Optimal Dividend and Capital Injection Strategies for a Risk Model under Force of Interest

Abstract

As a generalization of the classical Cramér-Lundberg risk model, we consider a risk model including a constant force of interest in the present paper. Most optimal dividend strategies which only consider the processes modeling the surplus of a risk business are absorbed at 0. However, in many cases, negative surplus does not necessarily mean that the business has to stop. Therefore, we assume that negative surplus is not allowed and the beneficiary of the dividends is required to inject capital into the insurance company to ensure that its risk process stays nonnegative. For this risk model, we show that the optimal dividend strategy which maximizes the discounted dividend payments minus the penalized discounted capital injections is a threshold strategy for the case of the dividend payout rate which is bounded by some positive constant and the optimal injection strategy is to inject capitals immediately to make the company's assets back to zero when the surplus of the company becomes negative.

1. Introduction

In the mathematical finance and the actuarial literature, the optimal dividend problem has attracted much attention. For example, there is a good deal of work on the problem of finding an optimal policy for paying out dividends. In the Brownian motion setting, it has been proved that the optimal dividend strategy is a threshold strategy for the case of the dividend payout rate is which bounded by some positive constant and a barrier strategy for the case with no restriction on the dividend payout rate by Asmussen and Taksar [1]. In the Cramér-Lundberg setting, it was first shown in [2] by a discrete approximation and then a limiting argument that the optimal dividend strategy is of the so-called band type. For exponentially distributed claim sizes this strategy simplifies to a barrier strategy. This result was rederived by stochastic control theory in [3, 4]. Recently, Albrecher and Thonhauser [5] studied the optimal dividend strategy by viscosity theory in the constant force of interest model. They pointed out that the optimal dividend strategy in the general case is again of band type and for exponential claim sizes collapses to a barrier strategy. In addition, Avram et al. [6] and Loeffen [7] considered the optimal dividend problem when the risk process is modeled by a spectrally negative Lévy process. They drew on the fluctuation theory of spectrally negative Lévy processes and gave sufficient conditions under which the optimal strategy is of barrier type.

Unfortunately, a drawback is that processes modeling the surplus of a risk business are absorbed at 0. In many cases, negative surplus does not necessarily mean that the business has to stop. Therefore, many authors suggested a model, where the above-mentioned fact should be taken into account.

One method is to make 0 as a reflecting barrier rather than absorbing barrier. For example, Shreve et al. [8] proposed that a diffusion process can be controlled by subtracting out a withdrawal (dividend) process and adding in a deposit (injection) process. The goal is then to maximize the expected discounted dividend payments minus the expected penalized discounted capital injections. If the surplus process is a Brownian motion with drift, they found that the optimal injection policy is to invest the minimum such that the controlled surplus remains positive. Furthermore, they also show that the optimal dividend policy is a barrier strategy. Løkka and Zervos [9] added proportional costs to the deposits. In the more general framework of spectrally negative Lévy processes, Avram et al. [6] studied the optimality of barrier strategies with capital injections. In the Cramér-Lundberg model, Kulenko and Schmidli [10] showed that the optimal dividend strategy is a barrier strategy and the optimal injection policy is to inject the minimum such that the controlled surplus remains nonnegative. In this paper, we will solve the problem for a risk process under a constant force of interest.

Consider the following risk model for the reserve process {𝑈(𝑡)} of an insurance portfolio: 𝑈_𝑡=𝑥𝑒^𝑟𝑡+𝑐^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑑𝑠−^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑑^𝑁𝑠_𝑖=1𝑋_𝑖, (1) where 𝑥≥0 is the initial capital, 𝑐>0 is the premium rate, 𝑟>0 is the constant force of interest, {𝑁_𝑡,𝑡≥0} is a homogeneous Poisson claim counting process with intensity 𝜆>0, and {𝑋_𝑖,𝑖=1,2,…} are i.i.d. claim size random variables, which are independent of {𝑁_𝑡,𝑡≥0} and have common distribution function 𝑃(⋅) with 𝑃(0)=0, density function 𝑝(⋅), and mean 𝐸[𝑋_𝑖]=𝜇. Let 𝜋 denote an admissible strategy consisting of an accumulated dividend process {𝐷^𝜋_𝑡} and an accumulated injection process {𝑍^𝜋_𝑡}. The accumulated dividend process {𝐷^𝜋_𝑡} is a nondecreasing and left continuous process with 𝐷^𝜋₀=0; the accumulated injection process {𝑍^𝜋_𝑡} is a nondecreasing and right continuous process with 𝑍^𝜋₀₋=0. Then the surplus process modified by policy 𝜋 becomes 𝑈^𝜋_𝑡=𝑈_𝑡−^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑑𝐷^𝜋_𝑠+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑑𝑍^𝜋_𝑠. (2) The set of admissible policies Π consists of those policies which satisfy(i)𝑈^𝜋_𝑡 is nonnegative for 𝑡>0,(ii)∫^∞₀𝑒^−𝛿𝑡𝑑𝑍^𝜋_𝑡<∞ almost surely. The value associated to the strategy 𝜋 is defined as 𝑉_𝜋(𝑥)=𝐸^∞₀𝑒^−𝛿𝑡𝑑𝐷^𝜋_𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍^𝜋_𝑡∣𝑈^𝜋₀=𝑥, (3) where 𝛿>0 is the force of interest for discounting the dividends and 𝜅 is the cost per unit injected capital or a penalizing factor. In this paper, we assume that 𝛿>𝑟 and 𝜅>1. The maximal value function is denoted by 𝑉(𝑥)=sup_𝜋∈Π𝑉_𝜋(𝑥). (4) Let us illustrate the reason why we choose 𝜅>1. If we had chosen 𝜅<1, the maximal value function would be infinite because it is cheaper to pay dividends by injection capital than from the reserve. If we had chosen 𝜅=1, the cost of paying incoming claims from the reserve is the same as by an injection capital. However, the discount factor 𝛿 is greater than the force of interest 𝑟. Then it is better to pay dividends earlier. Therefore, it is optimal to pay out all positive surplus immediately as dividends and to pay all claims by injection capital. That is, 𝑉(𝑥)=𝑥+^∞₀𝑒^−𝛿𝑡𝑐𝑑𝑡−𝐸^∞_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑋_𝑖=𝑥+𝑐−𝜆𝜇𝛿, (5) where 𝑇_𝑘 is the time of the 𝑘th claim.

