OPTIMAL DIVIDEND AND CAPITAL INJECTION STRATEGY WITH EXCESS-OF-LOSS REINSURANCE AND TRANSACTION COSTS

. This article deals with an optimal dividend, reinsurance and capital injection control problem in the diﬀusion risk model. Under the objective of maximizing the insurance company’s value, we aim at ﬁnding the joint optimal control strategy. We assume that there exist both the ﬁxed and proportional costs in control processes and the excess-of-loss reinsurance is “expensive”. We derive the closed-form solutions of the value function and optimal strategy by using stochastic control methods. Some economic interpretations of the obtained results are also given.


1.
Introduction. In mathematical insurance literature, the classical dividend problem consists in finding a dividend strategy that maximizes the total expected discounted dividends until the time of bankruptcy. Its origin can be traced to the work of [5]. Since then much research on this issue has been carried out for varieties of models. In practice, the company sometimes needs to raise new capital from market to continue the business. The expected present values of the dividends payout minus capital injections up to the time of ruin can be regarded as the company's value. The company seeks to find optimal dividend and capital injection strategy for maximizing its value. Recently, the combinational optimization of dividend and capital injection has attracted many attentions. Sometimes, transaction costs generated by the control processes are added in the risk models. The literature on this issue includes [2], [7], [9], [11], [15], [17], [19] and so on. As shown in above references, transaction costs usually include two parts: the fixed costs and the proportional costs. The former are generated by the advisory and consulting as well as the latter are generated by the tax. In general, the fixed costs can generate more difficult impulse control problem. To manage its risks and explore profit opportunities, the company needs to determine the times and amounts of dividend payments and capital injections, respectively.
As everyone knows, the insurance company usually cedes risks by purchasing reinsurance. An appropriate use of reinsurance protects it against unexpected, potentially large losses, and hence reduces the insurer's earnings volatility. In practice, the excess-of-loss reinsurance is one of most popular reinsurance policies. The optimal dividend problem in the presence of excess-of-loss reinsurance has attracted many interests for its potential application on insurance industry. For example, [1] studied the optimal excess-of-loss reinsurance and dividend strategies for maximizing the expected total discounted dividends received by shareholders until the time of bankruptcy. [3] further explored this problem by taking into account both the proportional and fixed costs incurred by dividends. [12] investigated the optimal dividend, capital injection and excess-of-loss reinsurance problem in a diffusion risk model. They also considered the influences of the fixed and proportional transaction costs. In discussed references, it was assumed that both insurer and reinsurer used the expectation premium principle with same safety loading to calculate premiums, i.e., the reinsurance is "cheap". However, the reinsurer may use a higher safety loading in practice, which results in the cedent paying larger amount of premium than the amount which is reinsured, i.e., the reinsurance is "expensive". In this case, the excess of premium is viewed as transaction costs. [10] studied the optimal dividend and capital injection problem with "expensive" excess-of-loss reinsurance. But the fixed transaction costs were not taken into account. With the exception of [10], very little work has considered such a joint optimal control problem. The literature on the "expensive" excess-of-loss reinsurance includes [8], [13], [16], [18] and so on.
Inspired by [10] and [12], this paper further studies the combined dividend, reinsurance and capital injection problem in the framework of diffusion risk model. Both the fixed and proportional transaction costs are included and the excess-of-loss reinsurance is assumed to be "expensive". We will solve the problem in 11 cases, which depend on the relationships among the parameters. Some existing results are extended. The rest of the paper is organized as follows. In Section 2, we introduce the risk model and raise the combined dividend-reinsurance-capital injection problem.
In Section 3, we analyze some properties of the value function and give the QVI (quasi-variational inequalities) associated with value function. Based on the costs of reinsurance, as measured by the safety loading, we establish the value function and associated optimal strategy in Section 4 and Section 5, respectively. Section 6 is a conclusion.

2.
