ,x≥0.Moreover, the functions f(x) defined as above are concave solutions of (14) and (15).


### 3.2. Optimal Policy

Based on sections above, in the following, we present our optimal policy. The admissible policyπ∗ for all t<τπ∗ is defined as follows.Case 1 (M≥μ/2+δσ2/μ).
(
1
) If (44) holds, then (86)aπ∗t≔aXtπ∗=μ1-γσ2Xtπ∗+PC11/γ,0≤Xtπ∗<x0,1,Xtπ∗≥x0,lπ∗t≔lXtπ∗=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x0 is given by (23) and x1 by (41).
(
2
) If (56) is true, then (87)aπ∗t≔aXtπ∗=1,Xtπ∗≥0,lπ∗t≔lXtπ∗=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x1 is given by (54).
(
3
) If (57) holds, then (88)aπ∗t≔aXtπ∗=1,Xtπ∗≥0,lπ∗t≔lXtπ∗=M,Xtπ∗≥0.Case 2 (M<μ/2+δσ2/μ).
(
1
) If (80) is true, then (89)aπ∗t≔aXtπ∗=μ1-γσ2Xtπ∗+PC11/γ,0≤Xtπ∗<x1,μMγδσ2,Xtπ∗≥x1,lπ∗t≔lXtπ∗=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x1 is given by (79).
(
2
) If (81) holds, then (90)aπ∗t≔aXtπ∗=μMγδσ2=2μMμ2+2δσ2,Xtπ∗≥0,lπ∗t≔lXtπ∗=M,Xtπ∗≥0.
Moreover, in the case ofM<μ/2+δσ2/μ, we have a(Xtπ∗)<1 for all Xtπ∗≥0.
In addition,Xtπ∗ is the solution of (91)Xtπ∗=X0π∗+∫0tμaπ∗s-lπ∗sds+∫0tσaπ∗sdWs.The following theorem shows that the solutionf(x) of (14) and (15) constructed above is the optimal value function.Theorem 7.
LetVP(x) be given by (7), and let f(x) be given by (59)–(61) for M≥μ/2+δσ2/μ and by (84)-(85) for M<μ/2+δσ2/μ. Then (92)VPx=fx.Proof.
The proof of this theorem can be carried out in the same way as that of Theorem2.3 in Højgaard and Taksar [29]. To avoid tedious repetition, we omit it here.


## 3.1. Constructing a Solution to the HJB Equation

In this section, we construct a solutionf(x) of (14) and (15), where for convenience VP(x) is replaced by f(x). Let x1≔inf{x:f′(x)=1}. Then, by concavity of f(x), we have (16)lx=0,x<x1,M,x>x1.Therefore, for all x<x1, the HJB equation (14) becomes(17)maxa∈0,112σ2a2f′′x+μaf′x-δfx=0.Let a(x) be the maximizer of the expression on the left-hand side of (17); then(18)ax=-μf′xσ2f′′x.Put (18) into (17); we obtain(19)-μ2f′x22σ2f′′x-δfx=0.A general solution of (19) with the boundary condition f(0)=P is given by(20)fx=C1x+PC11/γγ,where C1 is a free constant and γ is given by(21)γ=δδ+μ2/2σ2.This solution, however, is valid only when a(x)∈(0,1). From (18) and (20), it follows that(22)ax=μ1-γσ2x+PC11/γ.Since a(x) above is an increasing linear function of x, then a(x)≤1 if and only if x≤x0, where(23)x0=1-γσ2μ-PC11/γ.Comparingx0 and x1, there are two possible cases to be considered: x1≥x0 and x1<x0, whose necessary and sufficient conditions will be given later.


### 3.1.1. The Case:x1≥x0

From (23), we notice that x0 depends on the uncertain parameter C1, which would lead to two different cases for x0. So, in the following we consider two cases, x0>0 and x0≤0, and discuss them, respectively.Ifx0>0, then a(x)≥1 for x0≤x<x1. But as a(x)∈[0,1], we must have a(x)=1 for x∈[x0,x1). Consequently, (17) becomes(24)12σ2f′′x+μf′x-δfx=0.A general solution to (24) is given by (25)fx=C2eθ+x-x0+C3eθ-x-x0,x0≤x<x1,where θ+ and θ- are defined by(26)θ+=-μ+μ2+2δσ2σ2,θ-=-μ-μ2+2δσ2σ2,and C2, C3 are free constants.The continuity of the functionf(x) and its derivative f′(x) at x0 implies that (27)C2=C1a1,C3=C1a2,where(28)a1=βγ-1γ-θ-βθ+-θ-,a2=βγ-1θ+β-γθ+-θ-,and β=(1-γ)σ2/μ.Forx≥x1, we have a(x)=1 and l(x)=M. Thus, f(x) satisfies the following equation:(29)12σ2f′′x+μ-Mf′x-δfx+M=0.Concavity of f(x) implies that f(x)~ax+b when x→∞ with a,b>0. Thereby, a solution of (29) is given by (30)fx=Mδ+C4eθ^x-x1,x≥x1,where(31)θ^=-μ-M-μ-M2+2δσ2σ2.Summarizing the above, we get the structure of the solution to (14) and (15) for x0>0(32)fx=C1x+PC11/γγ,0≤x<x0,C1a1eθ+x-x0+C1a2eθ-x-x0,x0≤x<x1,Mδ+C4eθ^x-x1,x≥x1,where C1, C4, and x1 are unknown constants and x0 are determined by (23).By continuity off′(x) and f′′(x) at x1, it implies that (33)C4θ^=1,(34)C1a1θ+eθ+x1-x0+C1a2θ-eθ-x1-x0=1,(35)C1a1θ+2eθ+x1-x0+C1a2θ-2eθ-x1-x0=θ^,where we have used (33) in (35).In view of (33), we get(36)C4=1θ^.Dividing (35) by (34), we eliminate C1 and solve for x1, thereby obtaining(37)x1=x0+1θ+-θ-ln⁡a2θ-θ^-θ-a1θ+θ+-θ^.To simplify this expression, letα=μ2+2δσ2. Then θ+θ-=-2δ/σ2, γ=2δσ2/α2, and β=μσ2/α2. This yields (38)γ-θ-β=2δσ2-μσ2θ-α2=μ2+2δσ2+μαα2=α2+μαα2=1+μα,(39)a1=σ2βγ-11+μ/α2α=βγμ+α2μ=σ2βγ2μ-θ-.By the same argument as above, we have(40)a2=σ2βγ2μ-θ+.Substituting (39) and (40) into (37), we get(41)x1=x0+1θ+-θ-ln⁡θ^-θ-θ+-θ^.By (34), we easily obtain(42)C1=a1θ+eθ+x1-x0+a2θ-eθ-x1-x0-1=a1θ+θ^-θ-θ+-θ^θ+/θ+-θ-+a2θ-θ^-θ-θ+-θ^θ-/θ+-θ--1=μβ-γδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1.Consequently, all free constants have been determined. In addition, as in Lemma2.1 of Højgaard and Taksar [29], it shows that x1≥x0 if and only if M≥μ/2+δσ2/μ.Remark 3.
From (23), it easily shows that the assumption, x0>0, is equivalent to (43)P<C1βγ;that is,(44)P<μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1.On the other hand, for the case ofx0≤0, namely, (45)P≥μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1,we also need to discuss the following two cases by x1>0 and x1≤0, respectively.Ifx0≤0 and x1>0, it implies that a(x)≡1 for all x>0. Then, the HJB equation (14) becomes(46)12σ2f′′x+μf′x-δfx=0,0≤x<x1,12σ2f′′x+μ-Mf′x-δfx+M=0,x≥x1.Using the similar argument as above, we obtain a general solution of (46) as follows: (47)fx=C2eθ+x+C3eθ-x,0≤x<x1,Mδ+C4eθ^x-x1,x≥x1,where θ± is given by (26), θ^ by (31), and C2, C3, C4, and x1 need to be determined.The continuity off′(x) and f′′ at x1, together with f(0)=P, implies that(48)C2+C3=P,(49)C4θ^=1,(50)C2θ+eθ+x1+C3θ-eθ-x1=1,(51)C2θ+2eθ+x1+C3θ-2eθ-x1=θ^.By (49), we get(52)C4=1θ^.Combining (50) and (51), for given x1 we can solve for C2 and C3(53)C2=θ^-θ-θ+θ+-θ-e-θ+x1,C3=θ+-θ^θ-θ+-θ-e-θ-x1.Put (53) into (48); we obtain the following equation, where x1 must satisfy(54)θ^-θ-θ+θ+-θ-e-θ+x1+θ+-θ^θ-θ+-θ-e-θ-x1=P.By continuity of f(x) at x1, (52) and (53), it follows that(55)C2eθ+x1+C3eθ-x1=θ^-θ-θ+θ+-θ-+θ+-θ^θ-θ+-θ-=Mδ+1θ^.Remark 4.
Letgx=θ^-θ-/θ+θ+-θ-e-θ+x1+(θ+-θ^/θ-θ+-θ-)e-θ-x1. Under the condition of M≥μ/2+δσ2/μ, it is easily verified that g′(x)<0 on [0,∞). Thus, maxx∈[0,∞)g(x)=g(0)=(θ^-θ-)/θ+(θ+-θ-)+(θ+-θ^)/θ-(θ+-θ-)=M/δ+1/θ^ owing to (55). Suppose that P satisfies(56)μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1≤P<Mδ+1θ^;then by intermediate value theorem, it shows that (54) has a unique positive root x1. Thereby, the values of C2 and C3 can be determined certainly.Secondly, ifx0≤0 and x1≤0, that is,(57)P≥Mδ+1θ^,we get (29) and find as before a solution (58)fx=Mδ+C4eθ^x,x≥0,where C4=P-M/δ.Summarizing the above, in the case ofM≥μ/2+δσ2/μ, the solutions of (14) and (15) are presented by the following theorem.Theorem 5.
Assume thatM≥μ/2+δσ2/μ. Let θ+, θ-, θ^, and γ be defined as above.
If (44) holds, then the solution to (14) and (15) is given by(59)fx=C1x+PC11/γγ,0≤x<x0,C1a1eθ+x-x0+C1a2eθ-x-x0,x0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where C1 is given by (42), x0 by (23), and x1 by (41).
If (56) is true, then(60)fx=C2eθ+x+P-C2eθ-x,0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where C2 is given by (53) and x1 is determined by (54).
If (57) is true, then(61)fx=Mδ+P-Mδeθ^x,x≥0.Moreover, in any of the cases described above, f(x) is a concave solution of (14) and (15).Proof.
From the construction above, it is easily to verify that the expressions (59)–(61) are solutions to (14) and (15). In addition, the proof on concavity of f(x) is similar to that of Theorem 2.1 in Højgaard and Taksar [29], so we omit it here.


