Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process

We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy.


Introduction
In the literatures of actuarial science and finance, the optimal dividend problem is one of the key topics. For companies paying dividends to shareholders, a commonly encountered problem is to find a dividend strategy that maximizes the expected total discounted dividends until ruin. The pioneer work can be traced to de Finetti [1] who considered a discretetime risk model with step sizes ±1 and showed that a certain barrier strategy maximizes the expected discounted dividend payments. Since then, the problem of finding the optimal dividend strategy has become a popular topic in the actuarial literature. For diffusion models, see, for example, Jeanblanc-Picqué and Shiryaev [2], Asmussen and Taksar [3], Gerber and Shiu [4], Løkka and Zervos [5], Paulsen [6], He and Liang [7], and Bai and Paulsen [8]. For the Cramér-Lundberg risk model, some related works on this subject include, among others, Gerber [9], Azcue and Muler [10,11], Yuen et al. [12], Kulenko and Schmidli [13], Bai and Guo [14], and Hunting and Paulsen [15].
Analysis of optimal dividends for Lévy risk processes is of particular interest which have undergone an intensive development. For example, Avram et al. [16] considered a general spectrally negative Lévy process and gave a sufficient condition involving the generator of the Lévy process for the optimality of barrier strategy; Loeffen [17] showed that barrier strategy is optimal among all admissible strategies for general spectrally negative Lévy risk process with completely monotone jump density; Kyprianou et al. [18] relaxed this condition on the jump density; Yin and Wang [19] also studied the same problem and gave an alternate proof of the result; Loeffen [20,21] considered the optimal dividend problem with transaction costs and a terminal value for the spectrally negative Lévy process. Recently, Bayraktar et al. [22] using the fluctuation theory of spectrally positive Lévy processes show the optimality of barrier strategies for all such Lévy processes. See Yin and Wen [23] for a different approach. All of the above mentioned works are based on spectrally onesided models. There are, however, few papers that studied the analogous problems for Lévy process with two-sided jumps (cf. Bo et al. [24,25]). Inspired by the works of Avram et al. [16], Loeffen [17], and Kyprianou et al. [18], Yuen and Yin [26] considered the optimal dividend problem for a special Lévy process with both upward and downward jumps and showed that the optimal strategy takes the form of a barrier strategy if the Lévy measure (both negative and positive jumps) has a completely monotone density. The purpose of the present paper is to extend the result of Yuen and Yin [26] to the case with less restrictive conditions on the Lévy measure. Although the broader case definitely makes the optimization 2 The Scientific World Journal problem more challenging and complex, recent results on the theory of potential analysis of subordinators can be applied to handle it. In particular, our main results show that the optimal dividend strategy is still of a barrier type if the Lévy process has certain positive jumps and Lévy density of negative jumps is completely monotone or log-convex.
The paper is organized as follows. In Section 2, we introduce the mathematical formulation of the problem. In Section 3, we give a brief review on ladder processes and potential measure for general Lévy processes. The convexities of the ruin probability and the scale function are discussed in Sections 4 and 5 and the main results and their proofs are given in Section 6.