Model formulation and the optimal control problem. We work on a complete probability space (Ω, F, P) on which all processes are well defined. The information at time t is given by F t , in which {F t : t ≥ 0} is the complete filtration generated by the claim, dividend, reinsurance and capital injection processes. Our results will be formulated under the controlled diffusion model. However, for the purpose of motivation it is convenient to start from the classical Cramér-Lundberg model. In this model, the uncontrolled surplus process {U t } t≥0 of an insurance company follows that where U 0− = x ≥ 0 is the initial surplus, c > 0 is the rate of premiums, N t is a Poisson process with constant intensity λ, random variables Y i 's are positive i.i.d. claims with a common distribution function F (·). Define the quantity m = inf{y ≥ 0 : F (y) = 1} < ∞, the finite mean µ m = EY 1 < ∞ and finite second moment Assume that the premium c in (2.1) is calculated via the expected value principle, i.e., where θ 1 > 0 is the safety loading of the insurer. Let a denote the excess-of-loss retention level and whereF (y) = P (Y 1 > y) = 1 − F (y) and y ∧ a = min{y, a}. Then both functions µ(a) and ν(a) are increasing on [0, m], while on [m, ∞) they are constants equal to µ(m) = µ m and ν(m) = ν m . Assume that the reinsurer also employs the expected value principle but with a higher safety loading θ 2 ∈ (θ 1 , ∞) to calculate premiums, then represents the reinsurance premium that is payable by the insurer to the reinsurer. Then, the surplus process involving excess-of-loss reinsurance can be written as with U a 0− = x. According to [6], we approximate model (2.4) by a pure diffusion model {X a t } t≥0 with the same drift and volatility. Namely, X a t satisfies the following stochastic process with X a 0− = x. Suppose that a ∈ [0, m] can be adjusted dynamically to control the risk exposure. We use the process {a t } t≥0 to describe a reinsurance strategy. Moreover, we incorporate dividend distribution and capital injection in model (2.4).
the cumulative amount of dividends paid from time 0 to time t. It is determined by a sequence of increasing stopping times { i , i = 1, 2, · · · } and a sequence of non-negative random variables {ζ i , i = 1, 2, · · · }, which represent the times and the sizes of dividends, respectively. Furthermore, let R t = ∞ i=1 I {τi≤t} η i denote the cumulative amount of capital injections from time 0 to time t. It is described by a sequence of increasing stopping times {τ i , i = 1, 2, · · · } and a sequence of random variables {η i , i = 1, 2, · · · }, which represent the times and the amounts of capital injections, respectively. Given a joint strategy π = (a π , D π , R π ), the controlled surplus process follows with X π 0− = x. The insurance company selects a reinsurance strategy a π , a dividend strategy D π and a capital injection strategy R π at any time t based on information available up to and including time t, say F B t = σ{B s : s ≤ t}. Mathematically, we give the following definition of admissible strategy that can be selected by the insurer.
Definition 2.1. A strategy π = (a π , D π , R π ) is said to be admissible if it satisfies the following conditions: = 0 for all t ≥ 0, i.e., the insurance company can not pay dividend and raise new capital at the same time.
We write Π for the space of these admissible strategies. For each π ∈ Π, we define the time of bankruptcy as T π = inf{t ≥ 0 : X π t < 0}, which is an F B tstopping time. For each dividend payment, it incurs a fixed cost K 1 > 0, which is independent of the amount of the payment. Let β 1 ∈ (0, 1) be a positive constant, where 1 − β 1 is the tax rate at which the dividends are taxed. Consequently, if the amount ζ of liquid surplus is withdrawn, the net amount of money that the shareholders receive after transaction costs have been paid is β 1 ζ − K 1 . Similarly, the company needs to pay β 2 η + K 2 to meet the capital injection η, where β 2 > 1 measures the proportional costs and K 2 > 0 is the fixed cost. Then the value of the company is measured by the following performance function which is the expected present values of the dividends payout minus capital injections up to the time of bankruptcy. E x denotes the expectation conditional on X π 0− = x and δ > 0 is the discount factor. Problem 2.2 The optimization problem of the insurance company is to find the value function and associated optimal strategy π * ∈ Π such that V (x) = V (x; π * ).
3. Properties of the value function. In this section, we will establish the QVI associated with the optimization problem and derive some properties of the value function.
Proposition 1. The value function defined by (2.8) satisfies that with the boundedness condition Proof. Consider an admissible strategy π 1 with V (y; π 1 ) ≥ V (y) − ε for any ε > 0. For x ≥ y ≥ 0, we define a new admissible strategy as follows: x − y is paid immediately as dividend and then the strategy π 1 with initial surplus y is followed. Then for ε > 0, it holds that Because ε is arbitrary, The second inequality in (3.1) can be proved similarly.
The surplus process {X a t } t≥0 with only reinsurance is expressed by (2.5), then By Itô's formula, we have Since X π T π = 0 and X π t ≥ 0, for t ≤ T π , taking expectation on both sides yields Then, it has The last inequality is confirmed by (3.3) and (3.4). In addition, V (x) ≥ 0 is obvious. Therefore, (3.1) and (3.2) are proven.