In addition, notice that from (3) it can not be optimal to inject capital before they are really necessary because of 𝜅>1 and 𝛿>𝑟. Therefore, we only need to choose the dividend strategy {𝐷^𝜋_𝑡}. The corresponding injection process {𝑍^𝐷𝜋_𝑡} is defined as follows: when the surplus of the company with strategy {𝐷^𝜋_𝑡} becomes negative, the shareholders immediately inject the amount of the deficit such that the surplus can be restored to 0; when the surplus is nonnegative, no capitals are injected. If the initial capital is negative, then 𝑍₀=|𝑥|, thus 𝑉(𝑥)=𝑉(0)−𝜅|𝑥|.

In this paper, we impose restriction on the dividend stream to prevent ruin occurring almost surely. We assume that the dividends are paid at a dynamic rate 𝑑_𝑡 at time 𝑡. Stochastic process {𝑑_𝑡} is called a control process and can only vary in [0,𝛼] for some 0<𝛼<𝑐. This optimal dividend problem with additional constraints is postulated by Jeanblanc-Picqué and Shiryaev [11] and Asmussen and Taksar [1].

The outline of the remainder of the paper is organized as follows. In Section 2, we prove some properties of the maximal dividend-value function and derive the Hamilton-Jacobi-Bellman equation for the problem. In Section 3, we give the optimal strategy by the HJB equation and the concavity of the value function. We show that the optimal dividend strategy is a threshold strategy. Some concluding remarks are given in Section 4.

2. The Maximal Value Function

We first prove some properties of the maximal dividend-value function.

Proposition 1. The maximal value function 𝑉(𝑥) is increasing and converges to 𝛼/𝛿 as 𝑥 converges to ∞.

