# Classical and Impulse Stochastic Control on the Optimization of Dividends with Residual Capital at Bankruptcy

Authors: Peimin Chen; Bo Li
Journal: Discrete Dynamics in Nature and Society
 (2017)

---

## Abstract

In this paper, we consider the optimization problem of dividends for the terminal bankruptcy model, in which some money would be returned to shareholders at the state of terminal bankruptcy, while accounting for the tax rate and transaction cost for dividend payout. Maximization of both expected total discounted dividends before bankruptcy and expected discounted returned money at the state of terminal bankruptcy becomes a mixed classical-impulse stochastic control problem. In order to solve this problem, we reduce it to quasi-variational inequalities with a nonzero boundary condition. We explicitly construct and verify solutions of these inequalities and present the value function together with the optimal policy.


---

## Body

## 1. Introduction

Corporate dividend policy has long engaged the attention of financial economists. The classical paper [1] by Miller and Modigliani provides the valuation formula for an infinite horizon firm under perfect certainty. In a world of perfect capital markets, they show that the dividend policy is irrelevant as a firm can always raise funds to meet the need for continuing operation. However, under a more realistic condition with imperfections such as the presence of financial constraints, information asymmetry, agency costs, taxes, risk exposure under uncertainty, transaction costs, and other frictions, it has been shown that there exists an optimal dividend policy (see [2–5]). Thus, determining the optimal dividend payouts becomes an important issue as it affects firm value. More recent models have focused on the issue of how to set the optimal dividend policy in a dynamic uncertain environment.The valuation model used by Miller-Modigliani in 1961 can be extended to the situation of controllable business activities in a stochastic environment. During the recent decades, there have been increasing interests in applying diffusion models to financial decision problems, especially in (re)insurance modelling (see [6–17]). For most of these models, the liquid assets processes of the corporation contain a Brownian motion with drift and diffusion terms. The drift term corresponds to the expected profit per unit time, and the diffusion term represents risk exposure. By using diffusion models, many kinds of optimal dividend problems, such as in [7, 14–16], are discussed and optimal policies are presented in these papers. Particularly, in some papers (see [14–16, 18]), authors discuss much more practical problems by considering a fixed transaction cost for each dividend payout. In [14], the optimal dividend problem without bankruptcy for insurance firms is considered under the assumptions of constant tax rate and fixed cost for dividend payout. In [15, 16], the author considers the general income process Xt with drift term μ(Xt) and diffusion term σ(Xt) and the case of bankruptcy. Moreover, numerical methods, such as the Runge-Kutta method, are implemented to simulate the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear differential equation. In [19], the author studies the dual risk model with a barrier strategy under the concept of bankruptcy, in which one has a positive probability to continue business despite temporary negative surplus. In [20], the author considers an insurance entity endowed with an initial capital and an income, modelled as a Brownian motion with drift and finds an explicit expression for the value function and for the optimal strategy in the first but not in the second case, where one has to switch to the viscosity ansatz. In [21], the authors suppose that a large insurance company can control its surplus process by reinsurance, paying dividends, or injecting capitals and obtain the explicit solutions for value function and optimal strategy.In these papers, the value function is typically assumed to be zero when there is a bankruptcy. But in the real world, some shareholders, especially for preferred shareholders, can get some money back when a terminal bankruptcy occurs. That means for this case the value function is not zero at bankruptcy. Thus, it is very useful and necessary for us to consider this kind of problem. In this paper, we postulate that the amount of money, shareholders can obtain for the terminal bankruptcy, is a positive constant,a. Moreover, we assume that the liquid assets Xt follows a process with constant drift and diffusion coefficients.In the model of this paper, as that in [14], the dividend distribution policy is given by a purely discontinuous increasing functional. The net amount of money received by shareholders is kξi-K for the i-th dividends, where ξi is the amount of the dividend payments, 1-k is the tax rate the shareholder pays, and K is the fixed cost whenever the dividends are paid out. Further, τi represents the moments of dividend payments and λ is the discount rate. Based on these assumptions, we transform the value function V(x) into quasi-variational inequalities (QVI) and list out a candidate solution v(x) to QVI with a positive boundary condition v(0)=a. Subsequently, we show that the value function can be given by v(x) and the optimal policy can be presented based on the solution v(x). A natural question is how to point out whether there is a bankruptcy or not in order to obtain the optimal policy under some conditions. To answer this question, some criteria are provided.A major difficulty in this paper is that the structure of the candidate solution is uncertain since the existing interval of it has unfixed endpoints, which depends on some unknown parameters. This phenomenon does not appear in [14] and other related papers. Enlightened by the derivatives of candidate solutions, we construct the integral I(C) and then discuss it by several cases for μ, σ, k, K, and a. For the model mentioned above, which is restricted to stay at the bankruptcy state, it is denoted by terminal bankruptcy model as in [22].The structure of this paper is as follows. In the next section, we provide a rigorous mathematical model for the optimal dividend problem. Then the stochastic control problem is transformed to a QVI. Moreover, some definitions and an important verification are presented. Following this, the detailed structure of candidate solutions is given under different situations in Section3. In Section 4, the uniqueness of some unfixed parameters is verified; some formulas to calculate these parameters are proposed and some numerical examples are shown to support our theoretical results. In Section 5, we demonstrate that the candidate solutions satisfy QVI and obtain the optimal dividend policy. In the last section, we summarize our results and suggest a direction for future research.


## 2. The Mathematical Model

The proposed model considers the dividend optimization problem for a firm which can control its business activities that affect its risk and potential profit. It extends the classical Miller-Modigliani model of firm valuation to the situation of controllable business activities in a stochastic environment with a possibility of bankruptcy and a positive residual value to shareholders upon bankruptcy. The model is quite general and can be applied to any firm in which management has control on the dividend stream as well as the risk exposure. Without loss of generality, the model is cast in the framework of a large insurance company, as it possesses many nice features to best illustrate the model.


### 2.1. Value Function

Let(Ω,F,P) be a probability space with a filtration Ftt≥0 and W={Wt;t≥0} be a standard Brownian motion adapted to that filtration. Moreover, the reserve process X={X(t);t≥0} is a state variable, which denotes the liquid assets of the company. For an insurance company, in order to reduce risk, the risk control takes up the form of proportional reinsurance, which mathematically corresponds to decreasing the drift and diffusion coefficient by multiplying both quantities by the same factor u(t)∈[0,1]. The time of dividends is described by a sequence of increasing stopping times {τi;i=1,2,…} and the amounts of the dividends paid out to the shareholders, associated with the times, are represented by a sequence of random variables {ξi;i=1,2,…}. Then the controlled state process X(t) before bankruptcy is given by(1)Xt=x+∫0tμusds+∫0tσusdWs-∑n=1∞Iτn<tξn,where x≥0 is the initial reserve and I(τn<t) is an indicator function.Let the time of bankruptcy be given by(2)τ=inf⁡t≥0:Xt=0.Definition 1.
Letu:Ω×[0,∞)→[0,1] be an Ftt≥0-adapted process; let τi, i=1,2,…, be a stopping time with respect to Ftt≥0, and let the random variable ξi, i=1,2,…, be {Fτi} measurable with 0≤ξi≤X(τi-); then(3)π=u,T,ξ,τ=u;τ1,τ2,…,τn,…;ξ1,ξ2,…,ξn,…;τis called an admissible control or an admissible policy. The class of all admissible controls is denoted by A(x).In addition, we denote the net amount of money that shareholders receive by a functiong:[0,∞)→(-∞,∞) as(4)gω=kω-K,where the constant K>0 is a fixed setup cost incurred each time that a dividend is paid out, and the constant 1-k∈(0,1) is the tax rate at which the dividends are taxed, and ω is a real value variable with respect to the amount of liquid assets withdrawn.A performance functionalJ with each admissible control π is defined by(5)Jx,π=Ex∑n=1∞e-λτngξnIτn<τ+e-λτa,which represents the total expected discounted value received by shareholders until the time of bankruptcy, where a is the known amount paid out to shareholders when the terminal bankruptcy happens.Define the value functionV(x) by(6)Vx=supπ∈AxEx∑n=1∞e-λτngξnIτn<τ+e-λτa.Then the optimal control π∗=(u∗,T∗,ξ∗,τ∗) is a policy for which the following equality can be satisfied(7)Vx=Jx,π∗.


### 2.2. Properties of the Value Function

In this section, the QVI associated with the stochastic control problem is provided. Moreover, we derive some properties of the value function.Proposition 2.
For everyx∈0,∞, the value function V(x) in (6) satisfies(8)Vx≤kx-e-λτa+μλ.Proof.
By the same method as in [14] and letting X(τ)=a instead of X(τ)=0, then the result can be obtained.Letg be given by (4); then, for a function ϕ:0,∞→R, define the maximum utility operator M of it by(9)Mϕx=supω⁡ϕx-ω+gω:0<ω≤x.Suppose that the payment of dividends occurs at time0 and the amount of it equals ω; then the reserve decreases from initial position x to x-ω. After that, if the optimal policy is followed, the total expected utility is kω-K+V(x-ω). Consequently, under such a policy, the total maximal expected utility would be equal to MV(x). On the other hand, for each initial positionx, suppose that there exists an optimal policy, which is optimal for the whole domain. Then the expected utility associated with this optimal policy is V(x), which is greater or equal to any expected utility associated with another different policy. So, we have(10)Vx≥MVx.Now, define(11)Luvx=12σ2u2v′′x+μuv′x-λvx.By the dynamic programming principle, we know that in the continuation region, V(x) satisfies(12)maxu∈0,1⁡LuVx=0.The arguments in (10) and (12) give us an intuition for the following two definitions and one theorem.Definition 3.
Assume that functionv(x):[0,∞)→[0,∞). For every x∈[0,∞) and u∈[0,1], if we have(13)QVIvx≥Mvx,Luvx≤0,vx-Mvxmaxu∈0,1⁡Luvx=0,v0=a,then we claim that v(x) satisfies the quasi-variational inequalities of the control problem.Definition 4.
The controlπv=uv,Tv,ξv,τv is called the QVI control associated with v if(14)Puvt≠argmaxu∈0,1⁡LuvXtv,Xtv∈C=0;τ0v=0,ξ0v=0;τ1v=inf⁡t≥0:vXvt=MvXvt,ξ1v=argsup0<η≤Xvτ1v⁡vXvτ1v-η+gηand, for every n≥2,(15)τnv=inf⁡t≥τn-1:vXvt=MvXvt,ξnv=argsup0<η≤Xvτnv⁡vXvτnv-η+gη;τv=inf⁡t≥0:Xvt=0.As in Cadenillas et al. [14], we also have the following theorem.Theorem 5.
Letv∈C10,∞ be a solution of QVI. Suppose there exists U>0 such that v is twice continuously differentiable on [0,U) and v is linear on [U,∞). Then, for any x∈[0,∞),(16)Vx≤vx.Furthermore, if the QVI control (uv,Tv,ξv,τv) associated with v is admissible, then v coincides with the value function and the QVI control associated with v is the optimal policy; that is,(17)Vx=vx=Jx;uv,Tv,ξv,τv.Proof.
The idea of this proof is very similar to that of Theorem3.4 in Cadenillas et al. [14]. So, we do not show it in this paper.


