Optimal dividends and capital injection under dividend restrictions

We study a singular stochastic control problem faced by the owner of an insurance company that dynamically pays dividends and raises capital in the presence of the restriction that the surplus process must be above a given dividend payout barrier in order for dividend payments to be allowed. Bankruptcy occurs if the surplus process becomes negative and there are proportional costs for capital injection. We show that one of the following strategies is optimal: (i) Pay dividends and inject capital in order to reflect the surplus process at an upper barrier and at 0, implying bankruptcy never occurs. (ii) Pay dividends in order to reflect the surplus process at an upper barrier and never inject capital --- corresponding to absorption at 0 --- implying bankruptcy occurs the first time the surplus reaches zero. We show that if the costs of capital injection are low, then a sufficiently high dividend payout barrier will change the optimal strategy from type (i) (without bankruptcy) to type (ii) (with bankruptcy). Moreover, if the costs are high, then the optimal strategy is of type (ii) regardless of the dividend payout barrier. The uncontrolled surplus process is a Wiener process with drift.


Introduction
Insurance risk was originally studied in terms of ruin probability. However, this approach may underestimate risk since insurance companies are realistically more interested in maximizing company value than minimizing risk and an alternative approach is therefore to study optimal dividend policies -in the sense of maximizing the expected value of the sum of discounted future dividend payments -as suggested by De Finetti in the 1950s. A vast literature on various versions of the classical optimal dividend problem has since emerged. Some of the more common modeling choices regard for example: • The dynamics of the (uncontrolled) surplus process; e.g. the classical Cramér-Lundberg model, the dual model, or an Itô diffusion model.
• If capital injection is allowed or not.
• If bankruptcy is allowed or not. Bankruptcy not being allowed corresponds to obligatory capital injection to avoid bankruptcy.
• If the insurance company is regulated; e.g. by capital requirements or dividend restrictions.
• Different kinds of market frictions; e.g. fixed or proportional costs for capital injection.
• Additional decision variables for the insurance company; e.g. reinsurance level or investment policy.
Corporations in financial and insurance markets in the real world typically have the possibility of both going bankrupt and raising equity capital from its owners (capital injection).
One of the first papers to take both of these characteristics into account simultaneously is [20] which studies a singular stochastic control problem corresponding to the optimal dividend problem with the possibility of both capital injection and bankruptcy, under the assumption that the uncontrolled surplus process is a Wiener process with drift. The authors find that depending on the parameters of the model it is either optimal to pay dividends in order to reflect the surplus process at an upper barrier and never inject capital, or to pay dividends and inject capital in order to reflect the surplus process at an upper barrier and at 0.
However, corporations in financial and insurance markets are also regulated. In order to take this characteristic into account [21] studies the optimal dividend problem in a model with solvency constraints, meaning that it is not allowed to pay dividends unless the surplus process exceeds a given constant -in the present paper called dividend payout barrier. Capital injection is not considered. The author finds that it is optimal to use a reflection strategy with the barrier being the maximum of the dividend payout barrier and the reflection barrier that would have been optimal without regulation. The uncontrolled surplus process is a general Itô diffusion.
The main contribution and objective of the present paper is to consider all three mentioned characteristics by studying a singular stochastic control problem which allows capital injection as well as bankruptcy under regulation of the dividend payout barrier type. Section 1.1 contains a survey of some of the related literature. In Section 2 we formulate the main problem. In Section 3 we formulate and solve two problems which are intimately connected to the main problem. In Section 4 we use the results of Section 3 to solve the main problem; the main result is Theorem 4.1. Section 4.1 contains graphical illustrations. Section 5 contains conclusions and ideas for future research.