Proof. Let 𝜋^𝑥_𝜀 be an 𝜀-optimal strategy for initial capital 𝑥, that is, 𝑉_𝜋𝑥𝜀(𝑥)≥𝑉(𝑥)−𝜀. Let {𝑑^𝜋𝑥𝜀_𝑡} be the corresponding dividend rate process. For initial capital 𝑦(𝑦>𝑥), we define a policy 𝜋^𝑦 with dividend rate 𝑑^𝜋𝑦_𝑡=𝑑^𝜋𝑥𝜀_𝑡. (6) Note that this strategy is admissible for initial capital 𝑦 and its injection process has the following relationship with 𝑍^{𝑑𝜋𝑥𝜀}: 𝑍^𝑑𝜋𝑦_𝑡≤𝑍^{𝑑𝜋𝑥𝜀}_𝑡. (7) Hence we have 𝑉𝑦≥𝐸^∞₀𝑒^−𝛿𝑡𝑑^𝜋𝑦_𝑡𝑑𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍^𝑑𝜋𝑦_𝑡≥𝐸^∞₀𝑒^−𝛿𝑡𝑑^𝜋𝑥𝜀_𝑡𝑑𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍^{𝑑𝜋𝑥𝜀}_𝑡≥𝑉(𝑥)−𝜀. (8) Since this inequality holds for all 𝜀>0, then we have 𝑉(𝑦)≥𝑉(𝑥).
Consider a strategy 𝜋 with 𝑑𝜋_𝑡=𝛼. Then the time of the first injection under this strategy 𝑇𝜋=inf𝑡≥0∶𝑥𝑒^𝑟𝑡+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}()−𝑐−𝛼𝑑𝑠_𝑡0𝑒^{𝑟(𝑡−𝑠)}𝑑𝑆_𝑠<0 (9) converges to infinity as 𝑥→∞. Recall that capitals are injected only if the surplus process is below 0, then 𝐸^∞₀𝑒^−𝛿𝑡𝑑𝑍𝜋_𝑡≤𝐸𝑒^−𝛿𝑇𝜋𝑈𝜋_𝑇𝜋+^∞_{𝑖=𝑁𝑇}𝜋⁺¹𝑒^{−𝛿𝑇𝑖}𝑋_𝑖=𝐸𝑒^−𝛿𝑇𝜋𝑈𝜋_𝑇𝜋+^∞_{𝑖=𝑁𝑇}𝜋⁺¹𝑒^{−𝛿(𝑇𝑖−𝑇}𝜋⁾𝑋_𝑖=𝐸𝑒^−𝛿𝑇𝜋𝑈𝜋_𝑇𝜋+𝐸𝑒^−𝛿𝑇𝜋^∞_{𝑖=𝑁𝑇}𝜋⁺¹𝑒^{−𝛿(𝑇𝑖−𝑇}𝜋⁾𝑋_𝑖=𝐸𝑒^−𝛿𝑇𝜋𝑈𝜋_𝑇𝜋+𝐸𝑒^−𝛿𝑇𝜋𝐸^∞_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑋_𝑖≤𝐸𝑒^−𝛿𝑇𝜋𝑈𝜋_𝑇𝜋+𝜆𝜇𝛿𝐸𝑒^−𝛿𝑇𝜋. (10) Note that lim_𝑥→∞𝐸𝑒^−𝛿𝑇𝜋∣𝑈𝜋₀=𝑥=0,lim_𝑥→∞𝐸𝑒^−𝛿𝑇𝜋𝑈𝜋_𝑇𝜋∣𝑈𝜋₀=𝑥=0. (11) So, lim_𝑥→∞𝐸^∞₀𝑒^−𝛿𝑡𝑑𝑍𝜋_𝑡∣𝑈𝜋₀=𝑥=0. (12) Thus, as 𝑥→∞, 𝑉𝜋(𝑥)=𝐸^∞₀𝑒^−𝛿𝑡𝛼𝑑𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍𝜋_𝑡∣𝑈𝜋₀⟶𝛼=𝑥𝛿. (13) Naturally, it holds that 𝑉𝜋(𝑥)(𝑥)≤𝛼≤𝑉𝛿. (14) Hence, by squeeze theorem, we obtain lim_𝑥→∞𝑉(𝑥)=𝛼𝛿. (15)

Proposition 2. The maximal value function 𝑉(𝑥) is concave and Lipschitz continuous.

