1. Introduction
In recent decades, the optimization problem of dividend has received remarkable attention in the financial mathematics and insurance mathematics. This problem goes back to Finetti [1] , who considered a discrete time random walk with step size ±1 and found that the optimal dividend strategy is a barrier strategy, that is, any surplus above a certain level would be paid as dividend. Since then, many researchers have studied the dividends problems for various risk models under a barrier strategy. For the compound Poisson model, this problem was solved by Gerber [2] , identifying so-called band strategies as the optimal ones. For exponentially distributed claim sizes this strategy simplifies to a barrier strategy. Azcue and Nuler [3] follows a viscosity approach to investigate optimal reinsurance and dividend strategies in the Cramér-Lundberg model. Albrecher and Thonhauser [4] showed that the optimality of barrier strategies in the classical model with exponential claims still holds if there is a constant force of interest. Avram et al. [5] considered the case where the risk process is given by a general spectrally negative Lévy process and gave a sufficient condition involving the generator of the Lévy process for optimality of the barrier strategy. Loeffen [6] showed that barrier strategy is optimal among all admissible strategies for general spectrally negative Lévy risk processes with completely monotone jump density, and Kyprianou et al. [7] relaxed this condition on the jump density to log-convex. An alternative proof is given in Yin and Wang [8] . Loeffen and Renaud [9] pushed this result further by assuming the weaker condition that the Lévy measure has a density which is log-convex. Azcue and Muler [10] examine the analogous questions in the compound Poisson risk model with investment.
The Poisson processes are the most basic and widely used stochastic model for modeling discrete data, it may provide a poor fit in the presence of over-dispersion. For example, the use of the Poisson distribution as a model describing the number of claims caused by individual policyholders (e.g. in automobile insurance) during to that certain period is usually rejected, since in practice the behavior of policyholders is heterogeneous. In such a case the standard Poisson model is inappropriate. For example, in collective risk theory, it is assumed that claims occur in bulk, where the number of bulks 
occurring in 
follows a Poisson process with parameter
. Each bulk consists of a random number of claims so that the total number of claims is of the form
where 
denotes the number of claims in the 
-th bulk. The aggregate claim payments made up to time
, called the generalized Poisson process, is given by 
where 
representing the individual claim amounts. In this paper, we formulate and solve an optimal dividends problem for a generalized Poisson risk model in which the aggregate claim payments are defined by a generalized Poisson process.
The rest of the paper is organized as follows. In Section 2, we give a rigorous mathematical formulation of the problem. Section 3 gives notion of log-convexity and complete monotonicity. We present our main results in Section 4 and prove them in Section 5.
2. Problem Setting
Consider the risk model
, defined on the filtered probability space 
and,
(2.1)
where 
is a standard Brownian motion with
, the claim sizes 
are positive independent and identically distributed random variables whose probability distribution function is given by
,
is a generalized Poisson process defined as 
where 
are discrete independent and identically distributed random variables whose probability distribution is given by
and 
is a homogeneous Poisson process with intensity
.
Moreover, it is assumed that
, 
and 
are mutually independent. In particular, when
, the process 
reduces to the homogeneous Poisson process with intensity
, and hence the risk model (2.1) reduces to the classical risk model perturbed by Brownian motion (see Chiu and Yin [11] ).
The probability mass function of 
is given by
(2.2)
where 
is the 
-fold convolution of
. In a few special cases it is possible to determine the probabilities
’s explicitly.
Example 2.1. Suppose that 
are geometrically distributed with parameter
, where
, i.e.
Then the compound Poisson process by geometric compounding leads to the Pólya-Aeppli process
with parameters 
and 
(cf. Minkova [12] ). That is for all
,
(2.3)
Note that the Pólya-Aeppli process is a time-homogeneous process, it is also called Poisson-geometric process in Chinese literature, for example see Mao and Liu [13] , where the ruin probability was studied for compound Poisson-geometric process. In the case of
, the Pólya-Aeppli process becomes a homogeneous Poisson process.
Example 2.2. (Quenouille [14] ) Let 
denote a sequence of independent and identically distributed random variables, each one having the logarithmic distribution (also known as the logarithmic series distribution)
, with probability mass function
Suppose that 
has a Poisson process with parameter
. Then the random sum
has the negative binomial distribution
:
(2.4)
In this way, the negative binomial distribution is seen to be a compound Poisson distribution.
We now consider the classical optimal dividend control problem. Let 
be a dividend strategy consisting of a non-decreasing left-continuous 
-adapted process 
with
, where 
represents the cumulative dividends paid out by the company till time 
under the control
. We define the controlled risk process 
by
. Let 
be the ruin time and define the value function of a dividend strategy 
by
where 
is an interest force for the calculation of the present value. Let 
be the set of all admissible dividend strategies, that is all strategies 
such that 
for
. The objective is to solve the following stochastic control problem:
(2.5)
and to find an optimal policy 
that satisfies 
for all
.