## 2.1. Value Function

Let(Ω,F,P) be a probability space with a filtration Ftt≥0 and W={Wt;t≥0} be a standard Brownian motion adapted to that filtration. Moreover, the reserve process X={X(t);t≥0} is a state variable, which denotes the liquid assets of the company. For an insurance company, in order to reduce risk, the risk control takes up the form of proportional reinsurance, which mathematically corresponds to decreasing the drift and diffusion coefficient by multiplying both quantities by the same factor u(t)∈[0,1]. The time of dividends is described by a sequence of increasing stopping times {τi;i=1,2,…} and the amounts of the dividends paid out to the shareholders, associated with the times, are represented by a sequence of random variables {ξi;i=1,2,…}. Then the controlled state process X(t) before bankruptcy is given by(1)Xt=x+∫0tμusds+∫0tσusdWs-∑n=1∞Iτn<tξn,where x≥0 is the initial reserve and I(τn<t) is an indicator function.Let the time of bankruptcy be given by(2)τ=inf⁡t≥0:Xt=0.Definition 1.
Letu:Ω×[0,∞)→[0,1] be an Ftt≥0-adapted process; let τi, i=1,2,…, be a stopping time with respect to Ftt≥0, and let the random variable ξi, i=1,2,…, be {Fτi} measurable with 0≤ξi≤X(τi-); then(3)π=u,T,ξ,τ=u;τ1,τ2,…,τn,…;ξ1,ξ2,…,ξn,…;τis called an admissible control or an admissible policy. The class of all admissible controls is denoted by A(x).In addition, we denote the net amount of money that shareholders receive by a functiong:[0,∞)→(-∞,∞) as(4)gω=kω-K,where the constant K>0 is a fixed setup cost incurred each time that a dividend is paid out, and the constant 1-k∈(0,1) is the tax rate at which the dividends are taxed, and ω is a real value variable with respect to the amount of liquid assets withdrawn.A performance functionalJ with each admissible control π is defined by(5)Jx,π=Ex∑n=1∞e-λτngξnIτn<τ+e-λτa,which represents the total expected discounted value received by shareholders until the time of bankruptcy, where a is the known amount paid out to shareholders when the terminal bankruptcy happens.Define the value functionV(x) by(6)Vx=supπ∈AxEx∑n=1∞e-λτngξnIτn<τ+e-λτa.Then the optimal control π∗=(u∗,T∗,ξ∗,τ∗) is a policy for which the following equality can be satisfied(7)Vx=Jx,π∗.


## 2.2. Properties of the Value Function

In this section, the QVI associated with the stochastic control problem is provided. Moreover, we derive some properties of the value function.Proposition 2.
For everyx∈0,∞, the value function V(x) in (6) satisfies(8)Vx≤kx-e-λτa+μλ.Proof.
By the same method as in [14] and letting X(τ)=a instead of X(τ)=0, then the result can be obtained.Letg be given by (4); then, for a function ϕ:0,∞→R, define the maximum utility operator M of it by(9)Mϕx=supω⁡ϕx-ω+gω:0<ω≤x.Suppose that the payment of dividends occurs at time0 and the amount of it equals ω; then the reserve decreases from initial position x to x-ω. After that, if the optimal policy is followed, the total expected utility is kω-K+V(x-ω). Consequently, under such a policy, the total maximal expected utility would be equal to MV(x). On the other hand, for each initial positionx, suppose that there exists an optimal policy, which is optimal for the whole domain. Then the expected utility associated with this optimal policy is V(x), which is greater or equal to any expected utility associated with another different policy. So, we have(10)Vx≥MVx.Now, define(11)Luvx=12σ2u2v′′x+μuv′x-λvx.By the dynamic programming principle, we know that in the continuation region, V(x) satisfies(12)maxu∈0,1⁡LuVx=0.The arguments in (10) and (12) give us an intuition for the following two definitions and one theorem.Definition 3.
Assume that functionv(x):[0,∞)→[0,∞). For every x∈[0,∞) and u∈[0,1], if we have(13)QVIvx≥Mvx,Luvx≤0,vx-Mvxmaxu∈0,1⁡Luvx=0,v0=a,then we claim that v(x) satisfies the quasi-variational inequalities of the control problem.Definition 4.
The controlπv=uv,Tv,ξv,τv is called the QVI control associated with v if(14)Puvt≠argmaxu∈0,1⁡LuvXtv,Xtv∈C=0;τ0v=0,ξ0v=0;τ1v=inf⁡t≥0:vXvt=MvXvt,ξ1v=argsup0<η≤Xvτ1v⁡vXvτ1v-η+gηand, for every n≥2,(15)τnv=inf⁡t≥τn-1:vXvt=MvXvt,ξnv=argsup0<η≤Xvτnv⁡vXvτnv-η+gη;τv=inf⁡t≥0:Xvt=0.As in Cadenillas et al. [14], we also have the following theorem.Theorem 5.
Letv∈C10,∞ be a solution of QVI. Suppose there exists U>0 such that v is twice continuously differentiable on [0,U) and v is linear on [U,∞). Then, for any x∈[0,∞),(16)Vx≤vx.Furthermore, if the QVI control (uv,Tv,ξv,τv) associated with v is admissible, then v coincides with the value function and the QVI control associated with v is the optimal policy; that is,(17)Vx=vx=Jx;uv,Tv,ξv,τv.Proof.
The idea of this proof is very similar to that of Theorem3.4 in Cadenillas et al. [14]. So, we do not show it in this paper.


## 3. Smooth Solutions to the QVI Properties

In this section, we first recall the zero boundary (no recovery) problem in Cadenillas et al. [14], and then by the similar method of this solved problem, we obtain the smooth solutions of QVI properties.


### 3.1. Solution for the Problem with Zero Boundary Condition

Let us consider the similar problem of QVI as follows:(18)QVI0v0x≥Mv0x,Luv0x≤0,v0x-Mv0xmaxu∈0,1Luv0x=0,v00=0.Let(19)γ=λλ+μ2/2σ2,and define(20)X0=1-γσ2μ,X1=inf⁡x≥0:v0x=Mv0x.Then from Cadenillas et al. [14], we can obtain the structure of the solution of QVI0:(21)v0x=C0xγ,x∈0,X0,C0a1eθ+x-X0+C0a2eθ-x-X0,x∈X0,X1,v0X~+kx-X~-K,x∈X1,∞,where C0 is a free constant and θ+, θ-, a1, and a2 are given by(22)θ+=-μ+μ2+2λσ2σ2,θ-=-μ-μ2+2λσ2σ2,(23)a1=γX0γ-1-θ-X0γθ+-θ-,a2=θ+X0γ-γX0γ-1θ+-θ-,and X~ is the unique solution of the following equation:(24)v0′x=k,x∈0,X1.In Cadenillas et al. [14], it has been shown that a1>0 and a2<0. Define a function H(x) by(25)Hx=γxγ-1,x∈0,X0,a1θ+eθ+x-X0+a2θ-eθ-x-X0,x∈X0,∞.Let(26)I0C~0=∫X~C~0X1C~0k-C~0Hxdx;then I0(C~0) is a decreasing function of C~0 with the range [0,+∞), and there exists unique C~0, X~C~0, and X1C~0, such that I0(C~0)=K.Further, Cadenillas et al. [14] show that the solution of QVI0 can be given by (21) with C0=C~0, X~=X~C~0, and X1=X1C~0.


### 3.2. Smooth Solutions of QVI Properties on[0,x1]

The difference between QVI andQVI0 is just the boundary condition. Thus, we conjecture that solutions of them may have some similarities.