Proof. Consider 𝑥≥0, 𝑦≥0, and 𝜃∈(0,1). Let 𝜋^𝑥=(𝑑^𝑥,𝑍^𝑑𝑥) be an admissible strategy for the initial capital 𝑥 and let 𝜋^𝑦=(𝑑^𝑦,𝑍^𝑑𝑦) be an admissible strategy for the initial capital 𝑦. We define strategy (𝜃𝑑^𝑥+(1−𝜃)𝑑^𝑦,𝜃𝑍^𝑑𝑥+(1−𝜃)𝑍^𝑑𝑦) and find that 𝜃𝑥+(1−𝜃)𝑦+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑐𝑑𝑠−^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑑^𝑁𝑠_𝑖=1𝑋_𝑖−^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝜃𝑑^𝑥_𝑠+(1−𝜃)𝑑^𝑦_𝑠+𝑑𝑠_𝑡0𝑒^{𝑟(𝑡−𝑠)}𝑑𝜃𝑍^𝑑𝑥_𝑠+(1−𝜃)𝑍^𝑑𝑦_𝑠=𝜃𝑥+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑐−𝑑^𝑥_𝑠𝑑𝑠−^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑑^𝑁𝑠_𝑖=1𝑋_𝑖+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑑𝑍^𝑑𝑥_𝑠+(1−𝜃)𝑦+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑐−𝑑^𝑦_𝑠−𝑑𝑠_𝑡0𝑒^{𝑟(𝑡−𝑠)}𝑑^𝑁𝑠_𝑖=1𝑋_𝑖+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑑𝑍^𝑑𝑦_𝑠≥0, (16) which implies that policy (𝜃𝑑^𝑥+(1−𝜃)𝑑^𝑦,𝜃𝑍^𝑑𝑥+(1−𝜃)𝑍^𝑑𝑦) is an admissible dividend strategy for the initial capital 𝜃𝑥+(1−𝜃)𝑦. Denote 𝜃𝑑^𝑥+(1−𝜃)𝑑^𝑦=𝑑^{𝜃𝑥+(1−𝜃)𝑦} and let 𝑍^{𝑑𝜃𝑥+(1−𝜃)𝑦} be the corresponding injection process. Then we must have 𝑍^{𝑑𝜃𝑥+(1−𝜃)𝑦}_𝑡≤𝜃𝑍^𝑑𝑥_𝑡+(1−𝜃)𝑍^𝑑𝑦_𝑡. Therefore, it follows that 𝑉𝜃𝑥+(1−𝜃)𝑦≥𝐸^∞₀𝑒^−𝛿𝑡𝑑_𝑡^{𝜃𝑥+(1−𝜃)𝑦}𝑑𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍^{𝑑𝜃𝑥+(1−𝜃)𝑦}_𝑡≥𝐸^∞₀𝑒^−𝛿𝑡𝜃𝑑^𝑥_𝑡+(1−𝜃)𝑑^𝑦_𝑡𝑑𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝜃𝑍^𝑑𝑥_𝑡+(1−𝜃)𝑍^𝑑𝑦_𝑡=𝜃𝑉^𝑑𝑥(𝑥)+(1−𝜃)𝑉^𝑑𝑦𝑦. (17) Taking the supremum over all admissible strategies in Π, we find that 𝑉𝜃𝑥+(1−𝜃)𝑦≥𝜃𝑉(𝑥)+(1−𝜃)𝑉𝑦. (18) Then the concavity of the maximal value function 𝑉(𝑥) follows.
Let 𝜋_𝑦={𝑑^𝑦_𝑡,𝑍^𝑑𝑦_𝑡} be an arbitrary admissible strategy for initial capital 𝑦. Let ℎ denote the time that the surplus reaches 𝑦 from 𝑥 if no claims occur, that is, 1ℎ=𝑟ln𝑐+𝑟𝑦.𝑐+𝑟𝑥 (19) For initial capital 𝑥, consider a dividend strategy 𝑑^𝑥_𝑡=0,for𝑡≤ℎor𝑇₁𝑑≤ℎ,_{𝑦𝑡−ℎ},for𝑇₁∧𝑡>ℎ. (20) Then the probability that there is dividend payment is the probability that the surplus reaches 𝑦 without claims occurring first. This probability is 𝑒^−𝜆ℎ. So, we have 𝑉(𝑥)≥𝐸^∞₀𝑒^−𝛿𝑡𝑑^𝑥_𝑡𝑑𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍^𝑑𝑥_𝑡=𝑃𝑇₁>ℎ𝐸^∞₀𝑒^−𝛿𝑡𝑑^𝑥_𝑡𝑑𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍^𝑑𝑥_𝑡∣𝑇₁𝑇>ℎ+𝑃₁≤ℎ𝐸^∞₀𝑒^−𝛿𝑡𝑑^𝑥_𝑡𝑑𝑡−𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍^𝑑𝑥_𝑡∣𝑇₁≤ℎ=𝑒^−𝜆ℎ𝑒^−𝛿ℎ𝑉_𝜋𝑦𝑦−1−𝑒^−𝜆ℎ𝐸𝜅^∞₀𝑒^−𝛿𝑡𝑑𝑍^𝑑𝑥_𝑡∣𝑇₁≤ℎ≥𝑒^{−(𝜆+𝛿)ℎ}𝑉_𝜋𝑦𝑦−𝜅1−𝑒^−𝜆ℎ𝐸^∞_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑋_𝑖=𝑒^{−(𝜆+𝛿)ℎ}𝑉_𝜋𝑦𝑦−𝜅𝜆𝜇𝛿1−𝑒^−𝜆ℎ. (21) Taking the supremum over all Π yields 𝑉(𝑥)≥𝑒^{−(𝜆+𝛿)ℎ}𝑉𝑦−𝜅𝜆𝜇𝛿1−𝑒^−𝜆ℎ. (22) Note that 1−𝑒^{−(𝜆+𝛿)ℎ}≤(𝜆+𝛿)ℎ≤𝜆+𝛿𝑐𝑦−𝑥,1−𝑒^−𝜆ℎ𝜆≤𝜆ℎ≤𝑐𝑦−𝑥. (23) Thus, by (22) and (23), we obtain 0≤𝑉𝑦−𝑉(𝑥)≤1−𝑒^{−(𝜆+𝛿)ℎ}𝑉𝑦+𝜅𝜆𝜇𝛿1−𝑒^−𝜆ℎ≤(𝜆+𝛿)𝛼+𝜅𝜆²𝜇,𝑐𝛿𝑦−𝑥 (24) which implies that Lipschitz continuity holds. Therefore, 𝑉(𝑥) is absolutely continuous.

Lemma 3. The maximal value function 𝑉(𝑥) is differentiable from the right, and the right derivative 𝑉^′(𝑥+) satisfies HJB equation: max_{0≤𝑑≤𝛼}(𝑐−𝑑+𝑟𝑥)𝑉^′(𝑥+)−(𝜆+𝛿)𝑉(𝑥)+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦+𝑑=0. (25)