Previous literature
This section contains a brief survey of some of the literature on the optimal dividend problem related to the present paper. Further comparisons of the present paper to some of the references are made in the sections below. More complete surveys are [1,4,28]; see also [2].
The results of [20] (see the previous section) are in [33] extended to a general Itô diffusion model with a growth restriction for the drift function. The results of [21] (see the previous section) are in [15] extended by the introduction of the possibility of reinsurance.
In [16] both fixed and proportional transaction costs for capital injection, as well as reinsurance and bankruptcy are considered; the underlying model is a Wiener process with drift and there is no regulation. Other papers studying different models with capital injection and bankruptcy without regulation are [5,12,33]. In [32] proportional reinsurance and a maximum dividend rate restriction are studied in a particular diffusion model; capital injection and the possibility of bankruptcy are studied separately.
In [18] the classical Cramér-Lundberg model without bankruptcy is studied. A similar model with solvency constraints is studied in [31]. In [6] a spectrally negative Lévy process model without bankruptcy is considered. Other papers studying different models without bankruptcy are [22,23,25,26,30].
In [7] an Itô diffusion model with fixed transaction costs and solvency constraints without capital injection is studied. In [13,14] the dividend problem without capital injection is considered for the Wiener process with drift and a finite time horizon.
In [9] an optimal dividend problem given time-inconsistent preferences is studied using the game-theoretic approach to time-inconsistent stochastic control (general references for this approach include [8,10,11,19]).

Problem formulation and preliminaries
Consider a filtered probability space (Ω, F, P, F ) satisfying the usual conditions and supporting a Wiener process W . The controlled surplus process X of an insurance company is given by where the process D corresponds to accumulated dividends to the owner of the insurance company and the process C corresponds to accumulated capital injection from the owner. The initial surplus satisfies x ≥ 0 and the parameters satisfy µ > 0 and σ > 0. We suppose that for a given financing strategy (C, D) the value of the insurance company is the expected value of the sum of the discounted future cash flow to the owner and that the owner wants to maximize the value of the insurance company. In mathematical terms we thus consider the singular stochastic control problem where we interpret k > 1 as a proportional cost of injecting capital (equity issuance costs), α > 0 as a discount factor, τ as the random bankruptcy time and where A(x, b r ) is the set of admissible strategies: The main objective of the present paper is to study problem (2.1). This problem has according to the authors' knowledge not been studied before.
Obviously the parameters of the model are such that either condition (2.4) below holds, or (2.4) holds with reversed inequality, which is interpreted as proportional costs of capital injection k being low or high, respectively. In Theorem 4.1 we will see that which is the case determines the kind of solution problem (2.1) has. where Note that r 2 < 0 < r 1 and r 2 2 > r 2 1 .
(2.6) Remark 2.2. Formally we write dD t = dD c t + ∆D t where D c denotes the continuous part of D while ∆D t := D t+ − D t , for D t+ := lim hց0 D t+h , denotes jumps in D. The processes C and X are treated analogously. Remark 2.3. Condition (2.3) implies that if the surplus at time t is smaller than the dividend payout barrier b r then dividends are not allowed, i.e. dD t = 0. Thus, if X t+ < b r then a potential jump ∆X t cannot have been caused by D and must have been caused by C and hence X t ≤ X t+ < b r , implying that dD t = 0 also in this case; in mathematical terms this means that (2.3) implies that In other words, the surplus directly after a dividend payment cannot be lower than the dividend payout barrier b r either.
Remark 2.4. An inequality analogous to that of (2.4) is used in [20] when studying problem (2.1) without the presence of a dividend payout barrier, i.e. with b r = 0. Regulation of the type (2.3) was first studied in [21].