#### 3.2.1. Smooth Solutions of QVI Properties on[0,x1)

First, as in (20), we define(27)x1=inf⁡x≥0:vx=Mvx.Then, on the interval (0,x1), from QVI we have that(28)maxu∈0,1⁡Luvx=0,0<x<x1.Letu(x)∈R be the maximizer of the expression on the left-hand side of (28); then(29)ux=-μv′xσ2v′′x.Putting (29) into (28), we have(30)-μ2v′x22σ2v′′x-λvx=0.A general solution of (30) under the boundary condition v(0)=a is(31)vx=CaC1/γ+xγ,where C is a free constant and γ is presented by (19). From (29) and (31), it follows that(32)ux=μ1-γσ2x+aC1/γ.Sinceu(x) above is an increasing linear function, u(x)≤1 if and only if x≤x0, where(33)x0=1-γσ2μ-aC1/γ=X0-aC1/γ.Thus, if x0>0 and x0<x<x1, it follows that u(x)≥1. But since the range of u(x) is [0,1], then we must have u(x)=1 for x∈(x0,x1); consequently (28) becomes(34)12σ2v′′x+μv′x-λvx=0,x∈x0,x1.One general solution to (34) can be written by(35)vx=C1eθ+x-x0+C2eθ-x-x0,x∈x0,x1,where C1 and C2 are free constants and θ+ and θ- are given by (22). Continuity of the function v(x) and its derivative v′(x) at the point x0 implies that C1=Ca1 and C2=Ca2, where C is a free constant and a1 and a2 are defined by (23).On the other hand, ifx0≤0, then u(x)=1, for any x∈[0,x1), and (28) becomes(36)12σ2v′′x+μv′x-λvx=0,x∈0,x1,v0=a.Solving (36), we can obtain the general solution of (36) as follows:(37)vx=C1eθ+x+C2eθ-x,x∈0,x1,with C1+C2=a.Now, let us summarize the possible structure for the solution of (28) on [0,x1). If x0>0, then(38)vx=CaC1/γ+xγ,x∈0,x0,Ca1eθ+x-x0+Ca2eθ-x-x0,x∈x0,x1,where C is a free constant. If x0≤0, then the structure of v(x) on [0,x1) is given by (37).Remark 6.
From (33), we notice that x0 depends on the uncertain parameter C, which will be estimated later. For different C, the sign of x0 may be different. Moreover, it is easy to show that the assumption, x0>0, is equivalent to C>aX0-γ and x0→0+ is equivalent to C→aX0-γ+. In addition, for (37) and (38), they are consistent at x0=0. That is, for any x∈(0,x1),(39)limC→aX0-γ+⁡Ca1eθ+x-x0+Ca2eθ-x-x0=aX0-γa1eθ+x+aX0-γa2eθ-x=limC1→aX0-γa1-⁡C1eθ+x+a-C1eθ-x.At C=aX0-γ, from the consistency of two solutions, we have that C1=Ca1. In addition, the condition of existence for (37) implies C≤aX0-γ. Moreover, there is no conflict to denote C1 by Ca1 when C≤aX0-γ. So, we let C1=Ca1 for C≤aX0-γ.Remark 7.
Forv(x) in (38), from a1>0 and a2<0, it is easy to show that v′′′(x)>0 on [0,x1), which implies that v′(x) obtained from (38) have convexity on [0,x1).


#### 3.2.2. Smooth Solution of QVI Properties atx1

From the definition ofx1, we have that v(x1)=Mv(x1). Then, by(40)limη→0⁡vx1-η+kη-K=vx1-K<vx1,it follows that the maximizing sequence η for v(x1)=Mv(x1) cannot have zero as a limiting point. So, at x1, the supremum on the right-hand side of (9) can be taken over η∈[ϵ,x1] for some ϵ>0. Therefore, there exists η(x1)∈(0,x1], such that(41)vx1=vx1-ηx1+kηx1-K.Let x~=x1-η(x1); then 0≤x~<x1 and(42)vx1=vx~+kx1-x~-K.From (42), it follows that(43)vx1-vx~=∫x~x1v′xdx=kx1-x~-K;then we have(44)∫x~x1k-v′xdx=K.Remark 8.
Ifx~=0, then v(x1)=v(0)+kx1-K=a+kx1-K, which means the dividend happens once and then a bankruptcy follows. This is an important phenomenon that deserves to be discussed.


## 3.1. Solution for the Problem with Zero Boundary Condition

Let us consider the similar problem of QVI as follows:(18)QVI0v0x≥Mv0x,Luv0x≤0,v0x-Mv0xmaxu∈0,1Luv0x=0,v00=0.Let(19)γ=λλ+μ2/2σ2,and define(20)X0=1-γσ2μ,X1=inf⁡x≥0:v0x=Mv0x.Then from Cadenillas et al. [14], we can obtain the structure of the solution of QVI0:(21)v0x=C0xγ,x∈0,X0,C0a1eθ+x-X0+C0a2eθ-x-X0,x∈X0,X1,v0X~+kx-X~-K,x∈X1,∞,where C0 is a free constant and θ+, θ-, a1, and a2 are given by(22)θ+=-μ+μ2+2λσ2σ2,θ-=-μ-μ2+2λσ2σ2,(23)a1=γX0γ-1-θ-X0γθ+-θ-,a2=θ+X0γ-γX0γ-1θ+-θ-,and X~ is the unique solution of the following equation:(24)v0′x=k,x∈0,X1.In Cadenillas et al. [14], it has been shown that a1>0 and a2<0. Define a function H(x) by(25)Hx=γxγ-1,x∈0,X0,a1θ+eθ+x-X0+a2θ-eθ-x-X0,x∈X0,∞.Let(26)I0C~0=∫X~C~0X1C~0k-C~0Hxdx;then I0(C~0) is a decreasing function of C~0 with the range [0,+∞), and there exists unique C~0, X~C~0, and X1C~0, such that I0(C~0)=K.Further, Cadenillas et al. [14] show that the solution of QVI0 can be given by (21) with C0=C~0, X~=X~C~0, and X1=X1C~0.


## 3.2. Smooth Solutions of QVI Properties on[0,x1]

The difference between QVI andQVI0 is just the boundary condition. Thus, we conjecture that solutions of them may have some similarities.


### 3.2.1. Smooth Solutions of QVI Properties on[0,x1)

First, as in (20), we define(27)x1=inf⁡x≥0:vx=Mvx.Then, on the interval (0,x1), from QVI we have that(28)maxu∈0,1⁡Luvx=0,0<x<x1.Letu(x)∈R be the maximizer of the expression on the left-hand side of (28); then(29)ux=-μv′xσ2v′′x.Putting (29) into (28), we have(30)-μ2v′x22σ2v′′x-λvx=0.A general solution of (30) under the boundary condition v(0)=a is(31)vx=CaC1/γ+xγ,where C is a free constant and γ is presented by (19). From (29) and (31), it follows that(32)ux=μ1-γσ2x+aC1/γ.Sinceu(x) above is an increasing linear function, u(x)≤1 if and only if x≤x0, where(33)x0=1-γσ2μ-aC1/γ=X0-aC1/γ.Thus, if x0>0 and x0<x<x1, it follows that u(x)≥1. But since the range of u(x) is [0,1], then we must have u(x)=1 for x∈(x0,x1); consequently (28) becomes(34)12σ2v′′x+μv′x-λvx=0,x∈x0,x1.One general solution to (34) can be written by(35)vx=C1eθ+x-x0+C2eθ-x-x0,x∈x0,x1,where C1 and C2 are free constants and θ+ and θ- are given by (22). Continuity of the function v(x) and its derivative v′(x) at the point x0 implies that C1=Ca1 and C2=Ca2, where C is a free constant and a1 and a2 are defined by (23).On the other hand, ifx0≤0, then u(x)=1, for any x∈[0,x1), and (28) becomes(36)12σ2v′′x+μv′x-λvx=0,x∈0,x1,v0=a.Solving (36), we can obtain the general solution of (36) as follows:(37)vx=C1eθ+x+C2eθ-x,x∈0,x1,with C1+C2=a.Now, let us summarize the possible structure for the solution of (28) on [0,x1). If x0>0, then(38)vx=CaC1/γ+xγ,x∈0,x0,Ca1eθ+x-x0+Ca2eθ-x-x0,x∈x0,x1,where C is a free constant. If x0≤0, then the structure of v(x) on [0,x1) is given by (37).Remark 6.
From (33), we notice that x0 depends on the uncertain parameter C, which will be estimated later. For different C, the sign of x0 may be different. Moreover, it is easy to show that the assumption, x0>0, is equivalent to C>aX0-γ and x0→0+ is equivalent to C→aX0-γ+. In addition, for (37) and (38), they are consistent at x0=0. That is, for any x∈(0,x1),(39)limC→aX0-γ+⁡Ca1eθ+x-x0+Ca2eθ-x-x0=aX0-γa1eθ+x+aX0-γa2eθ-x=limC1→aX0-γa1-⁡C1eθ+x+a-C1eθ-x.At C=aX0-γ, from the consistency of two solutions, we have that C1=Ca1. In addition, the condition of existence for (37) implies C≤aX0-γ. Moreover, there is no conflict to denote C1 by Ca1 when C≤aX0-γ. So, we let C1=Ca1 for C≤aX0-γ.Remark 7.
Forv(x) in (38), from a1>0 and a2<0, it is easy to show that v′′′(x)>0 on [0,x1), which implies that v′(x) obtained from (38) have convexity on [0,x1).


### 3.2.2. Smooth Solution of QVI Properties atx1

From the definition ofx1, we have that v(x1)=Mv(x1). Then, by(40)limη→0⁡vx1-η+kη-K=vx1-K<vx1,it follows that the maximizing sequence η for v(x1)=Mv(x1) cannot have zero as a limiting point. So, at x1, the supremum on the right-hand side of (9) can be taken over η∈[ϵ,x1] for some ϵ>0. Therefore, there exists η(x1)∈(0,x1], such that(41)vx1=vx1-ηx1+kηx1-K.Let x~=x1-η(x1); then 0≤x~<x1 and(42)vx1=vx~+kx1-x~-K.From (42), it follows that(43)vx1-vx~=∫x~x1v′xdx=kx1-x~-K;then we have(44)∫x~x1k-v′xdx=K.Remark 8.
Ifx~=0, then v(x1)=v(0)+kx1-K=a+kx1-K, which means the dividend happens once and then a bankruptcy follows. This is an important phenomenon that deserves to be discussed.