Proof. Take a constant 𝑢∈[0,𝛼]. Let ℎ>0 be small enough and 𝜑(𝑥,ℎ)=𝑥𝑒^𝑟ℎ+∫^ℎ₀𝑒^{𝑟(ℎ−𝑠)}(𝑐−𝑢)𝑑𝑠. We can show analogously to the proof of Proposition 3.1 of [3, 12, 13] that the maximal dividend-value function fulfills dynamic programming principle for any stopping times 𝜏: 𝑉(𝑥)=sup_𝜋∈Π𝐸^{𝜏∧𝑇𝜋}₀𝑒^−𝛿𝑡𝑑_𝑡𝑑𝑡+𝑒^{−𝛿(𝜏∧𝑇𝜋)}𝑉𝑈^𝜋_{𝜏∧𝑇𝜋}. (26) Taking 𝜏=𝑇₁∧ℎ in (26), we derive 𝑉(𝑥)=sup_𝜋∈Π𝐸^𝑇1₀^∧ℎ𝑒^−𝛿𝑡𝑑_𝑡𝑑𝑡+𝑒^{−𝛿(𝑇1∧ℎ)}𝑉𝑈^𝜋_𝑇1∧ℎ≥𝑒^{−𝜆ℎℎ}₀𝑒^−𝛿𝑡𝑢𝑑𝑡+𝑒^−𝛿ℎ𝑉𝜑()+𝑥,ℎ_ℎ0𝜆𝑒^{−𝜆𝑡𝑡}₀𝑒^−𝛿𝑠𝑢𝑑𝑠+𝑒^{−𝛿𝑡∞}₀𝑉𝜑(𝑥,𝑡)−𝑦𝑝𝑦𝑑𝑦𝑑𝑡. (27) Rearranging the terms and dividing by ℎ yields 𝑉𝜑(𝑥,ℎ)−𝑉(𝑥)ℎ−1−𝑒^{−(𝜆+𝛿)ℎ}ℎ𝑉𝜑(+𝑒𝑥,ℎ)_−𝜆ℎℎ^ℎ₀𝑒^−𝛿𝑡+1𝑢𝑑𝑡ℎ^ℎ₀𝜆𝑒^{−𝜆𝑡𝑡}₀𝑒^−𝛿𝑠𝑢𝑑𝑠+𝑒^{−𝛿𝑡∞}₀𝑉𝜑(𝑥,𝑡)−𝑦𝑝𝑦𝑑𝑦𝑑𝑡≤0. (28) On the other hand, from the dynamic programming principle, there exists a strategy 𝜋^ℎ2={𝑑^ℎ2_𝑡} such that 𝐸^𝑇1₀^∧ℎ𝑒^−𝛿𝑡𝑑^ℎ2_𝑡𝑑𝑡+𝑒^{−𝛿(𝑇1∧ℎ)}𝑉𝑈^𝜋ℎ2_𝑇1∧ℎ≥𝑉(𝑥)−ℎ². (29) Let 𝑢_𝑡 denote 𝑑^ℎ2_𝑡 conditioned on 𝑇₁>𝑡 and 𝜙(𝑥,𝑡)=𝑥𝑒^𝑟𝑡+∫^𝑡₀𝑒^{𝑟(𝑡−𝑠)}(𝑐−𝑢_𝑠)𝑑𝑠. Using the same way as above, we get that 𝑉𝜙()(𝑥)ℎ+𝑥,ℎ−𝑉ℎ−1−𝑒^{−(𝜆+𝛿)ℎ}ℎ𝑉𝜙(+𝑒𝑥,ℎ)_−𝜆ℎℎ^ℎ₀𝑒^−𝛿𝑡𝑢_𝑡+1𝑑𝑡ℎ^ℎ₀𝜆𝑒^{−𝜆𝑡𝑡}₀𝑒^−𝛿𝑠𝑢_𝑠𝑑𝑠+𝑒^{−𝛿𝑡∞}₀𝑉𝜙(𝑥,𝑡)−𝑦𝑝𝑦𝑑𝑦𝑑𝑡≥0. (30) Notice that lim_ℎ→0+𝜑(𝑥,ℎ)=lim_ℎ→0+𝜙(𝑥,ℎ)=𝑥. (31) Let 𝐷⁺𝑉(𝑥)=limsup_𝑠→0+𝑉(𝑥+𝑠)−𝑉(𝑥)𝑠,𝐷⁻𝑉(𝑥)=liminf_𝑠→0+𝑉(𝑥+𝑠)−𝑉(𝑥)𝑠. (32) We know that 𝐷⁺𝑉(𝑥) and 𝐷⁻𝑉(𝑥) are finite by Lipschitz continuity and monotonicity. We assume that lim_ℎ→0+𝑢_ℎ=𝑑₀. (33) Then we get lim_ℎ→0+𝜙(𝑥,ℎ)−𝑥ℎ=𝑐−𝑑₀+𝑟𝑥. (34) Choose a sequence {ℎ_𝑛⁽¹⁾} such that lim_𝑛→∞𝑉𝜙𝑥,ℎ_𝑛⁽¹⁾−𝑉(𝑥)𝜙𝑥,ℎ_𝑛⁽¹⁾−𝑥=𝐷⁻𝑉(𝑥). (35) Then the limit is lim_𝑛→∞𝑉𝜙𝑥,ℎ_𝑛⁽¹⁾−𝑉(𝑥)ℎ_𝑛⁽¹⁾=lim_𝑛→∞𝜙𝑥,ℎ_𝑛⁽¹⁾−𝑥ℎ_𝑛⁽¹⁾⋅𝑉𝜙𝑥,ℎ_𝑛⁽¹⁾−𝑉(𝑥)𝜙𝑥,ℎ_𝑛⁽¹⁾=−𝑥𝑐−𝑑₀+𝑟𝑥𝐷⁻𝑉(𝑥). (36) Taking ℎ=ℎ_𝑛⁽¹⁾ in (30) and letting 𝑛→∞, we get 𝑐−𝑑₀+𝑟𝑥𝐷⁻𝑉(𝑥)−(𝜆+𝛿)𝑉(𝑥)+𝑑₀+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦≥0. (37) Using the same method as above, we choose a sequence {ℎ_𝑛⁽²⁾} such that lim_𝑛→∞𝑉𝜑𝑥,ℎ_𝑛⁽²⁾−𝑉(𝑥)𝜑𝑥,ℎ_𝑛⁽²⁾−𝑥=𝐷⁺𝑉(𝑥). (38) Then taking 𝑢=𝑑₀ and ℎ=ℎ_𝑛⁽²⁾ in (28) and letting 𝑛→∞, we have 𝑐−𝑑₀+𝑟𝑥𝐷⁺𝑉(𝑥)−(𝜆+𝛿)𝑉(𝑥)+𝑑₀+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦≤0. (39) From (37) and (39), we know that 𝐷⁺𝑉(𝑥)≤𝐷⁻𝑉(𝑥). (40) On the other hand 𝐷⁺𝑉(𝑥)≥𝐷⁻𝑉(𝑥). (41) So we have 𝐷⁺𝑉(𝑥)=𝐷⁻𝑉(𝑥)=𝑉^′(𝑥+). (42) Therefore, the maximal dividend-value function is differentiable from the right.
For the sequence {ℎ_𝑛⁽¹⁾} and 𝑢=𝑑₀, inequality (28) holds and lim_𝑛→∞𝑉𝜑𝑥,ℎ_𝑛⁽¹⁾−𝑉(𝑥)ℎ_𝑛⁽¹⁾=𝑐−𝑑₀+𝑟𝑥𝑉^′(𝑥+). (43) Then taking limit on both sides of inequalities (28) and (30), we find that 𝑐−𝑑₀+𝑟𝑥𝑉^′(𝑥+)−(𝜆+𝛿)𝑉(𝑥)+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦+𝑑₀=0. (44) From (28) we conclude that for any 𝑑(𝑐−𝑑+𝑟𝑥)𝑉^′(𝑥+)−(𝜆+𝛿)𝑉(𝑥)+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦+𝑑≤0. (45) Thus, we get the HJB equation for 𝑉^′(𝑥+)max_{0≤𝑑≤𝛼}(𝑐−𝑑+𝑟𝑥)𝑉^′(𝑥+)−(𝜆+𝛿)𝑉(𝑥)+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦+𝑑=0. (46)