### 3.1.2. The Case:x1<x0

Now we turn to the casex1<x0, which is equivalent to M<μ/2+δσ2/μ. From Højgaard and Taksar [29], it shows that there exists only one “switching point” x1 because of x0=∞. Therefore, the solution of (14) and (15) depends on whether x1>0 or x1≤0.Ifx1>0, for x<x1, we obtain (17) and find as before a solution as follows: (62)fx=C1x+PC11/γγ,where γ is given by (21) and C1 is a free constant.Forx≥x1, we have l(x)=M. Thus, the HJB equation (14) becomes(63)maxa∈0,112σ2a2f′′x+μa-Mf′x-δfx+M=0.Let f∗(x) be a solution of the following homogeneous equation:(64)maxa∈0,112σ2a2f′′x+μa-Mf′x-δfx=0.Then, a solution of (63) can be expressed by(65)fx=f∗x+Mδ.Differentiating (64) with respect to a yields the maximizer(66)ax=-μf′xσ2f′′x.Substituting (66) into (64), we obtain(67)-μ2f′x22σ2f′′x-Mf′x-δfx=0.A general solution of (67) can be given by(68)f∗x=C2eρx-x1,x≥x1,where C2(C2≠0) and ρ are free constants.Input (68) into (67); we obtain (69)C2-μ22σ2-Mρ-δeρx-x1=0.Then, it is easy to derive that(70)ρ=-δMγ,where γ is defined by (21).Thus, combining (65), (68), and (70), a solution of (63) is given by (71)fx=C2e-δ/Mγx-x1+Mδ,x≥x1.Consequently, for the case ofx1>0, the following solution to (14) and (15) is suggested (72)fx=C1x+PC11/γγ,0≤x<x1,C2e-δ/Mγx-x1+Mδ,x≥x1,where C1, C2, and x1 need to be specified.Making “smooth fit” atx1, we have the following equations:(73)-C2δMγ=1,(74)C1γx1+PC11/γγ-1=1,(75)C1γγ-1x1+PC11/γγ-2=-δMγ,where we have used (73) in (75).From (73), we obtain (76)C2=-Mγδ.Dividing (75) by (74) yields that(77)x1+PC11/γ=Mγ1-γδ.Put (77) into (74); we get(78)C1=1γMγ1-γδ1-γ.Combining (77) with (78), it follows that(79)x1=Mγ1-γδ1-δPM1-γ1/γ.In addition, by (79), it shows that x1>0 is equivalent to(80)P<M1-γδ.On the other hand, ifx1≤0, that is,(81)P≥M1-γδ,we get (63) and find as above a solution as follows: (82)fx=C2e-δ/Mγx+Mδ,x≥0.By the boundary condition f(0)=P, we have(83)fx=P-Mδe-δ/Mγx+Mδ,x≥0.Sincea(x)<1 for all x∈[0,∞), the following theorem follows easily from construction.Theorem 6.
Assume thatM<μ/2+δσ2/μ. Let γ be defined by (21).
If (80) is true, then the solution to (14) and (15) is given by(84)fx=C1x+PC11/γγ,0≤x<x1,Mδ1-γe-δ/Mγx-x1,x≥x1,where C1 is given by (78) and x1 by (79).
On the other hand, if (81) holds, then(85)fx=P-Mδe-δ/Mγx+Mδ,x≥0.Moreover, the functions f(x) defined as above are concave solutions of (14) and (15).