## 3.2.1. Smooth Solutions of QVI Properties on[0,x1)

First, as in (20), we define(27)x1=inf⁡x≥0:vx=Mvx.Then, on the interval (0,x1), from QVI we have that(28)maxu∈0,1⁡Luvx=0,0<x<x1.Letu(x)∈R be the maximizer of the expression on the left-hand side of (28); then(29)ux=-μv′xσ2v′′x.Putting (29) into (28), we have(30)-μ2v′x22σ2v′′x-λvx=0.A general solution of (30) under the boundary condition v(0)=a is(31)vx=CaC1/γ+xγ,where C is a free constant and γ is presented by (19). From (29) and (31), it follows that(32)ux=μ1-γσ2x+aC1/γ.Sinceu(x) above is an increasing linear function, u(x)≤1 if and only if x≤x0, where(33)x0=1-γσ2μ-aC1/γ=X0-aC1/γ.Thus, if x0>0 and x0<x<x1, it follows that u(x)≥1. But since the range of u(x) is [0,1], then we must have u(x)=1 for x∈(x0,x1); consequently (28) becomes(34)12σ2v′′x+μv′x-λvx=0,x∈x0,x1.One general solution to (34) can be written by(35)vx=C1eθ+x-x0+C2eθ-x-x0,x∈x0,x1,where C1 and C2 are free constants and θ+ and θ- are given by (22). Continuity of the function v(x) and its derivative v′(x) at the point x0 implies that C1=Ca1 and C2=Ca2, where C is a free constant and a1 and a2 are defined by (23).On the other hand, ifx0≤0, then u(x)=1, for any x∈[0,x1), and (28) becomes(36)12σ2v′′x+μv′x-λvx=0,x∈0,x1,v0=a.Solving (36), we can obtain the general solution of (36) as follows:(37)vx=C1eθ+x+C2eθ-x,x∈0,x1,with C1+C2=a.Now, let us summarize the possible structure for the solution of (28) on [0,x1). If x0>0, then(38)vx=CaC1/γ+xγ,x∈0,x0,Ca1eθ+x-x0+Ca2eθ-x-x0,x∈x0,x1,where C is a free constant. If x0≤0, then the structure of v(x) on [0,x1) is given by (37).Remark 6.
From (33), we notice that x0 depends on the uncertain parameter C, which will be estimated later. For different C, the sign of x0 may be different. Moreover, it is easy to show that the assumption, x0>0, is equivalent to C>aX0-γ and x0→0+ is equivalent to C→aX0-γ+. In addition, for (37) and (38), they are consistent at x0=0. That is, for any x∈(0,x1),(39)limC→aX0-γ+⁡Ca1eθ+x-x0+Ca2eθ-x-x0=aX0-γa1eθ+x+aX0-γa2eθ-x=limC1→aX0-γa1-⁡C1eθ+x+a-C1eθ-x.At C=aX0-γ, from the consistency of two solutions, we have that C1=Ca1. In addition, the condition of existence for (37) implies C≤aX0-γ. Moreover, there is no conflict to denote C1 by Ca1 when C≤aX0-γ. So, we let C1=Ca1 for C≤aX0-γ.Remark 7.
Forv(x) in (38), from a1>0 and a2<0, it is easy to show that v′′′(x)>0 on [0,x1), which implies that v′(x) obtained from (38) have convexity on [0,x1).


## 3.2.2. Smooth Solution of QVI Properties atx1

From the definition ofx1, we have that v(x1)=Mv(x1). Then, by(40)limη→0⁡vx1-η+kη-K=vx1-K<vx1,it follows that the maximizing sequence η for v(x1)=Mv(x1) cannot have zero as a limiting point. So, at x1, the supremum on the right-hand side of (9) can be taken over η∈[ϵ,x1] for some ϵ>0. Therefore, there exists η(x1)∈(0,x1], such that(41)vx1=vx1-ηx1+kηx1-K.Let x~=x1-η(x1); then 0≤x~<x1 and(42)vx1=vx~+kx1-x~-K.From (42), it follows that(43)vx1-vx~=∫x~x1v′xdx=kx1-x~-K;then we have(44)∫x~x1k-v′xdx=K.Remark 8.
Ifx~=0, then v(x1)=v(0)+kx1-K=a+kx1-K, which means the dividend happens once and then a bankruptcy follows. This is an important phenomenon that deserves to be discussed.


## 4. Uniqueness for the Unfixed Parameters

In Section3, some parameters, such as C, C1, and C2, are unfixed numbers. In this section, we discuss the uniqueness of their corresponding parameters by two useful integral functions, whose integrands can be used to obtain solutions of QVI. As in Remark 6, we claim that Ca1 can be used to denote C1 for C≤aX0-γ.


### 4.1. Definitions and Properties of Two Integral Functions

From Remark6, it is known that different C will lead to two possible cases for x0. The first case is x0>0 and the second is x0≤0. In the following, we discuss these two cases by two constructed integral functions, respectively.Case 1 (x0>0).
LetHC(x) be a function, with constant C, constructed by(45)HCx=γaC1/γ+xγ-1,x∈0,x0,a1θ+eθ+x-x0+a2θ-eθ-x-x0,x∈x0,∞,where x0 is also defined by (33) with C>aX0-γ.Define(46)I1C=∫x~C∨0x1Ck-CHCxdx,where x1C and x~C are two nonnegative roots of the equation k-CHC(x)=0 with x~C<x1C and x~C∨0 denotes max⁡{x~C,0}. If x~C does not exist on [0,∞), then let x~C∨0=0.From the definitions ofHC(x) and I1(C), obviously CHC(x) is a continuous function of C. Further, x~C∨0 and x1C are also continuous functions of C if the existence condition of them is satisfied. So, I1(C) is a continuous function.Proposition 9.
LetHC(x) be defined by (45); then it has convexity on [0,∞) by HC′′(x)>0. Moreover, CHC(x) is an increasing function with respect to C by CHCx′>0. For I1(C), if the condition of its existence is satisfied on some subintervals of (aX0-γ,∞), then it is a strictly decreasing function on these subintervals.Case 2 (x0≤0).
Let(47)HC1x=C1θ+eθ+x+a-C1θ-eθ-x,x∈0,∞,where 0<C1≤aX0-γa1.Define(48)I2C1=∫x~C1∨0x1C1k-HC1xdx,where x1C1 and x~C1 are two nonnegative roots of the equation k-HC1(x)=0 with x~C1<x1C1. If x~C1 does not exist on [0,∞), then let x~C1∨0=0.For (48), taking the derivative of I2(C1) with respect to C1 gives(49)I2′C1=∫x~C1∨0x1C1-θ+eθ+x+θ-eθ-xdx.Consequently, for any positive C1, I2′(C1)<0 due to the fact that θ+>0 and θ-<0. So, we have the following result.Proposition 10.
ForI2(C1), if its existence condition is satisfied on some subintervals of (0,aX0-γa1), then it is a continuous and strictly decreasing function on these subintervals. Moreover, we can show that limC1→0⁡x1C1=∞ and limC1→0⁡I2(C1)=∞.Remark 11.
For small enough and positiveC1, it is possible that a-C1>0. Under this situation, HC1(x) may have no convexity since HC1′′(x)>0 may not be satisfied.In the following, we discuss the property of two integral functions atx0=0.Atx0=0 there are some common properties of I1(C) and I2(Ca1). Firstly, it follows that C1=Ca1 and C=aX0-γ. Then from(50)a-C1=a-aX0-γa1=a1-X0-γa1=aX0-γX0γ-a1=aX0-γa2,we obtain that at x0=0 the integrands of I1C and I2C1 are the same. Consequently, we can conclude that if x1C1 exists at C1=aX0-γa1, then I2aX0-γa1=I1aX0-γ. To judge whether I2aX0-γa1>0, we have the following result.Proposition 12.
One equivalent condition ofI2aX0-γa1>0 is k>aM∗, where M∗ is given by(51)M∗=-θ-θ+θ-/θ+-θ-·2λμ-θ-.Proof.
LetB(x)=aX0-γa1θ+eθ+x+aX0-γa2θ-eθ-x. Notice that B′′(x)>0, and so B(x) has convexity. One equivalent condition of I2aX0-γa1>0 is that(52)minx∈0,∞⁡Bx<k.Solving B′(x)=0, we have that(53)x=1θ+-θ-ln⁡-a2θ-2a1θ+2.
It can check that-a2θ-2/a1θ+2>1, so x>0 in (53).
Putting (53) into B(x), we have(54)minx∈0,∞⁡Bx=aX0-γa1θ+-a2θ-2a1θ+2θ+/θ+-θ-+a2θ--a2θ-2a1θ+2θ-/θ+-θ-=aX0-γ-a2θ-2a1θ+2θ+/θ+-θ-·a1θ+θ--θ+θ-=a-θ-θ+θ-/θ+-θ-·2λμ-θ-=aM∗.
From (52) and (54), we can complete the proof.Remark 13.
An important inequality with respect toM∗ given by (51) is λ/μ<M∗<2λ/μ.Proof.
ForM∗ given by (51), taking its logarithm, we can get(55)ln⁡M∗=θ-θ+-θ-ln⁡-θ--ln⁡θ++ln⁡2λμ-θ-=-μ2μ2+2λσ2+12ln⁡μ+μ2+2λσ2-ln⁡-μ+μ2+2λσ2+ln⁡μ+μ2+2λσ2+12ln⁡μ2+2λσ2-ln⁡μ-ln⁡σ2.Let us check the derivative of M∗ on σ2 as follows:(56)∂ln⁡M∗∂σ2=-μ2·-12·μ2+2λσ2-3/2·2λ·ln⁡μ+μ2+2λσ2-μ+μ2+2λσ2-μ2μ2+2λσ2+12·λμ2+2λσ2-1/2μ+μ2+2λσ2-λμ2+2λσ2-1/2-μ+μ2+2λσ2+λμ2+2λσ2-1/2μ+μ2+2λσ2+λμ2+2λσ2-1σ2=λμ2·μ2+2λσ2-3/2·ln⁡μ+μ2+2λσ2-μ+μ2+2λσ2>0.Therefore, M∗ is an increasing function of σ2. From σ2∈(0,∞), we can take limits as σ2→0 and σ2→∞ to get the infimum and the supremum of M∗. For (55), it follows that(57)limσ2→0⁡ln⁡M∗=-ln⁡2μ+ln⁡2μ+ln⁡μ-ln⁡μ+limσ2→0ln⁡-μ+μ2+2λσ2σ2=ln⁡λμ,limσ2→∞⁡ln⁡M∗=-ln⁡μ+limσ2→∞⁡ln⁡-μ+μ2+2λσ2μ2+2λσ2σ2=ln2λμ.
Since the infimum ofM∗ is λ/μ and the supremum of M∗ is 2λ/μ, we can get λ/μ<M∗<2λ/μ.


