#### Abstract

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.

#### 1. 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 discrete-time risk model with step sizes 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 one-sided 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 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.

#### 2. The Model

To present the mathematical formulation of the problem of study, let us first introduce some notations and definitions. Let be a real-valued Lévy process on a filtered probability space where is generated by the process and satisfies the usual conditions of right continuity and completeness. Denote by the law of when . Let be the expectation associated with . For notational convenience, we write and when . Write the Lévy triplet of as , where are real constants and is a positive measure on which satisfies the integrability condition If , then we call the Lévy density. The characteristic exponent of is given by where is the indicator of set . Furthermore, define the Laplace exponent of bySuch a Lévy process is of bounded variation if and only if and . If , then the Lévy process with no positive jumps is called the spectrally negative Lévy process; if , then the Lévy process with no negative jumps is called the spectrally positive Lévy process. It is usual to assume that which says nothing other than . 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 be a dividend policy consisting of a right-continuous nonnegative nondecreasing process adapted to the filtration of with , where represents the cumulative dividends paid up to time . Given a control policy , the controlled reserve process with initial capital is given by wherewith . Let be the ruin time when dividend payments are taken into account. Define the value function associated to dividend policy by where is the discounted rate. The integral is understood pathwise in a Lebesgue-Stieltjes sense. Clearly, for . A dividend policy is called admissible if for and for . Denote by the set of all admissible dividend policies. Our objective is to find and an optimal policy such that for all . The function is called the optimal value function.

We denote by the barrier strategy at and let be the corresponding risk process; that is, . Note that . Also, if , then the process can be explicitly represented by If , then Denote by the dividend value function if barrier strategy is applied; that is,Applying Ito’s formula for semimartingale, we can prove that is the solution to where is the infinitesimal generator of withIn the sequel, we assume that, for any , the equation has a unique solution on , say . A typical example is that the Lévy measure of the positive jumps has the following gamma distribution ; that is, where is a positive number and is an even number.

Following similar reasoning to Yuen and Yin [26], can be expressed aswhereHere, let be the ruin probability for a Lévy process with Laplace exponent given by . Note that the process has the Lévy triplet , where , , and Moreover,

#### 3. 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 , with , where , . It is easy to see that the Lévy triplet of is , where . Let 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 denote the local time in the time period that spends at zero. Then is the inverse local time such that , where we take the infimum of the empty set as . Define an increasing process by , that is, the process of new maxima indexed by local time at the maximum. The processes and are both defective subordinators, and we call them the ascending ladder time and ladder height process of , respectively. It is understood that when . Throughout the paper, we denote the nondefective versions of , , and by , , and , respectively. In fact, the pair , is a bivariate subordinator. Define the descending ladder time and the ladder height processes in an analogous way. Note that is 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 and . The former is defined by Taking Laplace transforms shows that where is its joint Laplace exponent such that is the killing rate of so that if and only if , is the drift of , and is its jump measure. Denote the marginal measure of by The function is called the potential/renewal measure. As for the descending ladder process, and are defined similarly. Write and for the restrictions of and to . Furthermore, for , define

We next introduce the notions of a special Bernstein function and complete Bernstein function and two useful results. Recall that a function is called a Bernstein function if it admits a representation where is the killing term, is the drift, and is the Lévy measure concentrated on satisfying . A function is called a special Bernstein function if the function is again a Bernstein function. Let 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 . A function is called a complete Bernstein function if there exists a Bernstein function such that where 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 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 , , is log-convex. Then, the restriction of its potential measure to has a nonincreasing and convex density. Furthermore, if the drift of is strictly positive, then the density is in .*

Lemma 2. *Suppose that is a subordinator with Laplace exponent and potential measure . Then, has a density which is completely monotone on 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 ; or, equivalently, has a density which is completely monotone on if and only if has a completely monotone density.

#### 4. Convexity of Probability of Ruin

Define the probability of ruin by It follows from Bertoin and Doney [33] that , where , with given in (21).

For simplicity, we write the Lévy measure as Recall that an infinitely differentiable function is called completely monotone if for all and all .

Lemma 4 (see Vigon [34]). *For the Lévy process , one has **where and is the potential measure corresponding to .*

Theorem 5. *(i) Suppose is completely monotone on . Then, the probability of ruin is completely monotone on . In particular, .**(ii) Suppose is log-convex on . Then, *(a)* is convex on ;*(b)* is concave on ;*(c)*if has no Gaussian component, then is twice continuously differentiable except at finitely or countably many points on , else .*

* Proof. *We first prove (i). Since is completely monotone on , it follows from Lemma 4 that the tail of Lévy measure is a complete monotone function. Also, it follows from Lemma 2 that the potential measure has a density which is completely monotone on . Thus, the probability of ruin is completely monotone on as .

We now prove (ii). The log-convexity of implies the log-convexity of , and hence is log-convex on 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 , and hence (a) and (b) are proved. Since a convex function on is differentiable except at finitely or countably many points, we see that is twice continuously differentiable except at finitely or countably many points on 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 , and hence . Therefore, (c) is proved.