Lemma 4. When 𝑥>0, the maximal value function 𝑉(𝑥) is differentiable from the left and the left derivative 𝑉^′(𝑥−) satisfies HJB equation: max_{0≤𝑑≤𝛼}(𝑐−𝑑+𝑟𝑥)𝑉^′(𝑥−)−(𝜆+𝛿)𝑉(𝑥)+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦+𝑑=0. (47)

Proof. Let ℎ be small enough such that 𝜌()=𝑐𝑥,ℎ𝑥+𝑟𝑒^−𝑟ℎ−𝑐𝑟 (48) is positive. Changing the initial capital 𝑥 into 𝜌(𝑥,ℎ) and using the dynamic programming principle, we obtain 𝑉𝜑(𝑥,ℎ)−𝑉𝜌(𝑥,ℎ)ℎ−1−𝑒^{−(𝜆+𝛿)ℎ}ℎ𝑉𝜑(+𝑒𝑥,ℎ)_−𝜆ℎℎ^ℎ₀𝑒^−𝛿𝑡+1𝑢𝑑𝑡ℎ^ℎ₀𝜆𝑒^{−𝜆𝑡𝑡}₀𝑒^−𝛿𝑠𝑢𝑑𝑠+𝑒^{−𝛿𝑡∞}₀𝑉𝜑()𝑝𝑦𝑉𝜙()𝜌()𝑥,ℎ−𝑦𝑑𝑦𝑑𝑡≤0,ℎ+𝑥,ℎ−𝑉𝑥,ℎℎ−1−𝑒^{−(𝜆+𝛿)ℎ}ℎ𝑉𝜙(+𝑒𝑥,ℎ)_−𝜆ℎℎ^ℎ₀𝑒^−𝛿𝑡𝑢_𝑡+1𝑑𝑡ℎ^ℎ₀𝜆𝑒^{−𝜆𝑡𝑡}₀𝑒^−𝛿𝑠𝑢_𝑠𝑑𝑠+𝑒^{−𝛿𝑡∞}₀𝑉𝜙(𝑥,𝑡)−𝑦𝑝𝑦𝑑𝑦𝑑𝑡≥0, (49) where 𝜑()=𝑐𝑥,𝑡𝑥+𝑟𝑒^−𝑟ℎ−𝑐𝑟𝑒^𝑟𝑡+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}()𝜙()=𝑐𝑐−𝑢𝑑𝑠,𝑥,𝑡𝑥+𝑟𝑒^−𝑟ℎ−𝑐𝑟𝑒^𝑟𝑡+^𝑡₀𝑒^{𝑟(𝑡−𝑠)}𝑐−𝑢_𝑠𝑑𝑠. (50) Let 𝐷₊𝑉(𝑥)=limsup_𝑠→0−𝑉(𝑥+𝑠)−𝑉(𝑥)𝑠,𝐷₋𝑉(𝑥)=liminf_𝑠→0−𝑉(𝑥+𝑠)−𝑉(𝑥)𝑠. (51) Using the same method as Lemma 3, we obtain 𝐷₊𝑉(𝑥)=𝐷₋𝑉(𝑥)=𝑉^′(𝑥−), (52) which implies the maximal dividend-value function which is differentiable from the left. Furthermore, the left derivative 𝑉^′(𝑥−) fulfills HJB equation: max_{0≤𝑑≤𝛼}(𝑐−𝑑+𝑟𝑥)𝑉^′(𝑥−)−(𝜆+𝛿)𝑉(𝑥)+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦+𝑑=0. (53)