Preliminaries
Here we informally recall well-known results that are used throughout the present paper; cf. e.g. [17,24,27]. For any b > 0 and x ∈ [0, b] there exists an F-adapted non-decreasing continuous processesD b withD b 0 = 0 such that the process X defined by, is reflected at b and satisfies, withD b being constant on any interval where X t < b. There also exists a pair of Fadapted non-decreasing continuous processes (C 0 , D b ) with C 0 0 = D b 0 = 0 such that the process X defined by, is reflected at b and 0, and satisfies with D b being constant on any interval where X t < b and C b being constant on any interval where X t > 0. In the case x > b,D b and D b are defined so that the corresponding processes in (2.7) and (2.8) jump from x to b at t = 0.
The pair (C 0 , D b ) is in the present paper said to be a double barrier strategy, while (0,D b ), or simplyD b , is said to be an upper barrier strategy. For an upper barrier strategyD b and τ defined in (2.2), the value function The general solution together with (2.10) and the boundary conditions (2.11), and simple calculations, yield, for any b > 0, is welldefined and is the unique solution to (2.9) together with (2.10) and the boundary conditions, Remark 2.5. The processD b can be pathwise defined asD b t = max 0≤s≤t (x+ µs + σW s − b) + which can be seen using the corresponding Skorohod equation, cf. [17, Section 3.6 C] and [3]. The pair (C 0 , D b ) can be constructed pathwise in a procedure involving iteratively using the solutions to the Skorohod equations for reflection at b and at 0 in turn, as noted in [20, p. 960].

Two restricted problems
In order to solve problem (2.1) it is useful to first study two related problems for which the set of admissible strategies is further restricted.

Capital injection not allowed
Here we consider problem (2.1) under the additional restriction that capital injection is not allowed -this problem has been studied in the literature, see [21], and we here only recall the solution. That is, we consider admissible strategies (C, D) ∈ A(x, b r ) for which For this restricted problem we can write, using also that D is non-decreasing, the optimal value function (2.1) as, The solution to this problem is given by: Proof. The result can be proved using arguments similar to those in the proof of The value function corresponding to the strategy in Theorem 3.1 can, using the results of Section 2.1, be written as, We will usually write G(x) instead of G(x; b r ) for convenience. Lemma 3.2 presents properties of the function G that are used when solving problem (2.1) in Section 4. Proof. We use (2.6) repeatedly. Item (I) is directly verified. Let us prove (II). We find From the definition of b * follows that the derivative of the denominator in (3.5), i.e.
We remark that these facts and the function h will be used below.
Remark 3.3. Problem (3.1) was for a general Itô diffusion solved in [21]. In the case without a dividend payout barrier, i.e. with b r = 0, the problem (3.1) is the well-known absorption problem first studied, for a general Itô diffusion, in [27]. In particular, in [27,Theorem 4.3] it was shown -under appropriate assumptions -that an upper barrier strategyD b * is optimal, where the barrier b * is determined by the additional boundary condition (If no such b * exists, then no optimal strategy exists and the optimal value function is In particular, b * in Theorem 3.1 is found using condition (3.6) for the value function (2.13).

Bankruptcy not allowed
Here we consider the problem (2.1) under the restriction that bankruptcy is not allowed. That is, we consider admissible strategies (C, D) ∈ A(x, b r ) for which, We denote the set of such strategies by A R (x, b r ). For this restricted problem holds τ = ∞ a.s. and the optimal value function (2.1) can be written as, Problem (3.7) has according to the authors' knowledge not been considered before. The solution is given by: where b * * > 0 is the unique positive solution to the equation, Proof. The result can proved using arguments analogous to those in the proof of Theorem 4.1. The uniqueness of b * * is verified by noting that r 1 e −r 2 b −r 2 e −r 1 b is strictly increasing in b; to see this use differentiation and (2.6). It is easy to see that b * * must be positive. A proof in the case b r = 0 is found in [20,Sec. 4,5].
The value function corresponding to the strategy in Theorem 3.4 can, using the results of Section 2.1, be written as, (3.9) We will usually write H(x) instead of H(x; b r ). Lemma 3.5 presents properties of the function H that are used when solving the main problem (2.1) in Section 4. The proof of Lemma 3.5 relies on the same type of arguments as the proof of Lemma 3.2 and is found in the appendix.
Moreover, (2.4) with reversed inequality is equivalent to b * * ≥ b * , which is equivalent to (3.11) with reversed inequality.
Remark 3.6. In the case b r = 0 the problem (3.7) is the well-known reflection problem first studied, for a general Itô diffusion, in [27]; see also [27,Sec. 5] where the problem is studied for a Wiener process with drift. In particular, in [27,Theorem 4.5] it was for the case b r = 0 shown -under appropriate assumptions -that the double barrier strategy (C 0 , D b * * ) is optimal, where the barrier b * * is given by the additional boundary condition (If no such b * * exists, then no optimal strategy exists and the optimal value function is In particular, b * * in Theorem 3.4 is found using condition (3.13) for the value function (2.15).
Remark 3.7. The equivalences in (III) in Lemma 3.5 were in the context of studying problem (2.1) without a dividend payout barrier, i.e. with b r = 0, derived in [20].