The Model
To present the mathematical formulation of the problem of study, let us first introduce some notations and definitions. Let = { } ≥0 be a real-valued Lévy process on a filtered probability space (Ω, F, F, P) where F = (F ) ≥0 is generated by the process and satisfies the usual conditions of right continuity and completeness. Denote by the law of when 0 = . Let be the expectation associated with . For notational convenience, we write and when 0 = 0. Write the Lévy triplet of as ( , 2 , Π), where , ≥ 0 are real constants and Π is a positive measure on (−∞, ∞) \ {0} which satisfies the integrability condition If Π( ) = ( ) , then we call the Lévy density. The characteristic exponent of is given by where 1 is the indicator of set . Furthermore, define the Laplace exponent of by Such a Lévy process is of bounded variation if and only if = 0 and ∫ 1 −1 | |Π( ) < ∞. If Π{(0, ∞)} = 0, then the Lévy process with no positive jumps is called the spectrally negative Lévy process; if Π{(−∞, 0)} = 0, then the Lévy process with no negative jumps is called the spectrally positive Lévy process. It is usual to assume that (lim → ∞ = +∞) = 1 which says nothing other than Ψ (0+) > 0. For more information on Lévy processes we refer to the excellent book by Kyprianou [27]. Now, we consider an insurance company or investment company whose cash reserve process (also called risk process or surplus process) evolves according to the process before dividends are deducted. Let = { : ≥ 0} be a dividend policy consisting of a right-continuous nonnegative nondecreasing process adapted to the filtration {F } ≥0 of with 0− = 0, where represents the cumulative dividends paid up to time . Given a control policy , the controlled reserve process with initial capital ≥ 0 is given by with 0 = . Let = { > 0 : < 0} be the ruin time when dividend payments are taken into account. Define the value function associated to dividend policy by where > 0 is the discounted rate. The integral is understood pathwise in a Lebesgue-Stieltjes sense. Clearly, ( ) = 0 for < 0. A dividend policy is called admissible if − − ≤ for < and − − = 0 for < ∞. Denote by Ξ the set of all admissible dividend policies. Our objective is to find * ( ) = sup and an optimal policy * ∈ Ξ such that * ( ) = * ( ) for all ≥ 0. The function * is called the optimal value function. We denote by = { : ≥ 0} the barrier strategy at and let be the corresponding risk process; that is, = − . Note that ∈ Ξ. Also, if 0 ∈ [0, ], then the process can be explicitly represented by If 0 = > , then Denote by ( ) the dividend value function if barrier strategy is applied; that is, The Scientific World Journal 3 Applying Ito's formula for semimartingale, we can prove that is the solution to where Γ is the infinitesimal generator of with In the sequel, we assume that, for any > 0, the equation Ψ( ) = has a unique solution on (0, ∞), say ( ). A typical example is that the Lévy measure of the positive jumps has the following gamma distribution Γ( , 1/ ); that is, where is a positive number and is an even number. Following similar reasoning to Yuen and Yin [26], can be expressed as where Here, let̃( ) be the ruin probability for a Lévy processw ith Laplace exponent ( ) given by ( ) ( ) = Ψ( + ( )) − . Note that the process̃has the Lévy triplet (̃,̃2,Π), wherẽ2 = 2 ,Π( ) = ( ) Π( ), and

Some Results on Ladder Processes and Potential Measure
In this section, we recap some basic facts about ladder processes and potential measure. Consider the dual process be the processes of the first infimum and the last supremum of the Lévy process , respectively. Following Klüppelberg et al. [28], we now introduce the notion of ladder processes and potential measure. Let > }, where we take the infimum of the empty set as ∞. Define an increasing process by { = −1 : ≥ 0}, that is, the process of new maxima indexed by local time at the maximum. The processes −1 and are both defective subordinators, and we call them the ascending ladder time and ladder height process of , respectively. It is understood that = ∞ when −1 = ∞. Throughout the paper, we denote the nondefective versions of , −1 , and by L, L −1 , and H, respectively. In fact, the pair (L −1 , H) is a bivariate subordinator. Define (̂− 1 ,̂) the descending ladder time and the ladder height processes in an analogous way. Note that̂is a process which is negatively valued. Because drifts to −∞, the decreasing ladder height process is not defective. Associated with the ascending and descending ladder processes are the bivariate renewal functions and̂. The former is defined by Taking Laplace transforms shows that where ( , ) is its joint Laplace exponent such that ≥ 0 is the killing rate of so that > 0 if and only if lim → ∞ = −∞, ≥ 0 is the drift of , and Π is its jump measure. Denote the marginal measure of (⋅, ⋅) by The function is called the potential/renewal measure. As for the descending ladder process,̂and̂are defined similarly.

4
The Scientific World Journal Write Π + and Π − for the restrictions of Π( ) and Π(− ) to (0, ∞). Furthermore, for > 0, define We next introduce the notions of a special Bernstein function and complete Bernstein function and two useful results. Recall that a function : (0, ∞) → (0, ∞) is called a Bernstein function if it admits a representation where ≥ 0 is the killing term, ≥ 0 is the drift, and is the Lévy measure concentrated on (0, ∞) satisfying ∫ be the corresponding representation. It was shown in Song and Vondraček [29] that A possibly killed subordinator is called a special subordinator if its Laplace exponent is a special Bernstein function. Song and Vondraček [30] showed that a sufficient condition for to be a special subordinator is that ( , ∞) is log-convex on (0, ∞). A function : (0, ∞) → R is called a complete Bernstein function if there exists a Bernstein function such that where L stands for the Laplace transform. It is known that every complete Bernstein function is a Bernstein function and that the following three conditions are equivalent: (i) is a complete Bernstein function; (ii) ( ) = / ( ) is a complete Bernstein function; (iii) is a Bernstein function whose Lévy measure is given by where ] is a measure on (0, ∞) satisfying To end the section, we present two results which are useful in potential theory and will be used in later sections of the paper. The first due to Kyprianou et al. [18] (see also Song and Vondraček [30]) is summarized in Lemma 1 while the second due to Kingman [31] and Hawkes [32] is given in Lemma 2.