Theorem 5. The maximal value function 𝑉(𝑥) is continuously differentiable on [0,∞) and fulfills the following HJB equation: max_{0≤𝑑≤𝛼}(𝑐−𝑑+𝑟𝑥)𝑉^′(𝑥)−(𝜆+𝛿)𝑉(𝑥)+𝜆^∞₀𝑉𝑥−𝑦𝑝𝑦𝑑𝑦+𝑑=0, (54) where 𝑉^′(0) means the right derivative 𝑉^′(0+).

Proof. From Lemmas 3 and 4, we know that function 𝑉(𝑥) is continuously differentiable and fulfills the HJB equation (54).

3. Optimal Dividend Strategy

Since 𝑉(𝑥) is increasing, concave and converges to 𝛼/𝛿, there exists a finite constant: 𝑏^∗=inf𝑥∶𝑉^′(𝑥)≤1, (55) such that 𝑉^′(𝑥)>1,if𝑥<𝑏^∗,𝑉^′(𝑥)≤1,if𝑥≥𝑏^∗. (56) From Theorem 5, we construct a strategy 𝜋^∗={𝑑^∗_𝑡,𝑍^𝑑∗_𝑡} as follows: 𝑑^∗_𝑡+=0,if𝑈^𝜋∗_𝑡<𝑏^∗,𝛼,if𝑈^𝜋∗_𝑡≥𝑏^∗. (57)𝑍^𝑑∗_𝑡 is the corresponding capital injection process. This policy is a double level strategy with a lower barrier at zero and an upper threshold at level 𝑏^∗. Whenever the surplus is between 0 and 𝑏^∗, no dividends are paid and no capitals are injected; whenever the surplus is at or above 𝑏^∗, dividends will be paid at the maximal rate 𝛼, but no capitals are injected; whenever the surplus is below 0, a required amount of the capital which are equal to the amount of the deficit are injected immediately. Figure 1 gives an intuitional description for the strategy 𝜋^∗.

Figure 1 

A typical sample path of the surplus process with strategy 𝜋∗.

Theorem 6. The policy 𝜋^∗={𝑑^∗_𝑡,𝑍^𝑑∗_𝑡} which is given by (57) is optimal among all admissible policies.