Solution to the main problem
Since our model is Markovian it is reasonable to conjecture that the optimal strategy for problem (2.1) involves either that the owner always saves the insurance company from bankruptcy by injecting capital when the surplus process hits zero, or that the owner never does so. Indeed this is what we find in Theorem 4.1. The results in this section are illustrated in graphs in Section 4.1 and interpreted in Section 5.  Remark 4.5. The interpretation of Corollary 4.4 is that in the case capital injection costs are low (in the sense that (2.4) holds holds with strict inequality) holds that: if the dividend payout barrier satisfies b r <b then it is not optimal to allow bankruptcy and if the dividend payout barrier satisfies b r >b then it is not optimal to save the insurance company from bankruptcy.
The fundamental theorem of calculus gives Now suppose g is either the value function of the upper barrier strategyD b with b = b r ∨ b * , given by G in (3.3), or the value function of the double barrier strategy (C 0 , D b ) with b = b r ∨ b * * , given by H in (3.9) -the differentiability condition used above is directly verified in both cases. Then g satisfies (2.9) and (2.10) implying g ′ (x) = 1 for x ≥ b and g ′′ (x) = 0 for x > b.
• Use Lemma 3.5 and b r ≤b to find H(0) ≥ 0 and H ′ (x) > 0 for x ≥ 0. Hence, H(x) ≥ 0 for x ≥ 0. We conclude that the first term in (4.2) is non-negative.
• If we send t to infinity then the second term in (4.2) converges a.s. to a random variable with zero expectation (use that H ′ (x) is a bounded function).
Thus, sending t to infinity (lim sup) in (4.1)-(4.4) and taking expectation gives The terms in (4.2) are dealt with in the same way as above. Using the same limiting arguments as above we find (I.a) holds also in the case b = b r .
Using arguments analogous to those above we find that (II) holds in the case b = b * . Now suppose b r > b * . Observe: • G ′ (x) = 1 for x ≥ b r and condition (2.3) imply that the expressions in (4.3) vanish (as above).
The usual arguments now imply that (II) holds also in the case b = b r .
Case C: We have left to prove (I.b). If G ′ (0) ≤ k holds also in this case then the result follows by the exact same arguments as in Case B. Thus, it is enough to show that G ′ (0) ≤ k for b ≥b when (2.4) holds. In (3.5) we defined the function h by, Hence, G ′ (0) = k when b =b, by definition ofb in (3.12). Thus, in order to prove that G ′ (0) ≤ k for any b ≥b it is enough to show that h(b) is non-increasing in b for b ≥b. But h(b) is non-increasing exactly when b ≥ b * , as we saw in the proof of Lemma 3.2. Hence, it is enough to prove thatb ≥ b * . The right side of (2.4) is equal to h(b * ), cf. (3.4). Use this, the definition ofb in (3.12), and Lemma 3.
But since h is maximal at b * , see the proof of Lemma 3.2, follows
Recall that H(x; b r ) is the optimal value function without the possibility of bankruptcy, G(x; b r ) is the optimal value function without the possibility of capital injection and that the optimal value function when both bankruptcy and capital injection is allowed, i.e. V (x; b r ), is for any fixed b r given by either H(x; b r ) or G(x; b r ) according to which is dominating the other, see Corollary 4.3.

