On optimal dividends in the dual model

We revisit the dividend payment problem in the dual model of Avanzi et al. ([2], [1], and [3]). Using the fluctuation theory of spectrally positive L\'{e}vy processes, we give a short exposition in which we show the optimality of barrier strategies for all such L\'{e}vy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [3] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.


INTRODUCTION
In the so-called dual model, the surplus of a company is modeled by a Lévy process with positive jumps (spectrally positive Lévy processes); see [3], [7], [2], and [4]. This is an appropriate model for a company driven by inventions or discoveries. Our goal is to determine the optimal dividend strategy until the time of ruin for all spectrally positive Lévy processes.
In [2], Avanzi and Gerber consider the dividend payment problem when the Lévy process is assumed to be the sum of an independent Brownian motion and a compound Poisson process with i.i.d. positive hyper-exponential jumps; they determine the optimal strategy among the set of barrier strategies. (The special case in which the jumps are exponentially distributed was obtained by [7].) The optimality over all admissible strategies is later shown by [4] using the verification approach in [7].
In this paper, using the fluctuation theory, we give a short proof of the optimality of barrier strategies for all spectrally positive Lévy processes of bounded or unbounded variation. Moreover, the optimal barrier is characterized using a functional inverse of the scale functions. We also consider the cash injection problem considered in [4]: a variant of the dividend payment problem in which the shareholders are expected to give capital injection in order to avoid ruin. We observe that the form of the value function for this problem is very similar to the first problem we consider in which the horizon is the time of ruin.
Let P x be the conditional probability under which X 0 = x (also let P ≡ P 0 ), and let F := {F t : t ≥ 0} be the filtration generated by X. Using this, the drift of X is given by We also assume that µ < ∞ (and hence |ψ (0+)| < ∞) to ensure that the problem is nontrivial.
1.1. The dividend payment problem until the time of ruin. We first consider a control problem in which the goal is to maximize the expected net present value (NPV) of dividends until ruin. A (dividend) strategy π := {L π t ; t ≥ 0} is given by a nondecreasing, right-continuous and F-adapted process starting at zero. Corresponding to every strategy π, we associate a controlled surplus process U π = {U π t : t ≥ 0}, which is defined by where U π 0− = x is the initial surplus and L π 0− = 0. The time of ruin is defined to be A lump-sum payment must be smaller than the available funds and hence it is required that Let Π be the set of all admissible strategies satisfying (1.3). The problem is to compute, for q > 0, the expected NPV of dividends until ruin v π (x) := E x σ π 0 e −qt dL π t , π ∈ Π, and to obtain an admissible strategy that maximizes it, if such a strategy exists. Hence the problem is written as

1.2.
Dividend payment problem with capital injections. In this variant of the dividend payment problem, the time horizon is infinity, and the shareholders are required to inject just enough cash to keep the company alive. A strategy is now a pairπ := {Lπ t , Rπ t ; t ≥ 0} where Lπ is the cumulative amount of dividends as in the classical dividend problem and Rπ is again a nondecreasing, right-continuous and F-adapted process starting at zero, representing the cumulative amount of injected capital satisfying ∞ 0 e −qt dRπ t < ∞, a.s. (1.5) Assuming that ϕ > 1 is the cost per unit injected capital, we want to maximizē Hence the problem isv whereΠ is the set of all admissible strategies that satisfy (1.3) and (1.5).
1.3. Outline. In this note, we give a short proof showing that for a general spectrally positive Lévy process, barrier strategies are optimal for both problems, and we give a simple characterization of the optimal barriers in terms of the scale functions; see (2.13) and (3.4). It is interesting to note that the forms of the value functions (3.1) and (3.5) are the same, while the characterizations of barrier levels are in terms of different scale functions. Also, while, in the spectrally negative model, optimal strategies may not lie in the set of barrier strategies, our results show that the dual model can be solved in general by a barrier strategy regardless of the Lévy measure. Regarding the spectrally negative Lévy model, we refer the reader to [6] for examples where barrier strategies are suboptimal and to [15] for a sufficient condition for optimality.
The structure of the rest of the paper is as follows. In Section 2, we solve the optimal dividend problem in which the time horizon is the time of ruin. In this section, we first collect a few results about the scale functions for spectrally one-sided Lévy processes. We then construct a candidate optimal solution out of barrier strategies by C 1 (resp. C 2 ) conditions at the barrier when X is of bounded (resp. unbounded) variation, and verify its optimality. In Section 3, we solve the dividend payment problem with capital injections, where we follow the same plan to the one described for Section 2. We conclude the paper with numerical examples in Section 4.