## 3.1.1. The Case:x1≥x0

From (23), we notice that x0 depends on the uncertain parameter C1, which would lead to two different cases for x0. So, in the following we consider two cases, x0>0 and x0≤0, and discuss them, respectively.Ifx0>0, then a(x)≥1 for x0≤x<x1. But as a(x)∈[0,1], we must have a(x)=1 for x∈[x0,x1). Consequently, (17) becomes(24)12σ2f′′x+μf′x-δfx=0.A general solution to (24) is given by (25)fx=C2eθ+x-x0+C3eθ-x-x0,x0≤x<x1,where θ+ and θ- are defined by(26)θ+=-μ+μ2+2δσ2σ2,θ-=-μ-μ2+2δσ2σ2,and C2, C3 are free constants.The continuity of the functionf(x) and its derivative f′(x) at x0 implies that (27)C2=C1a1,C3=C1a2,where(28)a1=βγ-1γ-θ-βθ+-θ-,a2=βγ-1θ+β-γθ+-θ-,and β=(1-γ)σ2/μ.Forx≥x1, we have a(x)=1 and l(x)=M. Thus, f(x) satisfies the following equation:(29)12σ2f′′x+μ-Mf′x-δfx+M=0.Concavity of f(x) implies that f(x)~ax+b when x→∞ with a,b>0. Thereby, a solution of (29) is given by (30)fx=Mδ+C4eθ^x-x1,x≥x1,where(31)θ^=-μ-M-μ-M2+2δσ2σ2.Summarizing the above, we get the structure of the solution to (14) and (15) for x0>0(32)fx=C1x+PC11/γγ,0≤x<x0,C1a1eθ+x-x0+C1a2eθ-x-x0,x0≤x<x1,Mδ+C4eθ^x-x1,x≥x1,where C1, C4, and x1 are unknown constants and x0 are determined by (23).By continuity off′(x) and f′′(x) at x1, it implies that (33)C4θ^=1,(34)C1a1θ+eθ+x1-x0+C1a2θ-eθ-x1-x0=1,(35)C1a1θ+2eθ+x1-x0+C1a2θ-2eθ-x1-x0=θ^,where we have used (33) in (35).In view of (33), we get(36)C4=1θ^.Dividing (35) by (34), we eliminate C1 and solve for x1, thereby obtaining(37)x1=x0+1θ+-θ-ln⁡a2θ-θ^-θ-a1θ+θ+-θ^.To simplify this expression, letα=μ2+2δσ2. Then θ+θ-=-2δ/σ2, γ=2δσ2/α2, and β=μσ2/α2. This yields (38)γ-θ-β=2δσ2-μσ2θ-α2=μ2+2δσ2+μαα2=α2+μαα2=1+μα,(39)a1=σ2βγ-11+μ/α2α=βγμ+α2μ=σ2βγ2μ-θ-.By the same argument as above, we have(40)a2=σ2βγ2μ-θ+.Substituting (39) and (40) into (37), we get(41)x1=x0+1θ+-θ-ln⁡θ^-θ-θ+-θ^.By (34), we easily obtain(42)C1=a1θ+eθ+x1-x0+a2θ-eθ-x1-x0-1=a1θ+θ^-θ-θ+-θ^θ+/θ+-θ-+a2θ-θ^-θ-θ+-θ^θ-/θ+-θ--1=μβ-γδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1.Consequently, all free constants have been determined. In addition, as in Lemma2.1 of Højgaard and Taksar [29], it shows that x1≥x0 if and only if M≥μ/2+δσ2/μ.Remark 3.
From (23), it easily shows that the assumption, x0>0, is equivalent to (43)P<C1βγ;that is,(44)P<μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1.On the other hand, for the case ofx0≤0, namely, (45)P≥μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1,we also need to discuss the following two cases by x1>0 and x1≤0, respectively.Ifx0≤0 and x1>0, it implies that a(x)≡1 for all x>0. Then, the HJB equation (14) becomes(46)12σ2f′′x+μf′x-δfx=0,0≤x<x1,12σ2f′′x+μ-Mf′x-δfx+M=0,x≥x1.Using the similar argument as above, we obtain a general solution of (46) as follows: (47)fx=C2eθ+x+C3eθ-x,0≤x<x1,Mδ+C4eθ^x-x1,x≥x1,where θ± is given by (26), θ^ by (31), and C2, C3, C4, and x1 need to be determined.The continuity off′(x) and f′′ at x1, together with f(0)=P, implies that(48)C2+C3=P,(49)C4θ^=1,(50)C2θ+eθ+x1+C3θ-eθ-x1=1,(51)C2θ+2eθ+x1+C3θ-2eθ-x1=θ^.By (49), we get(52)C4=1θ^.Combining (50) and (51), for given x1 we can solve for C2 and C3(53)C2=θ^-θ-θ+θ+-θ-e-θ+x1,C3=θ+-θ^θ-θ+-θ-e-θ-x1.Put (53) into (48); we obtain the following equation, where x1 must satisfy(54)θ^-θ-θ+θ+-θ-e-θ+x1+θ+-θ^θ-θ+-θ-e-θ-x1=P.By continuity of f(x) at x1, (52) and (53), it follows that(55)C2eθ+x1+C3eθ-x1=θ^-θ-θ+θ+-θ-+θ+-θ^θ-θ+-θ-=Mδ+1θ^.Remark 4.
Letgx=θ^-θ-/θ+θ+-θ-e-θ+x1+(θ+-θ^/θ-θ+-θ-)e-θ-x1. Under the condition of M≥μ/2+δσ2/μ, it is easily verified that g′(x)<0 on [0,∞). Thus, maxx∈[0,∞)g(x)=g(0)=(θ^-θ-)/θ+(θ+-θ-)+(θ+-θ^)/θ-(θ+-θ-)=M/δ+1/θ^ owing to (55). Suppose that P satisfies(56)μδθ^-θ-θ+-θ^θ+/θ+-θ-+θ^-θ-θ+-θ^θ-/θ+-θ--1≤P<Mδ+1θ^;then by intermediate value theorem, it shows that (54) has a unique positive root x1. Thereby, the values of C2 and C3 can be determined certainly.Secondly, ifx0≤0 and x1≤0, that is,(57)P≥Mδ+1θ^,we get (29) and find as before a solution (58)fx=Mδ+C4eθ^x,x≥0,where C4=P-M/δ.Summarizing the above, in the case ofM≥μ/2+δσ2/μ, the solutions of (14) and (15) are presented by the following theorem.Theorem 5.
Assume thatM≥μ/2+δσ2/μ. Let θ+, θ-, θ^, and γ be defined as above.
If (44) holds, then the solution to (14) and (15) is given by(59)fx=C1x+PC11/γγ,0≤x<x0,C1a1eθ+x-x0+C1a2eθ-x-x0,x0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where C1 is given by (42), x0 by (23), and x1 by (41).
If (56) is true, then(60)fx=C2eθ+x+P-C2eθ-x,0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where C2 is given by (53) and x1 is determined by (54).
If (57) is true, then(61)fx=Mδ+P-Mδeθ^x,x≥0.Moreover, in any of the cases described above, f(x) is a concave solution of (14) and (15).Proof.
From the construction above, it is easily to verify that the expressions (59)–(61) are solutions to (14) and (15). In addition, the proof on concavity of f(x) is similar to that of Theorem 2.1 in Højgaard and Taksar [29], so we omit it here.