Conclusions and future research
The main interpretation of the results in the present paper, in particular of items (I.a) and (I.b) in Theorem 4.1, see also Figure 1 and Figure 2, is that if the proportional cost of injecting capital k is low, i.e. if (2.4) holds, then it is optimal to use a double barrier financing strategy and never allow the insurance company to go bankrupt as long as the dividend payout barrier b r is lower than the levelb, i.e. the optimal value function is given by V (x; b r ) = H(x; b r ). However, if the dividend payout barrier b r is set higher thanb then the optimal behavior switches to an upper barrier strategy that lets the insurance company go bankrupt the first time the surplus reaches zero, i.e. the optimal value function is given by V (x; b r ) = G(x; b r ). Moreover, the interpretation of Corollary 4.6, see also Figure 2, is that an increase in the dividend payout barrier decreases the optimal value function (i.e. the value of the insurance company), with the corresponding limit being zero.
The main economic conclusion of the present paper is that regulation may have the, perhaps unforeseen, effect that if a profitable insurance company (corresponding to µ > 0) has access to a well-functioning financial market (corresponding to the proportional costs for capital injection k satisfying (2.4)) then its owners will inject capital when needed in case the market is unregulated or at least not too heavily regulated (b r ≤b). However, if the regulation is sufficiently heavy (b r >b) then the owners of the same insurance company will change their behavior; specifically, they will never inject capital and instead let the insurance company go bankrupt in the case of financial distress, i.e. in the case of zero surplus.
A potential topic for future research is the investigation of these conclusions for other models considering e.g.: other surplus processes; other types of regulation; additional market frictions, for example fixed costs for capital injection; the presence of additional decision variables, for example reinsurance level and investment policy. Now, the definition of b * * is that the last inequality is an equality when b = b * * . Hence, if we can show that r 1 e −r 2 b − r 2 e −r 1 b is (strictly) increasing in b for b > b * * , then (6.4) is satisfied for b > b * * ; but this is easily verified using the derivative and (2.6). We have thus proved (6.2) also for H and b * * .
Thus, in the case b = b * * (i.e. b r ≤ b * * ) follows, from (II), that H(0) ≤ 0. Now, if we can prove that r 1 e r 1 b − r 2 e r 2 b is non-decreasing in b, for b ≥ b * * , then follows, from (II) that H(0) ≤ 0 also in the case b r > b * * and we are done. Hence, it is enough to show that its derivative, r 2 1 e r 1 b − r 2 2 e r 2 b , is non-negative for b ≥ b * * . But r 2 1 e r 1 b − r 2 2 e r 2 b ≥ 0 is equivalent to b ≥ b * (as we have seen) and since b * * ≥ b * (by (III)) follows therefore that r 1 e r 1 b − r 2 e r 2 b is non-decreasing in b, for b ≥ b * * .
Proof of (VI). (III) gives From the proof of Lemma 3.2 we know r 1 e r 1 b −r 2 e r 2 b is (strictly) increasing in b for b > b * and (strictly) decreasing in b for b < b * ; moreover, the left side of (6.6) clearly converges to ∞ as b → ∞. Hence, there exists a unique constantb ∈ [b * * , ∞) such that r 1 e r 1 b − r 2 e r 2 b ≤ r 1 − r 2 k for b ≤b, and r 1 e r 1 b − r 2 e r 2 b ≥ r 1 − r 2 k for b ≥b. From (3.3) we directly see that G(x; b r ) does not depend on b r for b r < b * . For b r > b * and 0 < x < b r it is easy to show that G(x; b r ) is strictly decreasing in b r (use differentiation and (2.6)). This also holds for b r > b * and x > b r . Hence, (i) follows from the continuity of G(x; b r ). Item (ii) is proved analogously.