### 4.2. Integral FunctionsI(C) under Different Conditions

By the convexity ofHC(x) on [0,∞), it shows that CHC(0)≤k, that is, C≤a1-γk/γγ, is equivalent to x~C∨0=0. From the definition of HC(x), it is known that C>aX0-γ. We compare aX0-γ to a1-γk/γγ as follows.Case 1 (aX0-γ<a1-γ(k/γ)γ).
By simplicity, this condition is equivalent tok>2aλ/μ. In addition, at point C=a1-γk/γγ, from (45) we have that CHC(0)=k and CHC′(0)<0. Combining with the convexity of HC(x), it follows that x1C>0 and k-CHC(x)>0 on (0,x1C).Proposition 14.
Ifk>2aλ/μ, then I1a1-γk/γγ>0 is satisfied. Consequently, from the monotonicity of I1(C), we have I2(aX0-γa1)=I1(aX0-γ)>0.From the above, it shows thataX0-γ can be the left domain endpoint for I1(C). On the other hand, from CHC(x)→∞ as C→∞ and I1(C) is a continuous function, then there exists C∗>a1-γk/γγ, such that I1(C∗)=0. Therefore, (aX0-γ,C∗) can be the domain of I1(C). Moreover, we have that 0,aX0-γa1 can be the domain of I2(C1) from Section 4.1.Remark 15.
Ifk>2aλ/μ, we have I2(Ca1)∈I2aX0-γa1,∞, for C∈0,aX0-γ and I1(C)∈0,I2aX0-γa1, for C∈aX0-γ,C∗, where C∗>a1-γk/γγ.Therefore, we can define(58)IC=I2Ca1,C∈0,aX0-γ,I1C,C∈aX0-γ,C∗as a useful integral function under aX0-γ<a1-γk/γγ. Moreover, I(C) is a strictly decreasing function.Case 2 (aX0-γ≥a1-γk/γγ).
By simplicity, this assumption is equivalent tok≤2aλ/μ. In addition, in I2(C1) the inequality HC1(0)≤k is equivalent to C1≤k-aθ-/θ+-θ-. From (22), we can obtain that θ-2(k-aθ+)/θ+2(k-aθ-)>1 is equivalent to k>aλ/μ. Then, for 0<x<1/θ+-θ-ln⁡θ-2k-aθ+/θ+2k-aθ-, it can be easily seen that HC1′(x)<0 at C1=k-aθ-/θ+-θ-. By the similar discussion as above, we can show that as follows.Proposition 16.
Ifaλ/μ<k≤2aλ/μ, then I2k-aθ-/θ+-θ->0. If k≤aλ/μ, then I2k-aθ-/θ+-θ-=0 and I2(C1)>0 for any C1∈0,k-aθ-/θ+-θ-.Remark 17.
Ifk≤aλ/μ, we have I2(Ca1)∈(0,+∞) for C∈(0,C∗), where C∗=k-aθ-/a1(θ+-θ-). If aλ/μ<k≤aM∗, we have I2(Ca1)∈(0,+∞) for C∈(0,C∗), where k-aθ-/a1(θ+-θ-)<C∗≤aX0-γ. If aM∗<k≤2aλ/μ, we have I2(Ca1)∈I2aX0-γa1,∞, for C∈0,aX0-γ, and I1(C)∈0,I2aX0-γa1, for C∈aX0-γ,C∗, where C∗>aX0-γ.Therefore, we can also useI1(C) and I2(Ca1) to define three types of I(C) similar to (58) as useful integral functions under aX0-γ≥a1-γ(k/γ)γ. Moreover, these types of I(C) are strictly decreasing functions.


### 4.3. Compute ParameterC and Numerical Examples

In the process of calculatingC, we can follow the same steps for these two cases. By the following steps, the uncertain parameter C can be obtained. Further, we can get other parameters, such as x1C, x1C1, x~C, and x~C1.Step 1.
CompareaM∗ with k.Step 2.
Ifk>aM∗, Compute I2(aX0-γa1), and then compare I2(aX0-γa1) with K.(i)
IfI2(aX0-γa1)>K, get C on (aX0-γ,∞), such that I1(C)=K.(ii)
IfI2(aX0-γa1)≤K, get C on (0,aX0-γ), such that I2(Ca1)=K.Step 3.
Ifk≤aM∗, then compare k with aλ/μ and solve I(C)=K.In practical problems,k∈(0,1), but the restriction for a is just a>0. In order to satisfy 0<k<1, for convenience, we just choose different values for a and never change the values of k, μ, λ, and σ in the following numerical examples. The values of these parameters are selected as μ=0.12, λ=0.002, σ=1.6, and k=0.13. For the value of a>0, we choose a=2,4,6,8 to satisfy four different conditions. In Figure 1, the graphs of I(C) are shown, and both forms of I(C), I1(C), and I2(Ca1) are listed out in their corresponding domains. Moreover, the right domain endpoint C∗ of I(C) is also presented. Figure 1 shows that I(C) is always a decreasing function.Figure 1
I ( C ) with respect to a.
(a)
a = 2 (b)
a = 4 (c)
a = 6 (d)
a = 8In Figure1(a), a=2, which satisfies k>2aλ/μ. We let C1∗/a1=aX0-γ and C0∗=a1-γk/γγ. We can see I(C) is composed of I2(Ca1) for C∈(0,C1∗/a1] and I1(C) for C∈(C1∗/a1,C∗) with I2(C1∗)=5.2494>0 and I1(C0∗)=3.5408>0. This phenomenon is consistent with Proposition 14. In Figure 1(b), a=4, which satisfies aM∗<k≤2aλ/μ, and C0∗ is given by C0∗=aX0-γ. Since k-aθ-/a1(θ+-θ-)=1.3935 is very close to C0∗=1.4018, we just draw the position of C0∗ in the graph. In fact, under aM∗<k≤2aλ/μ, this phenomenon happens almost on any other trial, for which we select different k, μ, λ, and σ. Figure 1(b) shows I2(C0∗a1)>0, which supports Proposition 12 correctly. In Figure 1(c), we choose a=6, which satisfies aλ/μ<k≤aM∗. I(C) is just composed of I2(Ca1) with I2k-aθ-/θ+-θ-=0.1631>0, which matches Proposition 16 very well. Since the value 0.1631, compared with the numbers of y-axis, is so small, it is not obvious in the graph. In Figure 1(d), a=8, which satisfies k≤aλ/μ. We can obtain k-aθ-/a1(θ+-θ-)=2.4651 by calculation. From numerical simulation we get C∗=2.4651, so C∗=k-aθ-/a1(θ+-θ-) is satisfied. Therefore, the conclusion in Proposition 16 is shown.Table1 summarizes the results for different parameter values. As shown in Table 1 (starting from row 4), for each case in Figure 1 where a>0, we choose three different values of K to calculate parameters such as C, x~∨0, x0∨0, and x1 from I(C)=K. For Figure 1(a) with a=2, from I2(C1∗)=5.2494 and I1(C0∗)=3.5408, three potential values of K are selected as K=0.15<I1(C0∗), I1(C0∗)<K=4<I2(C1∗), and K=8>I2(C1∗). By the same selection method, for Figure 1(b) with a=4, from I2k-aθ-/θ+-θ-=1.0669 and I2(C0∗a1)=1.0150, three trials for K are K=0.15,1.05,4. For Figure 1(c) with a=6, from I2k-aθ-/θ+-θ-=0.1631, three comparable values of K are given by K=0.15,1.05,4 and the same is applied in Figure 1(d) with a=8. This table shows that the results, whether the values of x~∨0 and x0∨0 are zeroes or not, are totally consistent with the theoretical results in Sections 4.1 and 4.2. In addition, C, x~∨0, and x0∨0 are decreasing and x1 is increasing when K is increasing under the same a. On the other hand, under the same K, C is increasing and x1 is decreasing as a is increasing.Table 1
Parameter estimations underμ=0.12, λ=0.002, k=0.13, and σ=1.6.
Conditions a K C x ~ ∨ 0 x 0 ∨ 0 x 1 I ( C ) k > 2 a λ μ 0 0.15 1.5748 15.9820 12.4675 49.9571 I 0 ( C ) k > 2 a λ μ 0 4 0.8671 5.7239 12.4675 91.9615 I 0 ( C )  k > 2 a λ μ 0 8 0.5196 2.3827 12.4675 127.4305 I 0 ( C )   k > 2 a λ μ 2 0.15 1.7489 18.4531 11.0865 40.0506 I 1 ( C ) k > 2 a λ μ 2 4 0.8648 0 4.9491 84.6289 I 1 ( C )  k > 2 a λ μ 2 8 0.5163 0 0 115.3948 I 2 ( C a 1 )   a M ∗ < k ≤ 2 a λ μ 4 0.15 1.7490 12.5186 5.1482 34.1052 I 1 ( C ) a M ∗ < k ≤ 2 a λ μ 4 1.05 1.3962 0.1815 0 46.2875 I 2 ( C a 1 )  a M ∗ < k ≤ 2 a λ μ 4 4 1.0055 0 0 69.2676 I 2 ( C a 1 )   a λ μ < k ≤ a M ∗ 6 0.15 1.9325 0.4645 0 22.0481 I 2 ( C a 1 ) a λ μ < k ≤ a M ∗ 6 1.05 1.7409 0 0 31.7341 I 2 ( C a 1 )  a λ μ < k ≤ a M ∗ 6 4 1.2550 0 0 54.2053 I 2 ( C a 1 )   k ≤ a λ μ 8 0.15 2.3953 0 0 11.0653 I 2 ( C a 1 ) k ≤ a λ μ 8 1.05 2.1493 0 0 21.6008 I 2 ( C a 1 )  k ≤ a λ μ 8 4 1.5614 0 0 40.7502 I 2 ( C a 1 )