Proof. First of all, the policy 𝜋^∗={𝑑^∗_𝑡,𝑍^𝑑∗_𝑡} is an admissible strategy. Let 𝑉_∗(𝑥) denote the corresponding value function. By an application of Itô’s formula to 𝑒^−𝛿𝑡𝑉(𝑈^𝜋∗_𝑡), it can be verified that 𝑒^−𝛿𝑡𝑉𝑈^𝜋∗_𝑡=𝑉(𝑥)+^𝑡₀𝑒^−𝛿𝑠𝑐−𝑑^𝜋∗_𝑠+𝑟𝑈^𝜋∗_𝑠−×𝑉^′𝑈^𝜋∗_𝑠−−𝛿𝑉𝑈^𝜋∗_𝑠−𝑑𝑠+_{𝑈0<𝑠≤𝑡𝜋∗𝑠−≠𝑈𝜋∗𝑠}𝑒^−𝛿𝑠𝑉𝑈^𝜋∗_𝑠−𝑉𝑈^𝜋∗_𝑠−. (58) Note that the injections occur only at the claim times. Let 𝑧^𝜋∗_𝑖 denote the amount of injected capital at the occurrence time of the 𝑖th claim. If 𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖≥0, no capitals are injected, that is, 𝑧^𝜋∗_𝑖=0. If 𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖<0, 𝑧^𝜋∗_𝑖=−(𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖) and 𝑈^𝜋∗_𝑇𝑖=𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖+𝑧^𝜋∗_𝑖=0. For both cases, the equality 𝑉𝑈^𝜋∗_𝑇𝑖=𝑉𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖+𝜅𝑧^𝜋∗_𝑖 (59) holds. Since 𝑈^𝜋∗_𝑠−≠𝑈^𝜋∗_𝑠 only at the occurrence time of a claim, then _{𝑈0<𝑠≤𝑡𝜋∗𝑠−≠𝑈𝜋∗𝑠}𝑒^−𝛿𝑠𝑉𝑈^𝜋∗_𝑠−𝑉𝑈^𝜋∗_𝑠−=^𝑁𝑡_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑉𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖+𝑧^𝜋∗_𝑖−𝑉𝑈^𝜋∗_𝑇𝑖−=^𝑁𝑡_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑉𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖−𝑉𝑈^𝜋∗_𝑇𝑖−+𝜅^𝑁𝑡_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑧^𝜋∗_𝑖=^𝑁𝑡_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑉𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖−𝑉𝑈^𝜋∗_𝑇𝑖−+𝜅^𝑡₀𝑒^−𝛿𝑠𝑑𝑍^𝑑∗_𝑠. (60) Taking expectations in (58), we obtain 𝐸𝑒^−𝛿𝑡𝑉𝑈^𝜋∗_𝑡=𝑉(𝑥)+𝐸^𝑡₀𝑒^−𝛿𝑠𝑐−𝑑^𝜋∗_𝑠+𝑟𝑈^𝜋∗_𝑠−𝑉^′𝑈^𝜋∗_𝑠−−𝛿𝑉𝑈^𝜋∗_𝑠−𝑑𝑠+𝜅𝐸^𝑡₀𝑒^−𝛿𝑠𝑑𝑍^𝑑∗_𝑠+𝐸^𝑁𝑡_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑉𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖−𝑉𝑈^𝜋∗_𝑇𝑖−. (61) Note that ^𝑁𝑡_𝑖=1𝑒^{−𝛿𝑇𝑖}𝑉𝑈^𝜋∗_𝑇𝑖−−𝑋_𝑖−𝑉𝑈^𝜋∗_𝑇𝑖−−𝜆^𝑡₀𝑒^{−𝛿𝑠∞}₀𝑉𝑈^𝜋∗_𝑠−−𝑦−𝑉𝑈^𝜋∗_𝑠−𝑝𝑦𝑑𝑦𝑑𝑠 (62) is a martingale with zero-expectation. Then 𝐸𝑒^−𝛿𝑡𝑉𝑈^𝜋∗_𝑡=𝑉(𝑥)+𝐸^𝑡₀𝑒^−𝛿𝑠𝑐−𝑑^𝜋∗_𝑠+𝑟𝑈^𝜋∗_𝑠−𝑉^′𝑈^𝜋∗_𝑠−−(𝜆+𝛿)𝑉𝑈^𝜋∗_𝑠−+𝜆^∞₀𝑉𝑈^𝜋∗_𝑠−−𝑦𝑝𝑦𝑑𝑦𝑑𝑠+𝜅𝐸^𝑡₀𝑒^−𝛿𝑠𝑑𝑍^𝑑∗_𝑠. (63) By HJB equation (54), we find that 𝑐−𝑑^𝜋∗_𝑠+𝑟𝑈^𝜋∗_𝑠−𝑉^′𝑈^𝜋∗_𝑠−−(𝜆+𝛿)𝑉𝑈^𝜋∗_𝑠−+𝜆^{𝑈𝜋∗𝑠−}₀𝑉𝑈^𝜋∗_𝑠−−𝑦𝑝𝑦𝑑𝑦=−𝑑^𝜋∗_𝑠. (64) Thus, we gather that 𝐸𝑒^−𝛿𝑡𝑉𝑈^𝜋∗_𝑡+𝐸^𝑡₀𝑒^−𝛿𝑠𝑑^𝜋∗_𝑠𝑑𝑠−𝜅𝐸^𝑡₀𝑒^−𝛿𝑠𝑑𝑍^𝑑∗_𝑠=𝑉(𝑥). (65) Taking limit in 𝑡→∞, by the boundedness of 𝑉, we obtain that 𝑉_∗(𝑥)=𝐸^∞₀𝑒^−𝛿𝑠𝑑^𝜋∗_𝑠𝑑𝑠−𝜅^∞₀𝑒^−𝛿𝑠𝑑𝑍^𝑑∗_𝑠=𝑉(𝑥). (66) This shows that the policy 𝜋^∗ is the optimal strategy.

4. Concluding Remarks

In summary, this paper gives the optimal strategy which maximizes the discounted dividend payments minus the penalized discounted capital injections for a risk model including a constant force of interest. The optimal dividend strategy is a threshold strategy for the case of the dividend payout rate which is bounded by some positive constant, and the optimal injection strategy is to inject capitals immediately to make the company’s assets back to zero when the surplus of the company becomes negative.

We also wish to point out that further research is needed. Further extensions of the analysis of the problems in the paper could remove the restriction on the dividend payout rate. We hope that this open problem can be addressed in future research.

Acknowledgments

The authors would like to thank the referees for their helpful comments. The research was supported by National Natural Science Foundation of China (Grant nos. 11201271, 11126093, and 71071088).