Optimal dividends and capital injections for a spectrally positive Lévy process

This paper investigates an optimal dividend and capital injection problem for a spectrally positive Levy process, where the dividend rate is restricted. Both the ruin penalty and the costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends, the penalized discounted capital injections before ruin, and the expected discounted ruin penalty. By the fluctuation theory of Levy processes, the optimal dividend and capital injection strategy is obtained. We also find that the optimal return function can be expressed in terms of the scale functions of Levy processes. Besides, a series of numerical examples are provided to illustrate our consults.


1.
Introduction. In recent years, quite a few papers deal with the optimal dividend problems in the spectrally positive Lévy risk model, which is also called the dual risk model. The dual model is an appropriate model for a company driven by inventions or discoveries. Other examples are commission-based business, such as real estate agent offices or insurance annuity business. In [1] and [2], the authors studied how the expectation of the discounted dividends until ruin can be calculated in the dual compound Poisson risk model. Recently, in [4], [5] and [18], the optimal dividend problems were studied in a general spectrally positive Lévy risk model.
Besides dividend payment, capital injection is another important approach to control the surplus process. In [3], [16] and [17], the optimal dividend and capital injection problem was considered in the compound Poisson dual model. For the general spectrally positive Lévy process, the optimal dividend and capital injection problems were studied in [4], [5] and [19], where there was no constraint on dividend rate. In [18], the optimal dividend with restricted dividend rate was considered for a general spectrally positive Lévy process, however, the capital injection was not involved the control problem. In addition, transaction cost, which usually where 1 A is an indicator function of a set A. The process X has paths of bounded variation if and only if σ = 0 and 1 0 xν(dx) < ∞. Correspondingly, the Laplace exponent (1) can be written as where c 0 = c + 1 0 xν(dx). We rule out the case that X has monotone paths, and so c 0 > 0 is necessary when X is of bounded variation. The drift of X is given by µ = EX 1 = −ψ (0+). It is well known that if ∞ 1 yν(dy) < ∞, then µ = −c + ∞ n=1 1 {τn≤t} ξ n } t≥0 is described by a sequence of increasing stopping times {τ n , n = 1, 2, · · · } and a sequence of random variables {ξ n , n = 1, 2, · · · }, which represent the times and the sizes of capital injections, respectively. A control policy π is described by π = (L π ; G π ) = (L π ; τ π 1 , · · · , τ π n , · · · ; ξ π 1 , · · · , ξ π n , · · · ). The controlled asset process associated with π is modeled as ξ π n 1 {τ π n ≤t} , t ≥ 0.
Denote the set of admissible control strategies by Π. Define the time of ruin by T π = inf{t ≥ 0 : X π t ≤ 0}. Considering the fixed and proportional transaction costs when capital injection occurs, we assume that (φ − 1)ξ and K > 0 are respectively proportional cost and fixed cost to meet the capital injection of amount ξ, where φ > 1. The force of interest δ > 0 reflects the time preference of investors. In addition, we consider a constant penalty P ∈ R that must be paid (collected, if P is negative) if and when ruin occurs. When P > 0, we think of P as a penalty for ruin, the monetary cost when ruin occurs. When P = 0, we are in the case of no penalty. When P < 0, we think of P as the salvage value of the company; for example, a company's brand name or agency network might be of value to a potential customer. Under these assumptions, the performance function with the strategy π ∈ Π is defined by Our objective is to find the optimal return function, or the value function, defined as and the optimal strategy π * ∈ Π such that V (x) = V (x; π * ).

2.3.
Property of the value function.
Proof. We first prove the increase of the value function V in the case of P ≥ − l0 δ . For the initial surplus x ≥ 0 and the admissible strategy π, we denote the ruin time by T π x . If T π x < ∞, for y > x, we construct a strategyπ as follows: Lπ t = L π t and Gπ t = G π t for 0 ≤ t ≤ T π x , Lπ t = l 0 Tπ y T π x e −δs ds and Gπ t = 0 for T π where Tπ y is the ruin time for the initial capital y and the strategyπ. Thenπ ∈ Π, and X π t + (y − x) = Xπ t a.s. for 0 ≤ t ≤ T π x . By (3), we have Due to P ≥ − l0 δ and Tπ y ≥ T π x a.s., we obtain V (x; π) ≤ V (y;π) ≤ V (y).