## 3.1.2. The Case:x1<x0

Now we turn to the casex1<x0, which is equivalent to M<μ/2+δσ2/μ. From Højgaard and Taksar [29], it shows that there exists only one “switching point” x1 because of x0=∞. Therefore, the solution of (14) and (15) depends on whether x1>0 or x1≤0.Ifx1>0, for x<x1, we obtain (17) and find as before a solution as follows: (62)fx=C1x+PC11/γγ,where γ is given by (21) and C1 is a free constant.Forx≥x1, we have l(x)=M. Thus, the HJB equation (14) becomes(63)maxa∈0,112σ2a2f′′x+μa-Mf′x-δfx+M=0.Let f∗(x) be a solution of the following homogeneous equation:(64)maxa∈0,112σ2a2f′′x+μa-Mf′x-δfx=0.Then, a solution of (63) can be expressed by(65)fx=f∗x+Mδ.Differentiating (64) with respect to a yields the maximizer(66)ax=-μf′xσ2f′′x.Substituting (66) into (64), we obtain(67)-μ2f′x22σ2f′′x-Mf′x-δfx=0.A general solution of (67) can be given by(68)f∗x=C2eρx-x1,x≥x1,where C2(C2≠0) and ρ are free constants.Input (68) into (67); we obtain (69)C2-μ22σ2-Mρ-δeρx-x1=0.Then, it is easy to derive that(70)ρ=-δMγ,where γ is defined by (21).Thus, combining (65), (68), and (70), a solution of (63) is given by (71)fx=C2e-δ/Mγx-x1+Mδ,x≥x1.Consequently, for the case ofx1>0, the following solution to (14) and (15) is suggested (72)fx=C1x+PC11/γγ,0≤x<x1,C2e-δ/Mγx-x1+Mδ,x≥x1,where C1, C2, and x1 need to be specified.Making “smooth fit” atx1, we have the following equations:(73)-C2δMγ=1,(74)C1γx1+PC11/γγ-1=1,(75)C1γγ-1x1+PC11/γγ-2=-δMγ,where we have used (73) in (75).From (73), we obtain (76)C2=-Mγδ.Dividing (75) by (74) yields that(77)x1+PC11/γ=Mγ1-γδ.Put (77) into (74); we get(78)C1=1γMγ1-γδ1-γ.Combining (77) with (78), it follows that(79)x1=Mγ1-γδ1-δPM1-γ1/γ.In addition, by (79), it shows that x1>0 is equivalent to(80)P<M1-γδ.On the other hand, ifx1≤0, that is,(81)P≥M1-γδ,we get (63) and find as above a solution as follows: (82)fx=C2e-δ/Mγx+Mδ,x≥0.By the boundary condition f(0)=P, we have(83)fx=P-Mδe-δ/Mγx+Mδ,x≥0.Sincea(x)<1 for all x∈[0,∞), the following theorem follows easily from construction.Theorem 6.
Assume thatM<μ/2+δσ2/μ. Let γ be defined by (21).
If (80) is true, then the solution to (14) and (15) is given by(84)fx=C1x+PC11/γγ,0≤x<x1,Mδ1-γe-δ/Mγx-x1,x≥x1,where C1 is given by (78) and x1 by (79).
On the other hand, if (81) holds, then(85)fx=P-Mδe-δ/Mγx+Mδ,x≥0.Moreover, the functions f(x) defined as above are concave solutions of (14) and (15).


## 3.2. Optimal Policy

Based on sections above, in the following, we present our optimal policy. The admissible policyπ∗ for all t<τπ∗ is defined as follows.Case 1 (M≥μ/2+δσ2/μ).
(
1
) If (44) holds, then (86)aπ∗t≔aXtπ∗=μ1-γσ2Xtπ∗+PC11/γ,0≤Xtπ∗<x0,1,Xtπ∗≥x0,lπ∗t≔lXtπ∗=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x0 is given by (23) and x1 by (41).
(
2
) If (56) is true, then (87)aπ∗t≔aXtπ∗=1,Xtπ∗≥0,lπ∗t≔lXtπ∗=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x1 is given by (54).
(
3
) If (57) holds, then (88)aπ∗t≔aXtπ∗=1,Xtπ∗≥0,lπ∗t≔lXtπ∗=M,Xtπ∗≥0.Case 2 (M<μ/2+δσ2/μ).
(
1
) If (80) is true, then (89)aπ∗t≔aXtπ∗=μ1-γσ2Xtπ∗+PC11/γ,0≤Xtπ∗<x1,μMγδσ2,Xtπ∗≥x1,lπ∗t≔lXtπ∗=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x1 is given by (79).
(
2
) If (81) holds, then (90)aπ∗t≔aXtπ∗=μMγδσ2=2μMμ2+2δσ2,Xtπ∗≥0,lπ∗t≔lXtπ∗=M,Xtπ∗≥0.
Moreover, in the case ofM<μ/2+δσ2/μ, we have a(Xtπ∗)<1 for all Xtπ∗≥0.
In addition,Xtπ∗ is the solution of (91)Xtπ∗=X0π∗+∫0tμaπ∗s-lπ∗sds+∫0tσaπ∗sdWs.The following theorem shows that the solutionf(x) of (14) and (15) constructed above is the optimal value function.Theorem 7.
LetVP(x) be given by (7), and let f(x) be given by (59)–(61) for M≥μ/2+δσ2/μ and by (84)-(85) for M<μ/2+δσ2/μ. Then (92)VPx=fx.Proof.
The proof of this theorem can be carried out in the same way as that of Theorem2.3 in Højgaard and Taksar [29]. To avoid tedious repetition, we omit it here.


## 4. Nonterminal Bankruptcy Model with Bailouts

### 4.1. Nonterminal Bankruptcy Model and Its HJB Equation

In the real world, firms may receive financial bailouts from the government or other corporations when they are at the edge of bankruptcy. Therefore, it is necessary for us to consider the optimal dividend problem with capital injections when a firm may encounter bankruptcy. In this section, based on the solutions of (14) and (15), the nonterminal bankruptcy model with bailouts is developed and further its solution is obtained under an appropriate boundary condition.In order to specify the development of the reserve process in our model whenXt=0, we consider the following discrete time model of bankruptcy. Assume that at time t the company is at the stage of bankruptcy, and meanwhile the probability it receives an ϵ amount of external capital inflow is ph, while it remains in bankruptcy with probability 1-ph. Thus, the capital injections of the reserve process at time t are Δt=Xt+h-Xt, where Δt(t=0,h,2h,…) are i.i.d. random variables with the distribution(93)Δt=0,withprobability1-ph,ϵ,withprobabilityph.Thus, (94)EΔt=ηh,whereη=pϵ.Again, we adopt the continuous analog(95)dXt=ηdt,atXt=0.Combining (4), (5), and (95), we obtain(96)dXtπ=μaπt-lπtIXtπ>0dt+ηIXtπ=0dt+σaπtIXtπ>0dWt,(97)X0π=x.Equation (96) shows that the recovery rate η can be viewed as a rate at which the company can receive new capital at the time when Xtπ=0. As in Sethi and Taksar [27], for the model considered in this part, we denote it bynonterminal bankruptcy (or bankruptcy with recovery). On the other hand, for the model, which is restricted to stay at the bankruptcy state, it is termed asterminal bankruptcy. Consequently, the optimal value function V(x) becomes as follows:(98)Vx=supπ∈ΠEx∫0∞e-δtlπtdt.Then, the HJB equation satisfied by (98) can be also given by (14). To find the function V(x), we need to specify the behavior of V(x) at the origin. Moreover, its boundary condition should be related to the capital injection rate η. Now, let us verify the following theorem, which may give us some enlightenment about its boundary condition.Theorem 8.
Suppose thatJ(x) is a nonnegative C2-function on [0,∞) with J′(x) bounded and J(x) satisfies the HJB equation(99)maxa∈0,1,l∈0,M12σ2a2J′′x+μa-lJ′x-δJx+l=0,x>0,with the mixed Dirichlet-Neumann boundary condition(100)ηJ′0-δJ0=0.Then, it follows that (101)Jx≥Vx,x>0.Proof.
LetX0π=x>0 and π∈Π be an admissible policy. For h>0, using the generalized Itô’s formula, we have (102)Jx≥Jx-Exe-δhJXhπ=Ex∫0he-δs-12σ2aπ2sJ′′Xsπ+μaπs-lπsJ′Xsπ-δJXhπIXsπ>0-ηJ′0-δJ0ds+Ex∫0he-δsσaπsJ′XsπIXsπ>0dWs.As J′(x) is bounded, the last term on the right-hand side is a zero-mean square integrable martingale. Then from (99) and (100), we obtain (103)Jx≥Ex∫0he-δslπsIXsπ>0ds.For Xsπ=0, it implies that lπ(s)=0. Thus,(104)Jx≥Ex∫0he-δslπsIXsπ>0+lπsIXsπ=0ds=Ex∫0he-δslπsds.Let h→∞; using Fatou’s lemma and taking the supremum over all policies π∈Π in (104), we obtain (105)Jx≥Vx,x>0.