To proceed with our work, let's define some operators for a function v: Similar to [4], using a standard application of the dynamic programming principle, we can give a characterization of the value function by the following definition.
Remark 1. It is optimal to postpone refinancing as long as possible, i.e., they may happen only at the moments when the surplus process hits the barrier 0. The result can be established by repeating a similar procedure to that in Lemma 3.2 of [14].
It is easy to understand that the insurer should buy less reinsurance with the increasing cost of reinsurance, which is measured by the safety loading θ 2 for reinsurer. It is expected that full retention will be taken once θ 2 exceeds some critical level. The following analysis shows that the critical level is 2mµm νm θ 1 ∈ (θ 1 , ∞). Consequently, we should solve the optimization problem in two different cases. 4. The case of θ 2 ∈ (θ 1 , 2mµm νm θ 1 ). In this section, let's consider the first case with In view of the structure of (3.9), we shall discuss the solution of QVI according to different boundary conditions in the next two subsections. Following the approach in optimal control theory, we assume that (3.5)-(3.9) have a appropriate solution, which is continuously differentiable on (0, ∞), twice continuously differentiable on (0, u 2 ] and linear on [u 2 , ∞) for some parameter u 2 . This assumption will be verified later.
4.1. The case without capital injection. Suppose that it is optimal to get out of the business whenever the surplus is null, then the corresponding boundary conditions are v(0) = 0 and C v(0) − v(0) ≤ 0. Following the approach in stochastic control theory, the candidate solution f (x) for v(x) in this case should satisfy that with some unknown parameters 0 < u 2 < ∞. Differentiating (4.2) with respect to a and setting the derivative to zero yield Plugging (4.6) in (4.2) yields Thus, (4.4) and (4.7) lead to Note that a is a function of x. Taking derivative with respect to x on both sides of (4.7) and using (4.6) again result in λθ 2 ν(y) λθ 2 2 (2yµ(y) − ν(y)) + 2λθ 2 (θ 1 − θ 2 )µ m y + 2δy 2 dy. It is not difficult to prove that the integrand is positive, so Q(x) is increasing strictly and Q(∞) < ∞. Consequently, the inverse Q −1 (x) of the function Q(x) exists. Since the function Q −1 (x) satisfies the same 1st order ordinary differential equation for x > 0 as the function a(x) (see (4.9)) and for x ≥ 0. We conjecture that there exists x 0 = Q(m) ≤ u 2 such that the insurer will not buy reinsurance once the surplus exceeds x 0 . Due to the strictly increasing property of Q(x), the inequality Q(m) > 0 is equivalent to m > Q −1 (0) = a(0) where the last equality follows by (4.10). It then follows by (4.8) that w −1 ( θ2−θ1 θ2 µ m ) = a(0) < m. Due to the strictly increasing property of w, we then obtain θ2−θ1 θ2 µ m < w(m) = µ m − νm 2m , where the first inequality is equivalent to (4.1) and the last equality follows by the definition of w. Under condition (4.1), in view of (4.4) and (4.6), we can express f (x) through a(x) by 2) turns to be a second-order ordinary differential equation Therefore, with The continuity of f (x) and f (x) at point x 0 yields that Above pair of equations gives with The inequality in (4.15) is confirmed by and a(x) ≡ m is optimal. Then the suggested solution to (4.2)-(4.5) is of the form with a(z) = Q −1 (z). The optimal reinsurance strategy is described by It now remains to determine k, u 1 and u 2 . Inspired by [4], we start with constructing a function ψ by (4.20) We can prove that ψ(x) ∈ C 2 is convex and has the following useful properties where The inequality in (4.21) holds iff r − + θ2 m < 0, which is confirmed by (4.1). See the proof process in Appendix A.
The function ψ(x) plays an important role in this paper. Next, we shall determine the values of k, u 1 and u 2 in different cases.
Case A. The case of 0 < K 1 < J 1 (β 1 /ρ), here the integral is defined by In view of the structure of ψ(x), we know that u k 1 is increasing, u k 2 is decreasing with respect to k. In particular, if k = β 1 /ρ then u k 1 = 0 and if k = β 1 /α then given by (4.18) and a π * (x) given by (4.19) satisfy (4.2) and (4.4). Moreover, we write that we need to check the inequality (4.5) in following subcases.
Obviously, (4.5) holds if and only if Obviously, (4.5) holds if and only if In addition, the following integral will be used later: so (4.5) follows.