If T π x = ∞, for y > x, we define a strategyπ = π. Then Tπ y = T π x = ∞, furthermore, V (x; π) = V (y;π) ≤ V (y). Combining with (5), we obtain V (x; π) ≤ V (y) for all π ∈ Π, and hence V (x) ≤ V (y). Now, we assume the penalty P < − l0 δ . For the initial surplus y > 0 and the admissible strategy π 1 , we denote the ruin time T π1 by T π1 y . For 0 ≤ x < y, let T π1 y−x = sup{t ≤ T π1 y : X π1 t ≤ y − x}. We construct the strategy π 2 = π 1 beforeT π1 y−x . For the initial surplus x and the strategy π 2 , the ruin time is denoted by T π2 IfT π1 y−x = ∞, then T π2 x = T π1 y = ∞, and so V (y; π 1 ) = V (x; π 2 ) ≤ V (x). Combining with (6), we have V (y; π 1 ) ≤ V (x) for any π 1 ∈ Π, and hence V (y) ≤ V (x). Remark 2.1. By the above proposition, if the penalty P < − l0 δ , the value function V (x) is decreasing, that is, larger initial surplus results in smaller profit. Then it is not necessary to operate the company as P < − l0 δ . Hence, we only discuss the case of P < − l0 δ in the following section.
Proof. For the initial surplus x = 0 and the admissible strategy π 1 : the company immediately declare ruin, the associated performance function V (0; π 1 ) = −P , and so V (0) ≥ −P . By Proposition 2.1, the value function is increasing, and so the first inequality in (7) is obtained. The second inequality in (7) is followed by 0 ≤ l π t ≤ l 0 for all π ∈ Π.
To derive the optimal strategy and the value function, the Quasi-Variational-Inequality (QVI) (see, e.g., [9]) approach is adopted. Before that, we introduce some operators as following. Throughout the paper, a function f : D → R is called sufficiently smooth meaning that it belongs to C 1 (D) if X is of bounded variation, otherwise it belongs to C 2 (D). Suppose that a sufficiently smooth function v is a candidate function for the value function. Let M denote the injection operator, defined by Then M V (x) represents the value of the strategy that consists in choosing the best immediate capital injection. In addition, the operators used in this paper Γ and L l are defined, respectively, by By Itô's formula for semimartingales, we obtain the following lemma, whose proof is presented in Appendix A.
Lemma 2.1. Let v(x) be an increasing, concave and sufficiently smooth function 3. Optimal control problem without capital injection.

3.1.
Formulation for the optimal problem without capital injection. We now study the optimal problem without capital injection. For this suboptimal problem, let Π p = {π p : π p = (L πp ; 0) ∈ Π} ⊂ Π denote the set of all admissible strategies. The value function associated with π p is defined by The objective is to find the value function V p (x) and the corresponding optimal strategy π * p ∈ Π p such that V p (x) = V (x; π * p ). Similar Proposition 2.2, V p (x) is bounded and satisfies the inequalities (7).
If the value function V p (x) is sufficiently smooth, by the stochastic control theory, it satisfies the following QVI The following Theorem 3.1 and Corollary 3.1 are proved in Appendix B.
Theorem 3.1. If an increasing, concave and bounded function g(x) is sufficiently smooth on (0, ∞) and continuous from the right at 0, and satisfies the QVI (11), we have the following statements: (i) For any strategy π p ∈ Π p , we have g(x) ≥ V (x; π p ), and so g( then g(x) = V p (x) = V (x; π * p ), where π * p = {L π * p , 0} ∈ Π p is the optimal strategy such that i.e. the optimal dividend strategy L π * p is a threshold dividend with parameters l 0 and x * p . Corollary 3.1. If an increasing, concave and bounded function g(x) is sufficiently smooth on (0, ∞) and continuous from the right at 0, and also satisfies, for some where π * p is given by (13). 3.2. Threshold dividend strategies. From the subsection 3.1, the optimal dividend strategy is a threshold dividend strategy under some hypotheses. Then we study the dividend problem when dividends are paid according to a threshold strategy with parameters l 0 and x p > 0, i.e. π p = {L πp t ; 0} in (13), where x * p is replaced by x p . Then the performance function associated with the strategy π p is Similar to Theorem 3.1 in [18], if V (x; π p ) is sufficiently smooth on (0, ∞)/{x p }, it satisfies the following integro-differential equation with the initial condition V (0; π p ) = −P and the continuity condition and furthermore, if X is of unbounded variation, V (x; π p ) also satisfies For the uncontrolled surplus process X, let The following lemma is Theorem 3.2 in [18].
The definitions of scale functions Using the method of Theorem 3.3 in [18], we give the following theorem.