### 4.2. Solution of the Nonterminal Bankruptcy Model

Although the HJB equation (14) with the mixed boundary condition (100) cannot be solved in a straightforward manner, we can establish a relation between V(x) and VP(x), in which the boundary condition (100) is replaced by (15), by comparing it with the nonzero terminal bankruptcy problem.Theorem 9.
LetV(x) be given by (98). Suppose that VP(x) is the optimal value function for problem (7) and that {aπ∗(t),lπ∗(t)} is the corresponding optimal policy. Suppose(106)η=δVP0VP′0,and let Xtπ∗ be a solution to the stochastic different equation (96) with η given by (106), aπ∗(t)=a(Xtπ∗) and lπ∗(t)=l(Xtπ∗). Then, the optimal value function V(x) can be expressed by VP(x). Moreover, the optimal policy in this model π∗ is given by(107)A∗t=aXtπ∗IXtπ>0,L∗t=lXtπ∗IXtπ>0.Proof.
According to Gihman and Skorohod [30], there exists a solution to (96) with aπ∗(t)=a(Xtπ∗) and lπ∗(t)=l(Xtπ∗). Thus, {A∗(t),L∗(t)} defined by (107) is admissible.
SinceVP(x) is a solution of the HJB (14), it follows that(108)La,lVPx=0.Equation (106) is equivalent to(109)ηVP′0-δVP0=0.Repeating the similar argument as the proof of Theorem 8, then by (108) and (109), we obtain(110)VPx-Exe-δhVPXhπ∗=Ex∫0he-δslπXsπ∗IXsπ>0ds.By boundedness of VP′, VP(y)≤K(1+y) for some K>0. Therefore,(111)e-δhVPXhπ∗≤e-δhK1+Xhπ∗.Furthermore, (112)ExXhπ∗=Exx+∫0hμaXsπ∗-lXsπ∗IXsπ∗>0ds+∫0hηIXsπ∗=0ds+∫0hσaXsπ∗IXsπ∗>0dWs=Exx+∫0hμaXsπ∗-lXsπ∗IXsπ∗>0ds+∫0hηIXsπ∗=0ds≤x+μ+ηh.Thus, the term on the left-hand side of (111) is majorized by a positive rand variable with a finite expectation.
Lettingh→∞ in (110) and using dominated and monotone convergence theorems, we have(113)VPx=Vπ∗x≤Vx.On the other hand, let P=VP(0)=ηVP′(0)/δ; then according to Theorem 8, it implies that(114)VPx≥Vx.Combining (113) and (114), it shows that V(x)=VP(x).Next, we will construct an optimal value functionV(x) of nonterminal bankruptcy problem with bailouts in the following way. Given a capital injection rate η, we wish to find a function P(·) depending on η such that the function VP(η)(x) satisfies (115)η=δVPη0VPη′0.Then, according to Theorem 9, we obtain (116)Vx=VPηx.For any givenη, a specific analysis of the behavior of P(·) is given as follows.Case 1 (M≥μ/2+δσ2/μ).
From (59), it is easy to derive that(117)η=δVP0VP′0=δγPC11/γ,where P<C1βγ, which implies that P/C1<βγ and η<δβ/γ=μ/2.
In view thatP=0 implies η=0, then for the case η∈[0,μ/2), by (117) we obtain(118)P=C1ηγδγ.
Similarly, from (61), we have(119)η=δVP0VP′0=δPP-M/δθ^,where P≥M/δ+1/θ^. It is easily verified that η(·) is a strictly increasing function of P, which implies η≥M+δ/θ^.
Thus, forη∈[M+δ/θ^,∞), (119) implies that (120)P=ηMθ^δηθ^-δ.
By continuity ofη, for η∈[μ/2,M+δ/θ^), then applying (60) we get (121)η=δVP0VP′0=δPC2θ++P-C2θ-.Thus,(122)P=C2pη,where C2 is given by (53) and pη=(θ+-θ-)/(δ/η-θ-).
Substituting (122) into (54) yields that(123)θ^-θ-θ+θ+-θ-e-θ+x1+θ+-θ^θ-θ+-θ-e-θ-x1=θ^-θ-θ+θ+-θ-e-θ+x1pη.According to (123), x1 is given by(124)x1=1θ+-θ-ln⁡θ-θ^-θ-θ+θ+-θ^pη-1.Combining (53), (122), and (124), it shows that P is uniquely determined by η.Remark 10.
To ensure thatx1>0 in (124), we need to verify (θ-(θ^-θ-)/θ+(θ+-θ^))(pη-1)>1 for η∈[μ/2,M+δ/θ^). Let h(η)=(θ-θ^-θ-/θ+θ+-θ^)(pη-1), then h(η) is a strictly decreasing function with respect to η on [μ/2,M+δ/θ^). Taking η=M+δ/θ^ and applying (55), it follows that (125)hM+δθ^=θ-θ^-θ-θ+θ+-θ^θ+-θ-1/M/δ+1/θ^-θ--1=θ-θ^-θ-θ+θ+-θ^θ+-θ-θ+θ-/θ++θ--θ^-θ--1=1.Therefore, h(η)>1 is surely true for η∈[μ/2,M+δ/θ^).Case 2 (M<μ/2+δσ2/μ).
In view of (84), we obtain(126)η=δVP0VP′0=δγPC11/γ,where C1 is given by (78). The validity of η<M(1-γ) follows from P<M(1-γ)/δ.
Thus, for the caseη∈[0,M(1-γ)), by (126) we get (127)P=C1ηγδγ.
On the other hand, forη∈[M(1-γ),∞), equivalent to P≥M(1-γ)/δ, (85) implies that (128)η=δVP0VP′0=δMγPM-δP.Then, (129)P=MηδMγ+η.
Consequently, it shows that for anyη∈[0,∞) there exists a unique P=P(η), satisfying η=δVP(0)/VP′(0). Then, based on Theorems 5, 6, and 9, we derive the following theorem, which presents an explicit expression for the optimal value function V(x).Theorem 11.
LetV(x) be given by (98).
Case  1 (M≥μ/2+δσ2/μ). (1) If η∈[0,μ/2), then V(x) is given by(130)Vx=C1x+ηγδγ,0≤x<x0,C1a1eθ+x-x0+C1a2eθ-x-x0,x0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where x0=(1-γ)σ2/μ-ηγ/δ and x1 and C1 are given by (41) and (42), respectively.
(2) Ifη∈[μ/2,M+δ/θ^), then(131)Vx=C2eθ+x+C2pη-1eθ-x,0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where x1 is given by (54) and C2 is determined by (53).
(3) Ifη∈[M+δ/θ^,∞), then(132)Vx=Mδ+Mηθ^-δeθ^x,x≥0.
Case  2 (M<μ/2+δσ2/μ). (1) If η∈[0,M(1-γ)), then V(x) is given by(133)Vx=1γMγ1-γδ1-γx+ηγδγ,0≤x<x1,Mδ1-γe-δ/Mγx-x1,x≥x1,where x1 is given by (79).
(2) Ifη∈[M(1-γ),∞), then(134)Vx=Mδ1-MγMγ+ηe-δ/Mγx,x≥0.Proof.
From Theorem9, (59) and (118), for η∈(0,μ/2), V(x) can be given by V(x)=VP(x) with P=C1(ηγ/δ)γ, that is, (130). By the same method, as the proof of the case η∈(0,μ/2), we can verify that V(x) is given by (131)–(134), respectively.