SOLUTION OF THE DIVIDEND PROBLEM UNTIL THE TIME OF RUIN
For the dividend problem we described in Section 1.1, a barrier strategy at level a ≥ 0 is denoted by and σ a : Our objective is to show that the optimal control lies in the class of barrier strategies and to identify a * such that v = v a * .
2.1. Scale functions. Fix q > 0. For any spectrally positive Lévy process, there exists a function called the q-scale function which is zero on (−∞, 0), continuous and strictly increasing on [0, ∞), and is characterized by the Laplace transform: Here, the Laplace exponent ψ in (1.1) is known to be zero at the origin, convex on R + ; therefore Φ(q) is well-defined and is strictly positive as q > 0. We also define and its anti-derivative Notice that because W (q) is uniformly zero on the negative half line, we have (1) If X is of unbounded variation, it is known that W (q) is C 1 (0, ∞); see, e.g., Chan et al. [9]. Hence, Z (q) is C 2 (0, ∞) and C 1 (R) for the bounded variation case, while it is C 3 (0, ∞) and C 2 (R) for the unbounded variation case. (2) Regarding the asymptotic behavior near zero, we have that Constructing a candidate value function. The following is a direct application of the results given in Theorem 1 of [5] (see, in particular, page 167 of this reference).
On the other hand, by for any negative y. Therefore, regardless of whether a is larger than x or not, we can write The function |k(x)|, x ≥ 0, is uniformly bounded by |k(0)| < ∞, which follows from the stochastic representation of this function in [5]. As a result, using the duality and Wiener-Hopf factorization of spectrally positive Lévy processes (see, e.g., pages 73-74 and 212-213 of [13]), where S t := sup 0≤s≤t (X s ∨ 0) and η(q) is an exponential random variable with parameter q > 0 that is independent of X. This asymptotic behavior is consistent with that of the expected NPV of dividendsṽ a , when X is a spectrally negative process, of a given barrier strategy starting at the barrier: which is equation (3.15) in [5].
We note that v a , for any a ≥ 0, is clearly continuous everywhere on [0, ∞) with v a (0) = 0. Here, we shall examine the smoothness of v a at x = a to obtain a candidate barrier level a * . In particular, we will choose a * so that v a * is C 1 for the case X is of bounded variation and C 2 for the case X is of unbounded variation.
Fix x = a. By differentiating (2.8), we obtain that and when X is of unbounded variation (see Remark 2.1 (1)) , a > 0.
On the other hand, since Z (q) (x) is strictly increasing, goes to ∞ as x ↑ ∞ and to −∞ as x ↓ 0, there exits a unique solution to (2.12). Because Z (q) (0) = 0, the solution is strictly positive if and only if µ > 0. We will denote our candidate barrier level by The following proposition states that with this choice of barrier level, the corresponding expected NPV function (2.2) is smooth enough to apply the verification arguments addressed below. In view of Remark 2.1 (1), the smoothness at barrier level a is the only point of concern.
(i) If X is of bounded variation, v a is continuously differentiable on (0, ∞) if and only if a = a * .
(ii) If X is of unbounded variation, v a is continuously differentiable on (0, ∞) for all a > 0. However, v a is twice continuously differentiable on (0, ∞) if and only if a = a * .
Regarding the twice differentiability, because W (q) and W (q) are continuous on R and R\{0}, respectively, it is clear in view of (2.10) that the twice differentiability holds anywhere on (0, ∞)\{a}. On the other hand, from which it follows that v a * (a * −) = 0 since W (q) (0) = 0. For any other choice of a, v a (a−) = 0, which follows from (2.6) and (2.10).
We shall show below that a * , as determined in (2.13), is indeed the optimal barrier level and (2.2) with a = a * , which can be written as for any x ≥ 0, is the value function of the dividend payment problem.
2.3. Verification. By Remark 2.1 (1) and Lemma 2.1, v a * defined in (2.15) is C 2 (0, ∞) (resp. C 1 (0, ∞)) when X is of unbounded (resp. bounded) variation. Moreover, it is clear that v a * (0) = 0 in both cases. Therefore, we can use Proposition 4 of [5], which is a generic verification theorem for the dividend payment problems of any Lévy process. (Also see Lemma 3.1 of [7].) From this theorem it follows that to prove the optimality of v a * it is sufficient to demonstrate the following variational inequality: Here L is the infinitesimal generator associated with the process X applied to a sufficiently smooth function f We show that v a * indeed satisfies (2.16) and its optimality over all admissible strategies Π.
Theorem 2.1. We have v = v a * as defined in (2.15) and π a * is the optimal strategy over Π.
Step 4. Suppose a * > 0. Thanks to the smoothness of v a * at x = a * , which we proved in Proposition