## 4.1. Definitions and Properties of Two Integral Functions

From Remark6, it is known that different C will lead to two possible cases for x0. The first case is x0>0 and the second is x0≤0. In the following, we discuss these two cases by two constructed integral functions, respectively.Case 1 (x0>0).
LetHC(x) be a function, with constant C, constructed by(45)HCx=γaC1/γ+xγ-1,x∈0,x0,a1θ+eθ+x-x0+a2θ-eθ-x-x0,x∈x0,∞,where x0 is also defined by (33) with C>aX0-γ.Define(46)I1C=∫x~C∨0x1Ck-CHCxdx,where x1C and x~C are two nonnegative roots of the equation k-CHC(x)=0 with x~C<x1C and x~C∨0 denotes max⁡{x~C,0}. If x~C does not exist on [0,∞), then let x~C∨0=0.From the definitions ofHC(x) and I1(C), obviously CHC(x) is a continuous function of C. Further, x~C∨0 and x1C are also continuous functions of C if the existence condition of them is satisfied. So, I1(C) is a continuous function.Proposition 9.
LetHC(x) be defined by (45); then it has convexity on [0,∞) by HC′′(x)>0. Moreover, CHC(x) is an increasing function with respect to C by CHCx′>0. For I1(C), if the condition of its existence is satisfied on some subintervals of (aX0-γ,∞), then it is a strictly decreasing function on these subintervals.Case 2 (x0≤0).
Let(47)HC1x=C1θ+eθ+x+a-C1θ-eθ-x,x∈0,∞,where 0<C1≤aX0-γa1.Define(48)I2C1=∫x~C1∨0x1C1k-HC1xdx,where x1C1 and x~C1 are two nonnegative roots of the equation k-HC1(x)=0 with x~C1<x1C1. If x~C1 does not exist on [0,∞), then let x~C1∨0=0.For (48), taking the derivative of I2(C1) with respect to C1 gives(49)I2′C1=∫x~C1∨0x1C1-θ+eθ+x+θ-eθ-xdx.Consequently, for any positive C1, I2′(C1)<0 due to the fact that θ+>0 and θ-<0. So, we have the following result.Proposition 10.
ForI2(C1), if its existence condition is satisfied on some subintervals of (0,aX0-γa1), then it is a continuous and strictly decreasing function on these subintervals. Moreover, we can show that limC1→0⁡x1C1=∞ and limC1→0⁡I2(C1)=∞.Remark 11.
For small enough and positiveC1, it is possible that a-C1>0. Under this situation, HC1(x) may have no convexity since HC1′′(x)>0 may not be satisfied.In the following, we discuss the property of two integral functions atx0=0.Atx0=0 there are some common properties of I1(C) and I2(Ca1). Firstly, it follows that C1=Ca1 and C=aX0-γ. Then from(50)a-C1=a-aX0-γa1=a1-X0-γa1=aX0-γX0γ-a1=aX0-γa2,we obtain that at x0=0 the integrands of I1C and I2C1 are the same. Consequently, we can conclude that if x1C1 exists at C1=aX0-γa1, then I2aX0-γa1=I1aX0-γ. To judge whether I2aX0-γa1>0, we have the following result.Proposition 12.
One equivalent condition ofI2aX0-γa1>0 is k>aM∗, where M∗ is given by(51)M∗=-θ-θ+θ-/θ+-θ-·2λμ-θ-.Proof.
LetB(x)=aX0-γa1θ+eθ+x+aX0-γa2θ-eθ-x. Notice that B′′(x)>0, and so B(x) has convexity. One equivalent condition of I2aX0-γa1>0 is that(52)minx∈0,∞⁡Bx<k.Solving B′(x)=0, we have that(53)x=1θ+-θ-ln⁡-a2θ-2a1θ+2.
It can check that-a2θ-2/a1θ+2>1, so x>0 in (53).
Putting (53) into B(x), we have(54)minx∈0,∞⁡Bx=aX0-γa1θ+-a2θ-2a1θ+2θ+/θ+-θ-+a2θ--a2θ-2a1θ+2θ-/θ+-θ-=aX0-γ-a2θ-2a1θ+2θ+/θ+-θ-·a1θ+θ--θ+θ-=a-θ-θ+θ-/θ+-θ-·2λμ-θ-=aM∗.
From (52) and (54), we can complete the proof.Remark 13.
An important inequality with respect toM∗ given by (51) is λ/μ<M∗<2λ/μ.Proof.
ForM∗ given by (51), taking its logarithm, we can get(55)ln⁡M∗=θ-θ+-θ-ln⁡-θ--ln⁡θ++ln⁡2λμ-θ-=-μ2μ2+2λσ2+12ln⁡μ+μ2+2λσ2-ln⁡-μ+μ2+2λσ2+ln⁡μ+μ2+2λσ2+12ln⁡μ2+2λσ2-ln⁡μ-ln⁡σ2.Let us check the derivative of M∗ on σ2 as follows:(56)∂ln⁡M∗∂σ2=-μ2·-12·μ2+2λσ2-3/2·2λ·ln⁡μ+μ2+2λσ2-μ+μ2+2λσ2-μ2μ2+2λσ2+12·λμ2+2λσ2-1/2μ+μ2+2λσ2-λμ2+2λσ2-1/2-μ+μ2+2λσ2+λμ2+2λσ2-1/2μ+μ2+2λσ2+λμ2+2λσ2-1σ2=λμ2·μ2+2λσ2-3/2·ln⁡μ+μ2+2λσ2-μ+μ2+2λσ2>0.Therefore, M∗ is an increasing function of σ2. From σ2∈(0,∞), we can take limits as σ2→0 and σ2→∞ to get the infimum and the supremum of M∗. For (55), it follows that(57)limσ2→0⁡ln⁡M∗=-ln⁡2μ+ln⁡2μ+ln⁡μ-ln⁡μ+limσ2→0ln⁡-μ+μ2+2λσ2σ2=ln⁡λμ,limσ2→∞⁡ln⁡M∗=-ln⁡μ+limσ2→∞⁡ln⁡-μ+μ2+2λσ2μ2+2λσ2σ2=ln2λμ.
Since the infimum ofM∗ is λ/μ and the supremum of M∗ is 2λ/μ, we can get λ/μ<M∗<2λ/μ.


## 4.2. Integral FunctionsI(C) under Different Conditions

By the convexity ofHC(x) on [0,∞), it shows that CHC(0)≤k, that is, C≤a1-γk/γγ, is equivalent to x~C∨0=0. From the definition of HC(x), it is known that C>aX0-γ. We compare aX0-γ to a1-γk/γγ as follows.Case 1 (aX0-γ<a1-γ(k/γ)γ).
By simplicity, this condition is equivalent tok>2aλ/μ. In addition, at point C=a1-γk/γγ, from (45) we have that CHC(0)=k and CHC′(0)<0. Combining with the convexity of HC(x), it follows that x1C>0 and k-CHC(x)>0 on (0,x1C).Proposition 14.
Ifk>2aλ/μ, then I1a1-γk/γγ>0 is satisfied. Consequently, from the monotonicity of I1(C), we have I2(aX0-γa1)=I1(aX0-γ)>0.From the above, it shows thataX0-γ can be the left domain endpoint for I1(C). On the other hand, from CHC(x)→∞ as C→∞ and I1(C) is a continuous function, then there exists C∗>a1-γk/γγ, such that I1(C∗)=0. Therefore, (aX0-γ,C∗) can be the domain of I1(C). Moreover, we have that 0,aX0-γa1 can be the domain of I2(C1) from Section 4.1.Remark 15.
Ifk>2aλ/μ, we have I2(Ca1)∈I2aX0-γa1,∞, for C∈0,aX0-γ and I1(C)∈0,I2aX0-γa1, for C∈aX0-γ,C∗, where C∗>a1-γk/γγ.Therefore, we can define(58)IC=I2Ca1,C∈0,aX0-γ,I1C,C∈aX0-γ,C∗as a useful integral function under aX0-γ<a1-γk/γγ. Moreover, I(C) is a strictly decreasing function.Case 2 (aX0-γ≥a1-γk/γγ).
By simplicity, this assumption is equivalent tok≤2aλ/μ. In addition, in I2(C1) the inequality HC1(0)≤k is equivalent to C1≤k-aθ-/θ+-θ-. From (22), we can obtain that θ-2(k-aθ+)/θ+2(k-aθ-)>1 is equivalent to k>aλ/μ. Then, for 0<x<1/θ+-θ-ln⁡θ-2k-aθ+/θ+2k-aθ-, it can be easily seen that HC1′(x)<0 at C1=k-aθ-/θ+-θ-. By the similar discussion as above, we can show that as follows.Proposition 16.
Ifaλ/μ<k≤2aλ/μ, then I2k-aθ-/θ+-θ->0. If k≤aλ/μ, then I2k-aθ-/θ+-θ-=0 and I2(C1)>0 for any C1∈0,k-aθ-/θ+-θ-.Remark 17.
Ifk≤aλ/μ, we have I2(Ca1)∈(0,+∞) for C∈(0,C∗), where C∗=k-aθ-/a1(θ+-θ-). If aλ/μ<k≤aM∗, we have I2(Ca1)∈(0,+∞) for C∈(0,C∗), where k-aθ-/a1(θ+-θ-)<C∗≤aX0-γ. If aM∗<k≤2aλ/μ, we have I2(Ca1)∈I2aX0-γa1,∞, for C∈0,aX0-γ, and I1(C)∈0,I2aX0-γa1, for C∈aX0-γ,C∗, where C∗>aX0-γ.Therefore, we can also useI1(C) and I2(Ca1) to define three types of I(C) similar to (58) as useful integral functions under aX0-γ≥a1-γ(k/γ)γ. Moreover, these types of I(C) are strictly decreasing functions.