### 4.3. Optimal Policy

According to Section3.2 and (107), the corresponding admissible policy π∗ for the nonterminal bankruptcy model with bailouts is given as follows.Case 1 (M≥μ/2+δσ2/μ).
(1) Ifη∈[0,μ/2), then (135)A∗t=μ1-γσ2Xtπ∗+ηγδ,0≤Xtπ∗<x0,1,Xtπ∗≥x0,L∗t=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x0=(1-γ)σ2/μ-ηγ/δ and x1 is given by (41).
(2) Ifη∈[μ/2,M+δ/θ^), then (136)A∗t=1,Xtπ∗≥0,(137)L∗t=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x1 is given by (54).
(3) Ifη∈[M+δ/θ^,∞), then (138)A∗t=1,L∗t=M,Xtπ∗≥0.Case 2 (M<μ/2+δσ2/μ).
(1) Ifη∈[0,M(1-γ)), (139)A∗t=μ1-γσ2Xtπ∗+ηγδ,0≤Xtπ∗<x1,μMγδσ2,Xtπ∗≥x1,(140)L∗t=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x1 is given by (79).
(2) Ifη∈[M(1-γ),∞), then (141)A∗t=μMγδσ2=2μMμ2+2δσ2,L∗t=M,Xtπ∗≥0.
In the case ofM<μ/2+δσ2/μ, it is easily seen that A∗(t)<1 for all Xtπ∗≥0.
In addition,Xtπ∗ is the solution of (142)Xtπ∗=X0π∗+∫0hμaπ∗s-lπ∗sIXsπ∗>0ds+∫0hηIXsπ∗=0ds+∫0hσaπ∗sIXsπ∗>0dWs.


### 4.4. Numerical Examples

Based on the calculated parametersC1, C2, x0, and x1, several numerical examples are provided here to support our theoretical results. It is interesting to look at the relationship between the optimal control and the capital injection rate η. In Figures 1 and 2, the graphs of the optimal value function V(x) are shown for different η under M=2 and M=0.5, respectively. In these plots, parameters used are given by μ=1, σ=2, and δ=0.1. From these graphs, it shows that, for any given x, the value of V(x) increases with the capital injection rate η.Figure 1
The optimal value functions forμ=1, σ=2, δ=0.1, M=2 and η=0, η=0.1, η=0.6, η=0.9.Figure 2
The optimal value functions forμ=1, σ=2, δ=0.1, M=0.5 and η=0, η=0.1, η=0.3.Figures3 and 4 present the optimal risk exposure a(x) for different η at cases M=2 and M=0.5, respectively. In these graphs, the used parameters are also given by μ=1, σ=2, and δ=0.1 as in Figures 1 and 2. In the case of M=2 (M>μ/2+δσ2/μ), for η=0.1<μ/2, the optimal risk exposure a(x) is a linear function of the current reserve process as long as this process is less than x0=1.78, after which it is a constant function equal to one. For η=0.6>μ/2, the optimal risk exposure a(x) is constantly equal to one; that is, it is optimal for the company to take the maximum risk all the time and in essence to optimize only the dividend payments. Under the condition M=0.5 (M<μ/2+δσ2/μ), for η=0.1<M(1-γ) the optimal risk exposure a(x) monotonically increases from zero to maximum as a linear function of the reserve process and when this value is attained for x1 the risk exposure stays here for all x beyond x1=0.79. For η=0.3>M(1-γ), the optimal risk exposure a(x) is always equal to the maximum, but less than one. Moreover, it also shows that for any fixed x the optimal risk exposure a(x) is a nondecreasing function of η.Figure 3
The optimal risk exposure forμ=1, σ=2, δ=0.1, M=2 and η=0, η=0.1, η=0.6.Figure 4
The optimal risk exposure forμ=1, σ=2, δ=0.1, M=0.5 and η=0, η=0.1, η=0.3.In Figures5 and 6, parameters μ, σ, δ, and M are the same as in Figures 3 and 4, respectively. From the two graphs, it is easily seen that the dividend threshold level x1 is a decreasing function with respect to η.Figure 5
The dividend threshold levelx1 as a function of η with μ=1, σ=2, δ=0.1, and M=2.Figure 6
The dividend threshold levelx1 as a function of η with μ=1, σ=2, δ=0.1, and M=0.5.All of this has the following economic interpretation. In an attempt to pay dividend to the shareholders, at least the company is interested in extending the time before bankruptcy. For an extension of time before bankruptcy, a reduction of risk is needed which simultaneously reduces potential profit. However, with increasing capital injection rate, the company can be less sensitive to bankruptcy, since an increasing rate of capital injection hedges against potential losses of the future profits. The same explanation holds for the decrease ofx1, when η increases. The company can start distributing dividends earlier by lowering the threshold not being concerned that such a policy can bring the bankruptcy faster.IfM is large enough and the reserve or the capital injection rate is large enough, it is optimal for the company to take the maximal risk exposure all the time. However, if M is small enough, it will never be optimal to take the maximal risk exposure, no matter how large the reserve or the capital injection rate is. In particular, if the rate of capital injection is larger than the upper of the dividend rate, it is the best for a company not to do business at all but to “take money and run.”


## 4.1. Nonterminal Bankruptcy Model and Its HJB Equation

In the real world, firms may receive financial bailouts from the government or other corporations when they are at the edge of bankruptcy. Therefore, it is necessary for us to consider the optimal dividend problem with capital injections when a firm may encounter bankruptcy. In this section, based on the solutions of (14) and (15), the nonterminal bankruptcy model with bailouts is developed and further its solution is obtained under an appropriate boundary condition.In order to specify the development of the reserve process in our model whenXt=0, we consider the following discrete time model of bankruptcy. Assume that at time t the company is at the stage of bankruptcy, and meanwhile the probability it receives an ϵ amount of external capital inflow is ph, while it remains in bankruptcy with probability 1-ph. Thus, the capital injections of the reserve process at time t are Δt=Xt+h-Xt, where Δt(t=0,h,2h,…) are i.i.d. random variables with the distribution(93)Δt=0,withprobability1-ph,ϵ,withprobabilityph.Thus, (94)EΔt=ηh,whereη=pϵ.Again, we adopt the continuous analog(95)dXt=ηdt,atXt=0.Combining (4), (5), and (95), we obtain(96)dXtπ=μaπt-lπtIXtπ>0dt+ηIXtπ=0dt+σaπtIXtπ>0dWt,(97)X0π=x.Equation (96) shows that the recovery rate η can be viewed as a rate at which the company can receive new capital at the time when Xtπ=0. As in Sethi and Taksar [27], for the model considered in this part, we denote it bynonterminal bankruptcy (or bankruptcy with recovery). On the other hand, for the model, which is restricted to stay at the bankruptcy state, it is termed asterminal bankruptcy. Consequently, the optimal value function V(x) becomes as follows:(98)Vx=supπ∈ΠEx∫0∞e-δtlπtdt.Then, the HJB equation satisfied by (98) can be also given by (14). To find the function V(x), we need to specify the behavior of V(x) at the origin. Moreover, its boundary condition should be related to the capital injection rate η. Now, let us verify the following theorem, which may give us some enlightenment about its boundary condition.Theorem 8.
Suppose thatJ(x) is a nonnegative C2-function on [0,∞) with J′(x) bounded and J(x) satisfies the HJB equation(99)maxa∈0,1,l∈0,M12σ2a2J′′x+μa-lJ′x-δJx+l=0,x>0,with the mixed Dirichlet-Neumann boundary condition(100)ηJ′0-δJ0=0.Then, it follows that (101)Jx≥Vx,x>0.Proof.
LetX0π=x>0 and π∈Π be an admissible policy. For h>0, using the generalized Itô’s formula, we have (102)Jx≥Jx-Exe-δhJXhπ=Ex∫0he-δs-12σ2aπ2sJ′′Xsπ+μaπs-lπsJ′Xsπ-δJXhπIXsπ>0-ηJ′0-δJ0ds+Ex∫0he-δsσaπsJ′XsπIXsπ>0dWs.As J′(x) is bounded, the last term on the right-hand side is a zero-mean square integrable martingale. Then from (99) and (100), we obtain (103)Jx≥Ex∫0he-δslπsIXsπ>0ds.For Xsπ=0, it implies that lπ(s)=0. Thus,(104)Jx≥Ex∫0he-δslπsIXsπ>0+lπsIXsπ=0ds=Ex∫0he-δslπsds.Let h→∞; using Fatou’s lemma and taking the supremum over all policies π∈Π in (104), we obtain (105)Jx≥Vx,x>0.