Lemma 1.
Let be a subordinator whose Lévy density, say ( ), > 0, is log-convex. Then, the restriction of its potential measure to (0, ∞) has a nonincreasing and convex density. Furthermore, if the drift of is strictly positive, then the density is in 1 (0, ∞).

Lemma 2. Suppose that
is a subordinator with Laplace exponent and potential measure . Then, has a density which is completely monotone on (0, ∞) if and only if the tail of the Lévy measure is completely monotone.
Remark 3. Note that the tail of the Lévy measure is a completely monotone function if and only if has a completely monotone density. Thus, we have the following two equivalent statements: is a complete Bernstein function if and only if has a density which is completely monotone on (0, ∞); or, equivalently, has a density which is completely monotone on (0, ∞) if and only if has a completely monotone density.
Proof. We first prove (i). Since − is completely monotone on (0, ∞), it follows from Lemma 4 that the tail Π H ( , ∞) of Lévy measure Π H is a complete monotone function. Also, it follows from Lemma 2 that the potential measure has a density which is completely monotone on (0, ∞). Thus, the probability of ruin is completely monotone on (0, ∞) as ( ) = ( , ∞). We now prove (ii). The log-convexity of − implies the log-convexity of Π + , and hence Π H is log-convex on (0, ∞) due to Lemma 4 as log-convexity is preserved under mixing. It follows from Lemma 1 that the potential measure has a nonincreasing and convex density . Thus, = − is nondecreasing and concave on (0, ∞), and hence (a) and (b) are proved. Since a convex function on (0, ∞) is differentiable except at finitely or countably many points, we see that is twice continuously differentiable except at finitely or countably many points on (0, ∞) if has no Gaussian component. On the other hand, if has a Gaussian component or, equivalently, the drift of ascending ladder processes strictly positive, then it follows from Lemma 1 that ∈ 1 (0, ∞), and hence ∈ 2 (0, ∞). Therefore, (c) is proved.
LetH (Ĥ) be the ascending (descending) ladder height process of̃= −̃. By Lemma 4, we have wherêis the renewal measure corresponding tô. Then, The assumption of log-convexity of − implies that ] + is logconvex, and hence ΠH( ) is also log-convex. It follows from Lemma 1 of Kyprianou and Rivero [35] that the restriction of its potential measure to (0, ∞) of a subordinator with Lévy density ] + has a nonincreasing and convex density, say . Also, the restriction of its potential measure to (0, ∞) of a subordinator with Lévy density ΠH( ) has a nonincreasing and convex density, say ℎ . Moreover, ℎ ( ) = ( ) ( ).
This implies that ℎ ( ) tends to ∞ as tends to ∞ as lim → ∞ ℎ( ) = ∞. Thus * < ∞. Applying the same arguments as those in Kyprianou et al. [18], we can prove that ℎ and its derivative ℎ are strictly convex on ( * , ∞). Finally, the smoothness of ℎ is a direct consequence of Theorem 5. So, (ii) is proved.

Main Results and Proofs
We now present the main results of the paper about the optimality of the barrier strategy * for de Finetti's dividend problem for general Lévy processes. This is a continuation of the work of Yuen and Yin [26] in which a special Lévy process with both upward and downward jumps and a completely monotone density was considered.

Theorem 7.
Suppose that ] is a nonnegative function on (0, ∞) which is sufficiently smooth and satisfies the following:  Theorem 9. Suppose that − is completely monotone. Then, * ( ) = * ( ); that is, the barrier strategy at * is the optimal strategy among all admissible strategies.
Before proving the main results, we give two lemmas which are similar to those in Loeffen [17] for spectrally negative Lévy processes.

Proof . As lim
These give (ii).
We now present the proofs of Theorems 7-10.
where = { } ≥0 is a standard Brownian motion and is a martingale with 0 = 0.
Note that ] is smooth enough for an application of the appropriate version of Itô's formula and the change of variables formula. In fact, if is of bounded variation, then ] ∈ 1 (0, ∞) and we are allowed to use the change of variables formula [36,Theorem 31]; if has a Gaussian exponent, then ] ∈ 2 (0, ∞) and we are allowed to use Itô's formula [36,Theorem 32]; and if has unbounded variation and = 0, then ] is twice continuously differentiable almost everywhere but is not in 2 (0, ∞) and we can use Meyer-Itô's formula [36,Theorem 70] and product rule formula. In any cases, for any appropriate localization sequence of stopping times { , ≥ 1}, we get under (46)