Theorem 3.2. Let π p be the threshold dividend strategy with the parameters l 0 and where Remark 3.1. From (21) and (22), we have When X is of bounded variation, and by (C.1), we have When X is of unbounded variation, we have Hence, if X is of unbounded variation, and by (C.2), we obtain V (x; π p ) ∈ C 1 ((0, ∞)) and 3.3. Optimal dividend threshold. By Corollary 3.1, we will search for a threshold strategy π * p such that V (x; π * p ) is increasing, concave and sufficiently smooth on (0, ∞). Due to (25) and (26), the optimal dividend threshold x * p satisfies By (24), the above equation is equivalent to We have Furthermore, we obtain θ (z) > 0 and lim z→∞ θ(z) = ∞. Thus the equation (28) has a unique solution where π * p is given by (13) and where Q(x) is defined by (23), π * p is given by (13) and x * p is determined by (28).
Remark 3.2. From (28), the optimal threshold x * p increases as the ruin penalty P increases. The value function V p in (31) is a decreasing function with respect to x * p . Combining with (30), the value function V p defined by (10) decreases with the increase of P . There is a reasonable economic explanation for this phenomenon. As the ruin penalty P increases, the company should raise the dividend threshold x * p in order to lower the risk of ruin. Certainly, the higher penalty results in the smaller the profits (value function).
4. Optimal problem with forced capital injections to prevent ruin.

4.1.
Formulation for the optimal problem without ruin. We assume that the company survives forever by forced capital injections. Let Π q denote the set of admissible strategies of this suboptimal problem, i.e., The objective is to find the value function V q (x) and the corresponding optimal δ . If the value function V q (x) is sufficiently smooth, by the stochastic control theory, it satisfies the following QVI, for We give the following theorem and corollary, which are proved in Appendix B.

4.2.
Optimal strategy for the optimal problem without ruin. From the above arguments, we consider the dividend strategies π q in (36) with parameters x q > 0 and η, where x * q is replaced by x q . The strategy π q is a band strategy with the upper threshold x q and the lower reflecting barrier 0. Following this strategy, the company's management takes no action as long as the asset process takes value in (0, x q ]; Whenever the asset value exceeds x q , dividends dividends are paid at the maximal rate l 0 ; The asset immediately jumps to η by injecting capitals once it reaches 0. The performance function associated with the strategy π q is Similar to the case of the threshold dividend without injecting capital, and by Itô's formula, if V (x; π p ) is sufficiently smooth on (0, ∞)/{x p }, it satisfies the following integro-differential equation with the continuity condition and furthermore, if X is of unbounded variation, V (x; π q ) also satisfies We use the hitting times T − 0 in (20) and T + xq = inf{t ≥ 0 : X t > x q }. Similar to the proof of Theorem 3.2 in [18], we can prove, for x > x q , Now, we consider the performance function V (x; π q ) in (38) for 0 ≤ x ≤ x q . By the law of total probability and the strong Markov property, for 0 < x < x q , we have where By the proof of Theorem 3.3 in [18], we obtain where Q(x) is defined by (23). By (8.8) in [11], we have h 2 (x) = W (δ) (xq) . Substituting h 1 (x), h 2 (x) into (43), and using the conditions in (40) and (41), we get By (42) and Remark 3.1, in order that the above V (x; π q ) is sufficiently smooth on (0, ∞), we set V (x q ; π q ) = l0 δ − 1 Φ1(δ) . Therefore, we obtain and Q is defined by (23), and the optimal strategy π * q is given in (36) with parameters x * q and η determined by Proof. By the discussions at the beginning of this subsection, if there exist 0 < η < x * q satisfying the equations (46) and (47), the performance function V (x; π * q ) associated with π * q (36) is given by (45). Similar to V (x; π * p ) in (31), we can show that V (x; π * q ) is increasing, concave and sufficiently smooth on (0, ∞). Moreover, by (39), (46) and (47), we know that V (x; π * q ) satisfies (35) and (37). The results are obtained by Corollary 4.1. Therefore, we only need to prove that there exists a pair of (η, x * q ) that (46) and (47) with 0 < η < x * q . Making the change of variable z = x q − η, we can rewrite (46) as Q(z) = φ. Recalling the definition in (23), we have Then there exists a unique x 1 > 0 such that Q(x 1 ) = φ. We define a function in x q as follows, for x q ≥ x 1 , Then the equation (47) can be rewritten as α(x q ) = 0. Due to we know that there exists an x * q > x 1 such that α(x * q ) = 0, so η = x * q − x 1 is also determined.

5.