## 4.2. Solution of the Nonterminal Bankruptcy Model

Although the HJB equation (14) with the mixed boundary condition (100) cannot be solved in a straightforward manner, we can establish a relation between V(x) and VP(x), in which the boundary condition (100) is replaced by (15), by comparing it with the nonzero terminal bankruptcy problem.Theorem 9.
LetV(x) be given by (98). Suppose that VP(x) is the optimal value function for problem (7) and that {aπ∗(t),lπ∗(t)} is the corresponding optimal policy. Suppose(106)η=δVP0VP′0,and let Xtπ∗ be a solution to the stochastic different equation (96) with η given by (106), aπ∗(t)=a(Xtπ∗) and lπ∗(t)=l(Xtπ∗). Then, the optimal value function V(x) can be expressed by VP(x). Moreover, the optimal policy in this model π∗ is given by(107)A∗t=aXtπ∗IXtπ>0,L∗t=lXtπ∗IXtπ>0.Proof.
According to Gihman and Skorohod [30], there exists a solution to (96) with aπ∗(t)=a(Xtπ∗) and lπ∗(t)=l(Xtπ∗). Thus, {A∗(t),L∗(t)} defined by (107) is admissible.
SinceVP(x) is a solution of the HJB (14), it follows that(108)La,lVPx=0.Equation (106) is equivalent to(109)ηVP′0-δVP0=0.Repeating the similar argument as the proof of Theorem 8, then by (108) and (109), we obtain(110)VPx-Exe-δhVPXhπ∗=Ex∫0he-δslπXsπ∗IXsπ>0ds.By boundedness of VP′, VP(y)≤K(1+y) for some K>0. Therefore,(111)e-δhVPXhπ∗≤e-δhK1+Xhπ∗.Furthermore, (112)ExXhπ∗=Exx+∫0hμaXsπ∗-lXsπ∗IXsπ∗>0ds+∫0hηIXsπ∗=0ds+∫0hσaXsπ∗IXsπ∗>0dWs=Exx+∫0hμaXsπ∗-lXsπ∗IXsπ∗>0ds+∫0hηIXsπ∗=0ds≤x+μ+ηh.Thus, the term on the left-hand side of (111) is majorized by a positive rand variable with a finite expectation.
Lettingh→∞ in (110) and using dominated and monotone convergence theorems, we have(113)VPx=Vπ∗x≤Vx.On the other hand, let P=VP(0)=ηVP′(0)/δ; then according to Theorem 8, it implies that(114)VPx≥Vx.Combining (113) and (114), it shows that V(x)=VP(x).Next, we will construct an optimal value functionV(x) of nonterminal bankruptcy problem with bailouts in the following way. Given a capital injection rate η, we wish to find a function P(·) depending on η such that the function VP(η)(x) satisfies (115)η=δVPη0VPη′0.Then, according to Theorem 9, we obtain (116)Vx=VPηx.For any givenη, a specific analysis of the behavior of P(·) is given as follows.Case 1 (M≥μ/2+δσ2/μ).
From (59), it is easy to derive that(117)η=δVP0VP′0=δγPC11/γ,where P<C1βγ, which implies that P/C1<βγ and η<δβ/γ=μ/2.
In view thatP=0 implies η=0, then for the case η∈[0,μ/2), by (117) we obtain(118)P=C1ηγδγ.
Similarly, from (61), we have(119)η=δVP0VP′0=δPP-M/δθ^,where P≥M/δ+1/θ^. It is easily verified that η(·) is a strictly increasing function of P, which implies η≥M+δ/θ^.
Thus, forη∈[M+δ/θ^,∞), (119) implies that (120)P=ηMθ^δηθ^-δ.
By continuity ofη, for η∈[μ/2,M+δ/θ^), then applying (60) we get (121)η=δVP0VP′0=δPC2θ++P-C2θ-.Thus,(122)P=C2pη,where C2 is given by (53) and pη=(θ+-θ-)/(δ/η-θ-).
Substituting (122) into (54) yields that(123)θ^-θ-θ+θ+-θ-e-θ+x1+θ+-θ^θ-θ+-θ-e-θ-x1=θ^-θ-θ+θ+-θ-e-θ+x1pη.According to (123), x1 is given by(124)x1=1θ+-θ-ln⁡θ-θ^-θ-θ+θ+-θ^pη-1.Combining (53), (122), and (124), it shows that P is uniquely determined by η.Remark 10.
To ensure thatx1>0 in (124), we need to verify (θ-(θ^-θ-)/θ+(θ+-θ^))(pη-1)>1 for η∈[μ/2,M+δ/θ^). Let h(η)=(θ-θ^-θ-/θ+θ+-θ^)(pη-1), then h(η) is a strictly decreasing function with respect to η on [μ/2,M+δ/θ^). Taking η=M+δ/θ^ and applying (55), it follows that (125)hM+δθ^=θ-θ^-θ-θ+θ+-θ^θ+-θ-1/M/δ+1/θ^-θ--1=θ-θ^-θ-θ+θ+-θ^θ+-θ-θ+θ-/θ++θ--θ^-θ--1=1.Therefore, h(η)>1 is surely true for η∈[μ/2,M+δ/θ^).Case 2 (M<μ/2+δσ2/μ).
In view of (84), we obtain(126)η=δVP0VP′0=δγPC11/γ,where C1 is given by (78). The validity of η<M(1-γ) follows from P<M(1-γ)/δ.
Thus, for the caseη∈[0,M(1-γ)), by (126) we get (127)P=C1ηγδγ.
On the other hand, forη∈[M(1-γ),∞), equivalent to P≥M(1-γ)/δ, (85) implies that (128)η=δVP0VP′0=δMγPM-δP.Then, (129)P=MηδMγ+η.
Consequently, it shows that for anyη∈[0,∞) there exists a unique P=P(η), satisfying η=δVP(0)/VP′(0). Then, based on Theorems 5, 6, and 9, we derive the following theorem, which presents an explicit expression for the optimal value function V(x).Theorem 11.
LetV(x) be given by (98).
Case  1 (M≥μ/2+δσ2/μ). (1) If η∈[0,μ/2), then V(x) is given by(130)Vx=C1x+ηγδγ,0≤x<x0,C1a1eθ+x-x0+C1a2eθ-x-x0,x0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where x0=(1-γ)σ2/μ-ηγ/δ and x1 and C1 are given by (41) and (42), respectively.
(2) Ifη∈[μ/2,M+δ/θ^), then(131)Vx=C2eθ+x+C2pη-1eθ-x,0≤x<x1,Mδ+1θ^eθ^x-x1,x≥x1,where x1 is given by (54) and C2 is determined by (53).
(3) Ifη∈[M+δ/θ^,∞), then(132)Vx=Mδ+Mηθ^-δeθ^x,x≥0.
Case  2 (M<μ/2+δσ2/μ). (1) If η∈[0,M(1-γ)), then V(x) is given by(133)Vx=1γMγ1-γδ1-γx+ηγδγ,0≤x<x1,Mδ1-γe-δ/Mγx-x1,x≥x1,where x1 is given by (79).
(2) Ifη∈[M(1-γ),∞), then(134)Vx=Mδ1-MγMγ+ηe-δ/Mγx,x≥0.Proof.
From Theorem9, (59) and (118), for η∈(0,μ/2), V(x) can be given by V(x)=VP(x) with P=C1(ηγ/δ)γ, that is, (130). By the same method, as the proof of the case η∈(0,μ/2), we can verify that V(x) is given by (131)–(134), respectively.