#### 5. Convexity of

For in (14), define a barrier level by where is understood to be the right-hand derivative at .

For a spectrally negative Lévy process, that is, in the case of , it was shown in Loeffen [17] that the derivative of the -scale function is convex for if is completely monotone. This implies that there exists an such that is concave on and convex on . Also, Kyprianou et al. [18] showed that if has a density on which is nonincreasing and log-convex; then, for each , the scale function and its first derivative are convex beyond some finite value of .

Parallel to the results of Loeffen [17] and Kyprianou et al. [18] for spectrally negative Lévy processes, we have the following results.

Theorem 6. *(i) Suppose is completely monotone on . Then, the derivative is strictly convex on and .**(ii) Suppose is log-convex on . Then, the and its derivative are strictly convex on . Moreover, if has no Gaussian component, is twice continuously differentiable except at finitely or countably many points on , else .*

*Proof. *Since is completely monotone on , we have which is also completely monotone on , where , . We can now apply Theorem 5 to deduce that the probability of ruin is completely monotone on . In particular, . It is easy to prove that is strictly convex on and . Hence, (i) is proved.

Let () be the ascending (descending) ladder height process of . By Lemma 4, we have where is the renewal measure corresponding to . Then, The assumption of log-convexity of implies that is log-convex, and hence is also log-convex. It follows from Lemma 1 of Kyprianou and Rivero [35] that the restriction of its potential measure to of a subordinator with Lévy density has a nonincreasing and convex density, say . Also, the restriction of its potential measure to of a subordinator with Lévy density has a nonincreasing and convex density, say . Moreover, . Thus, , where . Since , we have This implies that tends to as tends to as . Thus . Applying the same arguments as those in Kyprianou et al. [18], we can prove that and its derivative are strictly convex on . Finally, the smoothness of is a direct consequence of Theorem 5. So, (ii) is proved.

#### 6. 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 which is sufficiently smooth and satisfies the following: *(i)*, for almost every ;*(ii)* is concave on ;*(iii)*, **Then, .*

Theorem 8. *Suppose that defined in (13) is sufficiently smooth and satisfies *(i)*, for all ;*(ii)*, for all**Then, . In particular, if , for all , then .*

Theorem 9. *Suppose that is completely monotone. Then, ; that is, the barrier strategy at is the optimal strategy among all admissible strategies.*

Theorem 10. *Suppose that is log-convex on . 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.

Lemma 11. *Suppose that is sufficiently smooth and convex in the interval . Then, the following statements hold: *(i)*;*(ii)* for and for ;*(iii)* for *

*Proof . *As , we have (i). For (ii), for ; it follows from the definition of that for ; for because of ; and since . Finally, (iii) is due to for and (13).

Lemma 12. *Suppose that is sufficiently smooth and is convex in the interval . Then, for , *(i)* if ;*(ii)*;*(iii)*;*(iv)*.*

* Proof . *If , is clear. Also, since and is convex in the interval , we have Thus, (i) is proved.

For , by the definition of , we have On the other hand, for , by the convexity of on , we have These give (ii).

Note that and that is nondecreasing on because of (ii). Thus, ; that is, (iii) holds.

For , . For , we have Lemmas 11(ii) and 12(i) imply that , and Lemma 12(iii) implies that . For , we have Applying Lemmas 11(ii) and 12(ii) yields . For , we obtain which, together with Lemma 12(ii), imply that . These prove (iv).

We now present the proofs of Theorems 7–10.

* Proof of Theorem 7. *Define the jump measure of by and its compensator by . Then, the Lévy decomposition [36, Theorem 42] gives where is a standard Brownian motion and is a martingale with .

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 and we are allowed to use the change of variables formula [36, Theorem 31]; if has a Gaussian exponent, then and we are allowed to use Itô’s formula [36, Theorem 32]; and if has unbounded variation and , then is twice continuously differentiable almost everywhere but is not in 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 , we get under where is a local martingale. The concavity of implies that for any . Taking expectations on both sides of (43) and using conditions (i)–(iii), we obtainThen, letting in (45) and recalling that is an arbitrary strategy in , we get This ends the proof of Theorem 7.

* Proof of Theorem 8. *It follows from (13) and conditions (i) and (ii) that for and for . Similar to (43), one can show thatwhere is the continuous part of , and Note that and that on . Consequently, , and hence Also, for any appropriate localization sequence of stopping times , we haveLetting in (50) yields However, This ends the proof of Theorem 8.

* Proof of Theorem 9. *If is completely monotone, it follows from Theorem 6(i) that is strictly convex on . Then, is concave on because of (13). From Lemmas 11(ii) and (iii) and 12(iv), we see that the conditions in Theorem 7 are satisfied. Thus, . Consequently, and the proof is complete.

* Proof of Theorem 10. *If is log-convex on , it follows from Theorem 6(ii) that is strictly convex on . Then, applying Lemma 12(iv) gives for all . The result follows from Theorem 8.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The research of Chuancun Yin was supported by the National Natural Science Foundation of China (no. 11171179) and the Research Fund for the Doctoral Program of Higher Education of China (no. 20133705110002).