Optimal joint dividend and capital injection strategy. From the definitions of V p , V q and V , we can easily get the relationship Lemma 5.1. For initial capital x ≥ 0, if the functions g(x) and h(x) satisfy the conditions of Theorem 3.1 and Theorem 4.1, respectively, then (i) If M g(0) ≤ g(0), we have g(x) = V (x) and the optimal strategy π * = π * p given by (13); (ii) If h(0) ≥ −P , we have h(x) = V (x) and the optimal strategy π * = π * q given by (36); (iii) In particular, if the conditions in (i) and (ii) hold, we have g(x) = h(x) = V (x) and the optimal strategy π * = π * p or π * = π * q . Proof. (iii) In this case, there is no difference between injecting capital to rescue the company and declaring ruin on the edge of ruin. In fact, it is not necessary to rescue the company by capital injections when the surplus hits 0, because it can not profit from capital injections. Then the optimal strategy π * = π * p . Lemma 5.2. Let g(x) = V (x; π * p ) and h(x) = V (x; π * q ) given by (31) and (45), respectively, we have ] < x * p . By the concavity of g(x), we obtain η = 0 if and only if g (0) ≤ φ; 0 < η < x * p if and only if g ( η) = φ. We first consider the case of η = 0. Recalling the function Q in (23), we have By the increase of Q and Q(x 1 ) = φ, we conclude g (0) ≤ φ if and only if x * p ≤ x 1 , and so M g(0) − g(0) = g(0) − K − g(0) < 0 if and only if x * p ≤ x 1 ≤ x * q . In the case of 0 < η < x * p , we have g ( η) = φ, i.e., Q(x * p − η) = φ, and then Theorem 5.1. For the general optimal control problem in Section 2, given by (30) and the optimal control strategy π * = π * p given by (13) and the optimal threshold given by (31) and the optimal control strategy π * = π * p given by (13) and the optimal threshold given by (45) and the optimal control strategy π * = π * q given by (36) the optimal threshold x * = x * q > 0;. In other words, the value function V (x) = max{V p (x), V q (x)} and the optimal threshold x * = min{x * p , x * q }. (9). We obtain V p (x) ≥ V (x) from Lemma 2.1. The results in (i) are proved. By Lemma 5.1 and Lemma 5.2, the results in (ii) and (iii) are obtained. 6. Numerical examples. Recently, in [8], the authors considered the spectrally negative phase-type Lévy process, whose scale function admits an analytical expression; they proposed an approach to approximate the scale function for a general spectrally negative Lévy process. The numerical results of this paper are based on their approximation method. For simplicity, we discuss the cases of the Lévy process X with hyper-exponential and Gamma distributed compound Poisson positive jumps, respectively.
In the following, we assume ν(dy) = 3ye −y dy, y ≥ 0, to illustrate our results. That is, the size of jumps follows the Gamma(2,1) distribution, and the number of jumps follows the Poisson process with parameter 3. In the following section, we will discuss the influences of P , φ, K, l 0 , δ and σ on the optimal strategy and the value function.  Table 1 as the penalty P varies. It shows that x * p increases with the increase of P from Table 1. The optimal strategy π * switches from π * p to π * q with the increase of the penalty. The management would choose to avoid ruin by injecting capital for large penalty. Here, the maximum penalty P that the company can afford is 0.8380. In other word, when −∞ < P ≤ 0.8380, we have x * = x * p ≤ 1.7590 and V (x) = V p (x); and when P > 0.8380, we have x * = x * q = 1.7590 and V (x) = V q (x). Certainly, the value function decreases with the increase of the penalty because the value function is decreasing with respect to the level of threshold.
The influences of φ and K Let c 0 = 5, σ = 0, δ = 0.1 and l 0 = P = 1. It can verify that these parameters satisfy P > 1 Φ1(δ) − l0 δ . By Table 1, we know the optimal threshold x * p = 1.8830. From Table 2, the thresholds x * q and x * increase with the increase of φ or K, while the amount of capital injection η increases when φ decreases or K increases. Then the value function decreases as φ or K increases due to (31) and (45). Larger φ or K means higher costs of capital injection, the company would reserve more money instead of paying more dividends in order to reduce or avoid capital injection, which calls for higher dividend threshold. As K increases or φ decreases, it would increase the amount of capital injection η so as to cut down costs of capital injection. The optimal strategy π * switches from π * q to π * p with the increase of φ or K. Furthermore, when φ = 1.1 and K ≥ 0.1256; or when K = 0.1 and φ ≥ 1.1226, the optimal strategy π * ≡ π * p , i.e. the company prefer declaring ruin to injecting capital whenever it is on the edge of ruin.