SOLUTION OF THE DIVIDEND PROBLEM WITH CAPITAL INJECTION
For the capital injection problem as defined in Section 1.2, we consider the doubly reflected Lévy process with upper barrier b > 0 and lower barrier 0 of the form As shown by [16], this is a Markov process taking values only on [0, b]. By modifying Theorem 1 of [5], for any b > 0 and 0 ≤ x ≤ b, we obtain that Hence the expected payoff corresponding to the strategyπ b : Similarly to our observations in Remark 2.2, using (2.4), (3.1) holds even when x > b. Finally, we extend it to the negative line so thatv This result complements the observation in Remark 2.3; as b increases to ∞ the impact of ruin vanishes.

Ansatz and verification.
Analogously to the previous section, we choose our candidate barrier level using the C 1 (C 2 ) condition at the barrier. For x = b, by taking derivatives Hence it is clear that the C 1 (resp. C 2 ) condition at x = b for the bounded (resp. unbounded) variation case holds if and only if Z (q) (b) = ϕ. Since Z (q) is strictly increasing on (0, ∞), Z (q) (0) = 1 and lim x→∞ Z (q) (x) = ∞ (see e.g. Lemma 3.3 in [12]), there exists a unique The candidate value function simplifies to Remark 3.2. As ϕ ↓ 1, b * ↓ 0. This is consistent with the observation given in page 158 of [5]. On the other hand, b * ↑ ∞ as ϕ ↑ ∞; as ϕ increases, it gets more risky to pay dividends.
Thanks to Remark 2.1 (1) and the way b * is chosen to ensure the smoothness at b * , we can apply Proposition 4 (2) of [5], which tells us that it is sufficient to show thatv b * satisfies the following variational inequality: The steps of proving the verification are similar to the ones in Theorem 2.1. Therefore we will only verify (3.7) and (3.8). For 0 < x < b * , by (3.3), the monotonicity of Z (q) and (3.5) (3.8) is satisfied by (3.2). In summary, we have the following.

NUMERICAL EXAMPLES
We have shown that the dividend payment and cash injection problems both admit solutions written in terms of the scale function. In order to put this in practice, the only task left to do is to compute the scale function. There are several examples of Lévy processes whose scale functions are known explicitly; see [13], [14], [11] and [12]. In general, the scale function can be computed efficiently by inverting the Laplace transform (2.3) (see [17] and [12]), or alternatively it can be approximated by those of phasetype Lévy processes (see [1] and [10]). Here, we shall use the latter and confirm via numerical examples the results obtained in the previous sections.
Consider a spectrally positive Lévy process of the form is a Poisson process with arrival rate λ, and Z = {Z n ; n = 1, 2, . . .} is an i.i.d. sequence of phasetype-distributed random variables with representation (m, α, T ); see [1]. These processes are assumed mutually independent. Its Laplace exponent (1.1) is then which is analytic for every s ∈ C except at the eigenvalues of T . Suppose {−ξ i,q ; i ∈ I q } is the set of the (complex-valued) roots of the equality ψ(s) = q with negative real parts, and if these are assumed distinct, then the scale function can be written for the case σ > 0 and σ = 0, respectively for some {C i,q ; i ∈ I q }; see [10]. For the phase-type distribution, we use m = 6, This approximates (the absolute values of) the Gaussian distribution with mean zero and standard deviation 1, obtained using the EM-algorithm; see [10] for the approximation performance of the corresponding scale function. We also let q = 0.05 and λ = 3.5.
We shall first confirm the results obtained in Theorem 2.1. We consider both the bounded and unbounded variation cases with σ = 0 and σ = 1, respectively. In Figure 1, we show the value function v a * as well as the point (a * , v a * (a * )) for d = 2.0, 2.33, 2.67, 3.0 or equivalently µ = 0.80, 0.47, 0.13, −0.20. The value function as well as the value of a * decrease as d increases (or µ decreases); in particular a * = 0 for the case d = 3.0 (or µ = −0.20 ≤ 0). It is also observed that the value function is smooth at a * for both bounded and unbounded variation cases. Next we give results on the capital injection problem and confirm the results in Theorem 3.1. In Figure 2, we plot the value function as well as the point (b * ,v b * (b * )) = (b * , µ/q) for σ = 0, 1 and ϕ = 1.001, 1.5, 2, 5. Here we use the common value of d = 2.33 and hencev b * (b * ) is the same for each. The value function is indeed decreasing in the unit cost ϕ and the value of b * decreases to zero as ϕ decreases to 1. As in the case of dividend payment problem, we can again confirm the smoothness of the value function for all cases.