## 4.3. Optimal Policy

According to Section3.2 and (107), the corresponding admissible policy π∗ for the nonterminal bankruptcy model with bailouts is given as follows.Case 1 (M≥μ/2+δσ2/μ).
(1) Ifη∈[0,μ/2), then (135)A∗t=μ1-γσ2Xtπ∗+ηγδ,0≤Xtπ∗<x0,1,Xtπ∗≥x0,L∗t=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x0=(1-γ)σ2/μ-ηγ/δ and x1 is given by (41).
(2) Ifη∈[μ/2,M+δ/θ^), then (136)A∗t=1,Xtπ∗≥0,(137)L∗t=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x1 is given by (54).
(3) Ifη∈[M+δ/θ^,∞), then (138)A∗t=1,L∗t=M,Xtπ∗≥0.Case 2 (M<μ/2+δσ2/μ).
(1) Ifη∈[0,M(1-γ)), (139)A∗t=μ1-γσ2Xtπ∗+ηγδ,0≤Xtπ∗<x1,μMγδσ2,Xtπ∗≥x1,(140)L∗t=0,0≤Xtπ∗<x1,M,Xtπ∗≥x1,where x1 is given by (79).
(2) Ifη∈[M(1-γ),∞), then (141)A∗t=μMγδσ2=2μMμ2+2δσ2,L∗t=M,Xtπ∗≥0.
In the case ofM<μ/2+δσ2/μ, it is easily seen that A∗(t)<1 for all Xtπ∗≥0.
In addition,Xtπ∗ is the solution of (142)Xtπ∗=X0π∗+∫0hμaπ∗s-lπ∗sIXsπ∗>0ds+∫0hηIXsπ∗=0ds+∫0hσaπ∗sIXsπ∗>0dWs.


## 4.4. Numerical Examples

Based on the calculated parametersC1, C2, x0, and x1, several numerical examples are provided here to support our theoretical results. It is interesting to look at the relationship between the optimal control and the capital injection rate η. In Figures 1 and 2, the graphs of the optimal value function V(x) are shown for different η under M=2 and M=0.5, respectively. In these plots, parameters used are given by μ=1, σ=2, and δ=0.1. From these graphs, it shows that, for any given x, the value of V(x) increases with the capital injection rate η.Figure 1
The optimal value functions forμ=1, σ=2, δ=0.1, M=2 and η=0, η=0.1, η=0.6, η=0.9.Figure 2
The optimal value functions forμ=1, σ=2, δ=0.1, M=0.5 and η=0, η=0.1, η=0.3.Figures3 and 4 present the optimal risk exposure a(x) for different η at cases M=2 and M=0.5, respectively. In these graphs, the used parameters are also given by μ=1, σ=2, and δ=0.1 as in Figures 1 and 2. In the case of M=2 (M>μ/2+δσ2/μ), for η=0.1<μ/2, the optimal risk exposure a(x) is a linear function of the current reserve process as long as this process is less than x0=1.78, after which it is a constant function equal to one. For η=0.6>μ/2, the optimal risk exposure a(x) is constantly equal to one; that is, it is optimal for the company to take the maximum risk all the time and in essence to optimize only the dividend payments. Under the condition M=0.5 (M<μ/2+δσ2/μ), for η=0.1<M(1-γ) the optimal risk exposure a(x) monotonically increases from zero to maximum as a linear function of the reserve process and when this value is attained for x1 the risk exposure stays here for all x beyond x1=0.79. For η=0.3>M(1-γ), the optimal risk exposure a(x) is always equal to the maximum, but less than one. Moreover, it also shows that for any fixed x the optimal risk exposure a(x) is a nondecreasing function of η.Figure 3
The optimal risk exposure forμ=1, σ=2, δ=0.1, M=2 and η=0, η=0.1, η=0.6.Figure 4
The optimal risk exposure forμ=1, σ=2, δ=0.1, M=0.5 and η=0, η=0.1, η=0.3.In Figures5 and 6, parameters μ, σ, δ, and M are the same as in Figures 3 and 4, respectively. From the two graphs, it is easily seen that the dividend threshold level x1 is a decreasing function with respect to η.Figure 5
The dividend threshold levelx1 as a function of η with μ=1, σ=2, δ=0.1, and M=2.Figure 6
The dividend threshold levelx1 as a function of η with μ=1, σ=2, δ=0.1, and M=0.5.All of this has the following economic interpretation. In an attempt to pay dividend to the shareholders, at least the company is interested in extending the time before bankruptcy. For an extension of time before bankruptcy, a reduction of risk is needed which simultaneously reduces potential profit. However, with increasing capital injection rate, the company can be less sensitive to bankruptcy, since an increasing rate of capital injection hedges against potential losses of the future profits. The same explanation holds for the decrease ofx1, when η increases. The company can start distributing dividends earlier by lowering the threshold not being concerned that such a policy can bring the bankruptcy faster.IfM is large enough and the reserve or the capital injection rate is large enough, it is optimal for the company to take the maximal risk exposure all the time. However, if M is small enough, it will never be optimal to take the maximal risk exposure, no matter how large the reserve or the capital injection rate is. In particular, if the rate of capital injection is larger than the upper of the dividend rate, it is the best for a company not to do business at all but to “take money and run.”


## 5. Conclusions

In this paper, we develop a stochastic dynamic dividend model with terminal values at bankruptcy status and achieve the analytic solution for the optimal value function. Based on this result, the dividend problem with bailouts at bankruptcy can also be solved and its solution has been constructed. A wonderful result obtained is the concise necessary and sufficient condition for immediate dividend actions after bailouts. That is, if the rate of bailouts is greater than the upper of the dividend rate, then the dividend event would happen immediately after bailouts. It demonstrates that to prohibit dividend payout after bailouts the manager or governor should make the rate of bailouts less than the maximum dividend rate.In the future, following the researches of Zhu [31, 32], Zhu and Zhang [33], Wang and Zhu [34], and so on, we can address some dividend optimization problems for general diffusion models with the drift and diffusion coefficients being general functions of the surplus under the assumption that terminal bankruptcy values are nonzero. For this kind of problem, sometimes we cannot derive the explicit solutions of optimal value function. Then we need to use some suitable numerical methods as in Zhou et al. [35] and Zhou [36] to simulate.



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