## 4.3. Compute ParameterC and Numerical Examples

In the process of calculatingC, we can follow the same steps for these two cases. By the following steps, the uncertain parameter C can be obtained. Further, we can get other parameters, such as x1C, x1C1, x~C, and x~C1.Step 1.
CompareaM∗ with k.Step 2.
Ifk>aM∗, Compute I2(aX0-γa1), and then compare I2(aX0-γa1) with K.(i)
IfI2(aX0-γa1)>K, get C on (aX0-γ,∞), such that I1(C)=K.(ii)
IfI2(aX0-γa1)≤K, get C on (0,aX0-γ), such that I2(Ca1)=K.Step 3.
Ifk≤aM∗, then compare k with aλ/μ and solve I(C)=K.In practical problems,k∈(0,1), but the restriction for a is just a>0. In order to satisfy 0<k<1, for convenience, we just choose different values for a and never change the values of k, μ, λ, and σ in the following numerical examples. The values of these parameters are selected as μ=0.12, λ=0.002, σ=1.6, and k=0.13. For the value of a>0, we choose a=2,4,6,8 to satisfy four different conditions. In Figure 1, the graphs of I(C) are shown, and both forms of I(C), I1(C), and I2(Ca1) are listed out in their corresponding domains. Moreover, the right domain endpoint C∗ of I(C) is also presented. Figure 1 shows that I(C) is always a decreasing function.Figure 1
I ( C ) with respect to a.
(a)
a = 2 (b)
a = 4 (c)
a = 6 (d)
a = 8In Figure1(a), a=2, which satisfies k>2aλ/μ. We let C1∗/a1=aX0-γ and C0∗=a1-γk/γγ. We can see I(C) is composed of I2(Ca1) for C∈(0,C1∗/a1] and I1(C) for C∈(C1∗/a1,C∗) with I2(C1∗)=5.2494>0 and I1(C0∗)=3.5408>0. This phenomenon is consistent with Proposition 14. In Figure 1(b), a=4, which satisfies aM∗<k≤2aλ/μ, and C0∗ is given by C0∗=aX0-γ. Since k-aθ-/a1(θ+-θ-)=1.3935 is very close to C0∗=1.4018, we just draw the position of C0∗ in the graph. In fact, under aM∗<k≤2aλ/μ, this phenomenon happens almost on any other trial, for which we select different k, μ, λ, and σ. Figure 1(b) shows I2(C0∗a1)>0, which supports Proposition 12 correctly. In Figure 1(c), we choose a=6, which satisfies aλ/μ<k≤aM∗. I(C) is just composed of I2(Ca1) with I2k-aθ-/θ+-θ-=0.1631>0, which matches Proposition 16 very well. Since the value 0.1631, compared with the numbers of y-axis, is so small, it is not obvious in the graph. In Figure 1(d), a=8, which satisfies k≤aλ/μ. We can obtain k-aθ-/a1(θ+-θ-)=2.4651 by calculation. From numerical simulation we get C∗=2.4651, so C∗=k-aθ-/a1(θ+-θ-) is satisfied. Therefore, the conclusion in Proposition 16 is shown.Table1 summarizes the results for different parameter values. As shown in Table 1 (starting from row 4), for each case in Figure 1 where a>0, we choose three different values of K to calculate parameters such as C, x~∨0, x0∨0, and x1 from I(C)=K. For Figure 1(a) with a=2, from I2(C1∗)=5.2494 and I1(C0∗)=3.5408, three potential values of K are selected as K=0.15<I1(C0∗), I1(C0∗)<K=4<I2(C1∗), and K=8>I2(C1∗). By the same selection method, for Figure 1(b) with a=4, from I2k-aθ-/θ+-θ-=1.0669 and I2(C0∗a1)=1.0150, three trials for K are K=0.15,1.05,4. For Figure 1(c) with a=6, from I2k-aθ-/θ+-θ-=0.1631, three comparable values of K are given by K=0.15,1.05,4 and the same is applied in Figure 1(d) with a=8. This table shows that the results, whether the values of x~∨0 and x0∨0 are zeroes or not, are totally consistent with the theoretical results in Sections 4.1 and 4.2. In addition, C, x~∨0, and x0∨0 are decreasing and x1 is increasing when K is increasing under the same a. On the other hand, under the same K, C is increasing and x1 is decreasing as a is increasing.Table 1
Parameter estimations underμ=0.12, λ=0.002, k=0.13, and σ=1.6.
Conditions a K C x ~ ∨ 0 x 0 ∨ 0 x 1 I ( C ) k > 2 a λ μ 0 0.15 1.5748 15.9820 12.4675 49.9571 I 0 ( C ) k > 2 a λ μ 0 4 0.8671 5.7239 12.4675 91.9615 I 0 ( C )  k > 2 a λ μ 0 8 0.5196 2.3827 12.4675 127.4305 I 0 ( C )   k > 2 a λ μ 2 0.15 1.7489 18.4531 11.0865 40.0506 I 1 ( C ) k > 2 a λ μ 2 4 0.8648 0 4.9491 84.6289 I 1 ( C )  k > 2 a λ μ 2 8 0.5163 0 0 115.3948 I 2 ( C a 1 )   a M ∗ < k ≤ 2 a λ μ 4 0.15 1.7490 12.5186 5.1482 34.1052 I 1 ( C ) a M ∗ < k ≤ 2 a λ μ 4 1.05 1.3962 0.1815 0 46.2875 I 2 ( C a 1 )  a M ∗ < k ≤ 2 a λ μ 4 4 1.0055 0 0 69.2676 I 2 ( C a 1 )   a λ μ < k ≤ a M ∗ 6 0.15 1.9325 0.4645 0 22.0481 I 2 ( C a 1 ) a λ μ < k ≤ a M ∗ 6 1.05 1.7409 0 0 31.7341 I 2 ( C a 1 )  a λ μ < k ≤ a M ∗ 6 4 1.2550 0 0 54.2053 I 2 ( C a 1 )   k ≤ a λ μ 8 0.15 2.3953 0 0 11.0653 I 2 ( C a 1 ) k ≤ a λ μ 8 1.05 2.1493 0 0 21.6008 I 2 ( C a 1 )  k ≤ a λ μ 8 4 1.5614 0 0 40.7502 I 2 ( C a 1 )


## 5. Solutions to the QVI and the Optimal Policy

Notice that all possible structures to calculateC are I1(C) and I2(Ca1). Moreover, all integrands of these structures can be used to construct the candidate solutions to the QVI. In the following, we first demonstrate that all constructed candidate solutions are solutions to the QVI under different conditions and give several corresponding numerical examples by using the results of Section 4.3. Then, we provide and verify the optimal policy based on the solutions to the QVI.


### 5.1. Solutions to the QVI and Numerical Examples

Assume that we have gotC, such that I(C)=K, by those steps provided in Section 4.3. Then, other parameters, such as x1C, x1C1, x~C∨0, and x~C1∨0, can also be obtained. For simplicity, we use the same denotations as before, such as C, C1, x1C, x1C1, x~C∨0, and x~C1∨0, to represent all calculated parameters.ForC, such that I1(C)=K, we define a function T1(x) by(59)T1x=CHCx,x∈0,x1C,k,x∈x1C,∞.Then C>aX0-γ and T1(x) satisfy I1(C)=K.Definev1(x)=a+∫0xT1(y)dy; then we can obtain that(60)v1x=Cx+aC1/γγ,x∈0,x0,Ca1eθ+x-x0+Ca2eθ-x-x0,x∈x0,x1C,vx1C+kx-x1C,x∈x1C,∞,where x0=X0-a/C1/γ.In addition, forC, such that I2(Ca1)=K, define(61)T2x=HC1x,x∈0,x1C1,k,x∈x1C1,∞,where C1=Ca1. Then C≤aX0-γ and T2(x) satisfy I2(C1)=K.Definev2(x)=a+∫0x‍T2(y)dy; it follows that(62)v2x=C1eθ+x+a-C1eθ-x,x∈0,x1C1,vx1C1+kx-x1C1,x∈x1C1,∞.In the following, forv1(x) and v2(x), we use the same notations, x1, to denote both x1C and x1C1 and x~ to denote both x~C and x~C1. From the construction of v1(x) and v2(x), we have the following theorem by verifying the system QVI.Theorem 18.
Both functionsv1(x) and v2(x) given by (60) and (62) are continuously differentiable on [0,∞), and for each of v1 and v2, there exists U>0 such that this function is twice continuously differentiable on [0,U). Moreover, v1(x) and v2(x) provide a solution to QVI.In addition, from the discussion of Section4, we have an important result as follows to make a decision with respect to whether there is a bankruptcy or not.(1)
Under the assumptionk>2aλ/μ, if I1(a1-γk/γγ)≤K, then x~∨0=0 and there would be a bankruptcy. If I1(a1-γk/γγ)>K, then x~>0 and the optimal policy is continuation strategy.(2)
Under the assumptionaλ/μ<k≤2aλ/μ, if I2k-aθ-/θ+-θ-≤K, then x~∨0=0 and there would be a bankruptcy. If I2k-aθ-/θ+-θ->K, then x~>0 and the optimal policy is continuation strategy.(3)
Ifk≤aλ/μ, then x~∨0=0 and there would be a bankruptcy.Based on the calculated parameters,C, x0∨0, and x1 of Section 4.3, several numerical examples are provided here to support our theoretical results. In Figure 2, the graphs of v(x) are shown under a=2, a=4, a=6, and a=8, respectively. These figures show that at the same value of a, the graphs of v(x) with greater K are below those of v(x) with smaller K. This phenomenon means that the optimal dividends would be greater if the fixed costs become smaller, which accords with the actual situation in the real world. Moreover, given x, the value of the firm v(x) increases with the recovery rate a.Figure 2
v ( x ) with respect to a.
(a)
a = 2 (b)
a = 4 (c)
a = 6 (d)
a = 8In Figure3, the corresponding derivatives of v(x) in Figure 2 are plotted. For Figure 3(a) at K=0.15, Figure 3(b) at K=0.15 and K=1.05, and Figure 3(c) at K=0.15, it shows that they have v′(0)>k in corresponding graphs. In addition, for Figure 3(b) at K=1.05, the graph of it has convexity. Moreover, for Figure 3(b) at K=4, Figure 3(c) at K=1.05 and K=4, and Figure 3(d) at K=0.15, K=1.05, and K=4, all graphs of them are below the level line of k.Figure 3
v ′ ( x ) with respect to a.
(a)
a = 2 , k = 0.13 (b)
a = 4 , k = 0.13 (c)
a = 6 , k = 0.13 (d)
a = 8 , k = 0.13For comparative purposes, we also compute the results for the cases whena=0, which are considered by Cadenillas et al. [14]. The first three rows in Table 1 present the cases for a=0. As shown, the results for a=0 are quite different from those for a≠0 (see rows 4–15 of Table 1). An important difference is that, for a=0, x0 and x~ are all greater than zero, which are critical factors to reduce the complexity of verification for the uniqueness of solutions. The value of v(x) and its derivatives under a=0 are plotted in Figure 4. In fact, in Figure 4(a)v(x) is the lower limit lines of those corresponding lines with the same parameters except for a in Figure 2 as a→0. Comparing Figure 4(b) with Figure 3, it shows that the case a=0 has much simpler graphs than the case a≠0 for the derivative function v′(x). The former only has a convex shape, whereas the latter has both convex and concave shapes, which are consistent with the complicated properties of I(C) in Section 4.2. More importantly, for any given x, the value of the firm v(x) is always lower in the case with no recovery (a=0) than that with a recovery value (a≠0). Results show that the model with a=0 underestimates the firm value when there is a residual value at bankruptcy.Figure 4
v ( x ) and v′(x) under a=0.
(a)
a = 0 (b)
a = 0