The foregoing analysis shows that the conjecture 0 < x 0 < u ki 2 , i = 1, 2 is true and all forms of f (x) and a π * (x) can solve Eqs. (4.2)-(4.5). In addition, f (x) is increasing and continuously differentiable on (0, ∞). Logically, we need to further check that they satisfy QVI except for a single point, but we omit it here, please see a similar proof process forf 1 (x) in Appendix B.

GONGPIN CHENG, RONGMING WANG AND DINGJUN YAO
It is easy to see that Consequently, there exists a unique p * or, equivalently,f Then, is true, (4.38) follows. It proves thatf 1 (x) andã π * 1 (x) satisfy (4.35)-(4.38) under condition (4.34). Finally, we need to further check that they also solve QVI except for a single point b k * 1 2 . Please see the detailed proof procedure in Appendix B.
In this case, the value function V (x) and associated optimal strategy π * take the same forms as those in (1).
with a(z) = Q −1 (z). The optimal dividend strategy D π * is characterized by (4.61) This implies that it is optimal to distribute all surplus x as dividend and declare bankruptcy immediately once x ≥ u k * 2 2 . Whenever the surplus reaches the lower barrier 0, the company should declare bankruptcy at once. There is at most one lump sum dividend. It is unprofitable to raise new money, so The optimal reinsurance policy is characterized by (4.19). The surplus process controlled π * = (a π * , D π * , R π * ) satisfies that (4.63) The value function V (x) is identical withf 1 (x) in (4.39). The optimal dividend strategy D π * is characterized by (4.64) It is profitable to inject new capital when and only when the surplus is null, the bankruptcy should be avoided, and R π * is characterized by (4.65) The optimal reinsurance policyã π * 1 is given by (4.40). The surplus process controlled by π * = (ã π * 1 , D π * , R π * ) satisfies that The value function V (x) is identical withf 2 (x) in (4.47). The optimal dividend strategy D π * is characterized by (4.67) It is profitable to raise new capital when and only when the surplus is null, the bankruptcy should be avoided for ever, and R π * is characterized by (4.68) The optimal reinsurance policyã π * 2 is given by (4.50). The surplus process controlled by π * = (ã π * 2 , D π * , R π * ) satisfies that (4.69) (7) 0 < K 1 < J 1 (β 1 /ρ), k * 1 > β 2 and 0 < K 2 ≤ I 2 (k * 1 ). The value function V (x) is identical withf 3 (x) in (4.51). The optimal dividend strategy D π * is characterized by (4.70) Capital injection is optimal when and only when the surplus is null. Mathematically, R π * is characterized by (4.71) The optimal reinsurance policyã π * 3 is given by (4.54), namely, it is optimal choice to buy no reinsurance all the time. The surplus process controlled by π * = (ã π * 3 , D π * , R π * ) satisfies that Proof. We only give the proof of (5) as an example. The method is applicable to prove other results. We fix an arbitrary admissible strategy π = (a π , D π , R π ) ∈ Π. By the previous statement, the functionf 1 (x) is continuously differentiable on (0, ∞) and is twice continuously differentiable on (0, b . However, for x = b k * 1 2 , the continuity off 1 fails. In spite of this, since {0 ≤ t ≤ T π : X π t = b k * 1 2 } has zero Lebesgue measure almost surely under P, Itô's formula is applicable to such af 1 (x) as well. Thus, we have where the inequality follows from (3.5) on (0, b In view of the boundedness off 1 (x) ∈ [k * 1 α, k * 1 ψ(p * 1 )] on [0, ∞), we know that t∧T π 0 e −δs λν(a π )f 1 (X π s )dB s is a martingale. Taking expectations on both sides of (4.72) yields Because of {s : X π s− = X π s } = { π 1 , · · · π i , · · · ; τ π 1 , · · · , τ π i , · · · }, it has Since the functionf 1 (x) satisfies (3.6) and (3.7), we have (4.78) The first term of the left side is positive due tof 1 (x) = f (x + p * 1 ) > 0. Thus Letting t → ∞ in (4.78) and rearranging the inequality, we get which indicates thatf 1 (x) ≥ V (x). When the strategy π * ∈ Π described in case (5) is applied, all inequalities become equalities, that is which impliesf 1 (x) ≤ V (x). In summary, we obtainf 1 (x) = V (x) = V (x; π * ).

5.
The case of θ 2 ∈ [ 2mµm νm θ 1 , ∞). In this section, let's consider the other case with Similar to analysis in Section 4, we need to consider two suboptimal problems each corresponding to different boundary conditions. In particular, the proofs resemble those of Section 4, so we present the main results without giving the verification processes.