We now consider the effect of the dividend rate ceiling l 0 on the optimal control problem. Let c 0 = 5, δ = 0.1, σ = 0, φ = 1.2, K = 0.2 and P = 1.5. We plot the levels of the optimal dividend thresholds and the value functions under the condition P > 1 Φ1(δ) − l0 δ . From the left figure in Figure 1, larger l 0 results in the higher levels of η, x * q and x * p . When l 0 increases, the dividend may be paid at a relatively rapid speed. In contrast to early dividend payout, it is better to reserve more money to hedge against financial risks. The optimal strategy π * switches from π * p to π * q as l 0 increases. The management becomes optimistic over the company's prospects with the increase of l 0 , he would choose to inject capital if l 0 is large enough. The more benefits (larger value function) are obtained for the higher level of dividend rate, which is implied from the right figure of Figure 1.   Figure 2, it follows that the larger force of interest results in the lower levels of η, x * q and x * p , but the change of η is not sensible with the increase of δ. When the time discounted value is larger, it should be earlier to dividend, which calls for lower dividend thresholds. Certainly, the increase of discounted value decreases the profits, which is shown in the right figure of Figure 2. As δ increases, the optimal strategy π * switches from π * q to π * p . In another word, as δ is higher, it would be better not to inject capital on the edge of ruin since the present value of the costs by capital injections would far outweigh the present value of the dividends distributed.  • The influence of σ Finally, we discuss the impact of σ on the optimal control problem. Let c 0 = 5, δ = 0.1, l 0 = 1, P = 0.5, φ = 1.02 and K = 0.12. As σ increases, x * p decreases while x * q and η increase. Larger volatility brings the company higher risk which calls for a lower threshold x * p to distribute a greater proportion of the surplus as dividend. As σ increases, the cash flow of the company becomes more and more wobbly, if the ruin is forbidden, the threshold level x * q and η have to increase to delay the coming of injecting. The optimal strategy π * switches from π * q to π * p as σ increases. That is, if the volatility is large enough, it may prefer declaring ruin to rescuing it by capital injections whenever it is on the edge of ruin. Certainly, the value function decreases with the increase of volatility.  Lemma A.1. It is optimal to postpone the capital injection as long as possible, i.e., if capital injection occurs, it happens only at the moment when the surplus process hits the barrier 0.
Proof of Lemma 2.1 Due to Lemma A.1 and that there are only positive jumps for the reserve process, it only needs to consider the strategies such that the jumps of capital injection and the positive jumps of X do not occur at the same time. For a policy π ∈ Π, define Λ t = {s ≤ t : X s− = X s }; Let {T m } m≥1 be a sequence of stopping times defined by T m = inf{t ≥ 0 : X π t > m or X π t < 1 m }. Since the controlled surplus process X π is a semimartingale, and recalling the sufficient smoothness of v, we apply Itô's formula on e −(δ+γ)(t∧Tm) v(X π t∧Tm ) to have By the Lévy-Itô decomposition the expression between the first pair of curly brackets is a zero-mean martingale, and by the compensation formula the expression between the second pair of curly brackets is also a zero-mean martingale. Hence we derive that e −δ(t∧Tm) v(X π t∧Tm ) =v(x) + t∧Tm 0 e −δs [(Γ − δ)v(X π s ) − l π s v (X π s ) + l π s ]ds − t∧Tm 0 e −δs l π s ds + τ π n ≤t∧Tm e −δ+γτ π n [v(X π s− + ξ π n ) − v(X π s− )] + M t , which together with (14) yields The results are followed from Theorem 3.1.
Proof of Theorem 4.1 The proof of (i) is similar to that of Lemma 2.1. Now we only give the proof of (ii). Since h is increasing and concave, and satisfies (35), the function F (y) = h(x + y) − φy − K is increasing in [0, η − x] and decreasing in [η − x, ∞) for 0 ≤ x ≤ η. Then Under the strategy π * q given by (36), Taking expectation and letting t → ∞ on both sides of the above equation, it yields h(x) = V (x; π * q ) = V q (x).
C. Introduction of scale function. We now recall the definition of the q-scale function for the spectrally positive Lévy process X, whose Laplace exponent ψ is given by (1). For q > 0, there exists a continuous and strictly increasing function W (q) : R → [0, ∞), called the q-scale function defined in such a way that W (q) (x) = 0 for all x < 0 and on [0, ∞) its Laplace transform is given by where Φ(q) := sup{s ≥ 0 : ψ(s) = q}. Note that the Laplace exponent ψ in (1) is known to be zero at the origin, convex on R + . Then Φ(q) is well-defined and is strictly positive as q > 0. We give the function Z (q) (x), closely related to W (q) (x), by Z (q) (x) = 1 + q x 0 W (q) (y)dy, x ∈ R, and its anti-derivative Noting that W (q) (x) is uniformly zero on the negative half line, we have Z (q) (x) = 1 and Z (q) (x) = x for x ≤ 0.