### 5.2. Optimal Policy

In the following theorem, the optimal policy is presented and verified, and it is shown that the solution to QVI constructed above is the value function.Theorem 19.
Letτ0=0; then the control π∗=(u∗,T∗,ξ∗,τ∗) defined by(63)u∗t=μ1-γσ2X∗+aC1/γ,X∗t∈0,x0∨0,1,X∗t∈x0∨0,∞,(64)τ∗=inf⁡t≥0:X∗t=0,τ1∗=inft≥0:X∗t=x1,ξ1∗=x1-x~∨0;if x~∨0=0, then let τn∗=∞ and ξn∗=0; if x~∨0>0, then every n≥2,(65)τn∗=inf⁡t≥τn-1:X∗t=x1,ξn∗=x1-x~,where X∗ is the solution to the stochastic differential equation.(66)X∗t=X∗0+∫0tμu∗X∗sds+∫0tσu∗X∗sdWs-x1-x~∨0∑n=1∞Iτn∗<tis the QVI control associated with the functions v1 defined by (60) and v2 defined by (62). This control is optimal and the functions v1(x) and v2(x) coincide with the value function.Proof.
In Section5.1, for any functions v1 defined by (60) and v2 defined by (62), we can see that v1(x) and v2(x) satisfy all conditions of Theorem 5. From Definition 3 and the discussion in Sections 3 and 4, we show that the control π∗ defined in (63)–(65) is the control associated with v1 and v2. In addition, the control π∗ is admissible from Definition 1. So, by Theorem 18, the claim that v1(x) and v2(x) are the value functions and π∗ is the optimal policy is proved.


## 5.1. Solutions to the QVI and Numerical Examples

Assume that we have gotC, such that I(C)=K, by those steps provided in Section 4.3. Then, other parameters, such as x1C, x1C1, x~C∨0, and x~C1∨0, can also be obtained. For simplicity, we use the same denotations as before, such as C, C1, x1C, x1C1, x~C∨0, and x~C1∨0, to represent all calculated parameters.ForC, such that I1(C)=K, we define a function T1(x) by(59)T1x=CHCx,x∈0,x1C,k,x∈x1C,∞.Then C>aX0-γ and T1(x) satisfy I1(C)=K.Definev1(x)=a+∫0xT1(y)dy; then we can obtain that(60)v1x=Cx+aC1/γγ,x∈0,x0,Ca1eθ+x-x0+Ca2eθ-x-x0,x∈x0,x1C,vx1C+kx-x1C,x∈x1C,∞,where x0=X0-a/C1/γ.In addition, forC, such that I2(Ca1)=K, define(61)T2x=HC1x,x∈0,x1C1,k,x∈x1C1,∞,where C1=Ca1. Then C≤aX0-γ and T2(x) satisfy I2(C1)=K.Definev2(x)=a+∫0x‍T2(y)dy; it follows that(62)v2x=C1eθ+x+a-C1eθ-x,x∈0,x1C1,vx1C1+kx-x1C1,x∈x1C1,∞.In the following, forv1(x) and v2(x), we use the same notations, x1, to denote both x1C and x1C1 and x~ to denote both x~C and x~C1. From the construction of v1(x) and v2(x), we have the following theorem by verifying the system QVI.Theorem 18.
Both functionsv1(x) and v2(x) given by (60) and (62) are continuously differentiable on [0,∞), and for each of v1 and v2, there exists U>0 such that this function is twice continuously differentiable on [0,U). Moreover, v1(x) and v2(x) provide a solution to QVI.In addition, from the discussion of Section4, we have an important result as follows to make a decision with respect to whether there is a bankruptcy or not.(1)
Under the assumptionk>2aλ/μ, if I1(a1-γk/γγ)≤K, then x~∨0=0 and there would be a bankruptcy. If I1(a1-γk/γγ)>K, then x~>0 and the optimal policy is continuation strategy.(2)
Under the assumptionaλ/μ<k≤2aλ/μ, if I2k-aθ-/θ+-θ-≤K, then x~∨0=0 and there would be a bankruptcy. If I2k-aθ-/θ+-θ->K, then x~>0 and the optimal policy is continuation strategy.(3)
Ifk≤aλ/μ, then x~∨0=0 and there would be a bankruptcy.Based on the calculated parameters,C, x0∨0, and x1 of Section 4.3, several numerical examples are provided here to support our theoretical results. In Figure 2, the graphs of v(x) are shown under a=2, a=4, a=6, and a=8, respectively. These figures show that at the same value of a, the graphs of v(x) with greater K are below those of v(x) with smaller K. This phenomenon means that the optimal dividends would be greater if the fixed costs become smaller, which accords with the actual situation in the real world. Moreover, given x, the value of the firm v(x) increases with the recovery rate a.Figure 2
v ( x ) with respect to a.
(a)
a = 2 (b)
a = 4 (c)
a = 6 (d)
a = 8In Figure3, the corresponding derivatives of v(x) in Figure 2 are plotted. For Figure 3(a) at K=0.15, Figure 3(b) at K=0.15 and K=1.05, and Figure 3(c) at K=0.15, it shows that they have v′(0)>k in corresponding graphs. In addition, for Figure 3(b) at K=1.05, the graph of it has convexity. Moreover, for Figure 3(b) at K=4, Figure 3(c) at K=1.05 and K=4, and Figure 3(d) at K=0.15, K=1.05, and K=4, all graphs of them are below the level line of k.Figure 3
v ′ ( x ) with respect to a.
(a)
a = 2 , k = 0.13 (b)
a = 4 , k = 0.13 (c)
a = 6 , k = 0.13 (d)
a = 8 , k = 0.13For comparative purposes, we also compute the results for the cases whena=0, which are considered by Cadenillas et al. [14]. The first three rows in Table 1 present the cases for a=0. As shown, the results for a=0 are quite different from those for a≠0 (see rows 4–15 of Table 1). An important difference is that, for a=0, x0 and x~ are all greater than zero, which are critical factors to reduce the complexity of verification for the uniqueness of solutions. The value of v(x) and its derivatives under a=0 are plotted in Figure 4. In fact, in Figure 4(a)v(x) is the lower limit lines of those corresponding lines with the same parameters except for a in Figure 2 as a→0. Comparing Figure 4(b) with Figure 3, it shows that the case a=0 has much simpler graphs than the case a≠0 for the derivative function v′(x). The former only has a convex shape, whereas the latter has both convex and concave shapes, which are consistent with the complicated properties of I(C) in Section 4.2. More importantly, for any given x, the value of the firm v(x) is always lower in the case with no recovery (a=0) than that with a recovery value (a≠0). Results show that the model with a=0 underestimates the firm value when there is a residual value at bankruptcy.Figure 4
v ( x ) and v′(x) under a=0.
(a)
a = 0 (b)
a = 0


## 5.2. Optimal Policy

In the following theorem, the optimal policy is presented and verified, and it is shown that the solution to QVI constructed above is the value function.Theorem 19.
Letτ0=0; then the control π∗=(u∗,T∗,ξ∗,τ∗) defined by(63)u∗t=μ1-γσ2X∗+aC1/γ,X∗t∈0,x0∨0,1,X∗t∈x0∨0,∞,(64)τ∗=inf⁡t≥0:X∗t=0,τ1∗=inft≥0:X∗t=x1,ξ1∗=x1-x~∨0;if x~∨0=0, then let τn∗=∞ and ξn∗=0; if x~∨0>0, then every n≥2,(65)τn∗=inf⁡t≥τn-1:X∗t=x1,ξn∗=x1-x~,where X∗ is the solution to the stochastic differential equation.(66)X∗t=X∗0+∫0tμu∗X∗sds+∫0tσu∗X∗sdWs-x1-x~∨0∑n=1∞Iτn∗<tis the QVI control associated with the functions v1 defined by (60) and v2 defined by (62). This control is optimal and the functions v1(x) and v2(x) coincide with the value function.Proof.
In Section5.1, for any functions v1 defined by (60) and v2 defined by (62), we can see that v1(x) and v2(x) satisfy all conditions of Theorem 5. From Definition 3 and the discussion in Sections 3 and 4, we show that the control π∗ defined in (63)–(65) is the control associated with v1 and v2. In addition, the control π∗ is admissible from Definition 1. So, by Theorem 18, the claim that v1(x) and v2(x) are the value functions and π∗ is the optimal policy is proved.


## 6. Conclusion

In this paper, we consider the dividend optimization problem that some fixed money would be returned to shareholders at the state of terminal bankruptcy. For this problem, the optimal timing and the optimal amount of dividends paid out to the shareholders and the resulting firm values are provided. Moreover, we identify some conditions that there would be a terminal bankruptcy under the optimal policy. A challenge in this paper is that the structure of the candidate solution is not explicit since the existing interval of it has unfixed endpoints, which depends on some unknown parameters. We overcome this difficulty by using the integral,I(C), and dividing it into several cases. The importance of this paper is that it generalizes the terminal bankruptcy model and provides several new kinds of optimal dividend strategies, especially at the state of bankruptcy.In the future, we can address some problems with general drift and diffusion terms under the assumption that shareholders can get some fixed money when there is terminal bankruptcy. For this kind of problem, sometimes we cannot write out the explicit expression for the solutions. Then we need to use some suitable numerical methods to simulate some nonlinear equations in the quasi-variational inequalities. Moreover, we can compare two optimal policies derived from two choices, investing new money or terminating the business, and, based on the obtained result, the best corporate decision can be determined.



---

*Source: 2693568-2017-02-23.xml*