3. Log-Convexity and Complete Monotonicity
Before starting our main results, we introduce the definitions of log-convexity and complete monotonicity.
Definition 3.1. (Willmot and Lin [15] ). (1) A distribution 
on the non-negative integers is said to be log-convex if 
and 
is said to be strictly log-convex if 
A counting distribution 
is discrete completely monotone iff it is a mixture of geometric distributions, i.e.
where 
is a probability distribution on
.
(2) A function 
is log-convex if 
is a convex function. Let 
with
. We say 
is completely monotone if 
for all
.
(3) The distribution function 
is said to be decreasing (increasing) failure rate or DFR (IFR) if
is nondecreasing (nonincreasing) in 
for fixed
, i.e. if 
is log-convex
(log-concave).
Note that the completely monotone class is a subclass of the log-convex. For examples of continuous log-convex or completely monotone functions can be found in Yin and Wang [8] . Now, we give a discrete example.
Example 3.1. Let 
be a logarithmic random variable with
Then 
is completely monotone (see Van Harn ([16] , p. 58)). The generalized logarithmic series distribution is defined by
with 
and
. Then 
is strictly log-convex (see Hansen and Willekens [17] ).
4. Main Results
Denote by 
the constant barrier strategy at level 
which is defined by 
and
for all
. That is, for a level 
whenever surplus goes above
, the excess is paid as dividends to the shareholders of the company and, if the surplus is less than
, no dividends are paid out. We will now present the main results of this note which give sufficient conditions for optimality of a barrier strategy
. It is important to note that various dividend strategies can be employed by an insurance company. However, we will only focus on the conditions for the optimality of a dividend strategy.
After some tedious calculations, we get
where
(4.1)
Here 
is the 
-fold convolution of 
with itself. So that 
is a special spectrally negative Lévy process with the Laplace exponent
. Therefore, all the known results for spectrally negative Lévy process models can be applied to the model (2.1). However, since the distribution function 
is not explicit (depends on the distribution of 
and
), it can be of interest to study which assumptions on the probability distributions of 
and 
ensure that the optimal dividend strategy is barrier one.
We now recall the definition of the 
-scale function 
and some properties of this function. For each 
there exists a continuous and increasing function
, called the 
-scale function defined in such a way that 
for all 
and on 
its Laplace transform is given by
Here, 
is the unique root of equation 
in the half-plane
.
From Avram et al. [5] we get the expected discounted value of dividend payments of the barrier strategy at level 
is given by
(4.2)
Define
Theorem 4.1. For model (2.1), if 
has a completely monotone probability density function on 
and
is discrete completely monotone, then the barrier strategy with level 
is the optimal dividend strategy. Moreover, the 
defined by (2.5) is given by
.
Theorem 4.2. For model (2.1), if 
is discrete completely monotone and 
is DFR, then the barrier strategy with level 
is the optimal dividend strategy. Moreover, the 
defined by (2.5) is given by
.
Corollary 4.1. For model (2.1) with 
given by (2.3) or (2.4), if 
is DFR, then the barrier strategy with level 
is the optimal dividend strategy. Moreover, the 
defined by (2.5) is given by
.
Theorem 4.3. For model (2.1), if 
is a log-convex probability mass function and 
is the exponential distribution function with mean
, then the barrier strategy with level 
is the optimal dividend strategy. Moreover, the 
defined by (2.5) is given by
.
5. Proof of Main Results
Before proving the main results, we give several lemmas.
Lemma 5.1. (Loeffen [13] ) Suppose that the Lévy measure of a spectrally negative Lévy process 
has a completely monotone density on
, then the barrier strategy at 
is an optimal strategy.
Kyprianou, Rivero and Song [7] providing weaker conditions on the Lévy measure for the optimality of a barrier strategy. An alternative approach can be found in Yin and Wang [8] .
Lemma 5.2. Suppose that a spectrally negative Lévy process 
has a Lévy density 
on 
that is log-convex, then the barrier strategy at 
is an optimal strategy.
Note that for the Cramér-Lundberg model with or without a Brownian component, the requirement of log-convexity of the Lévy density 
on 
is equivalent to the log-convexity of the probability density function of the individual claim amount on
. Since the Lévy measure having a log-convex (or completely monotone) density implies that tail of the Lévy measure is log-convex and the converse is not true (cf. Loeffen and Renaud [9] ), the following result improves the results in Lemmas 3.1 and 3.2.
Lemma 5.3. (Loeffen and Renaud [9] ) Suppose that the tail of the Lévy measure of a spectrally negative Lévy process 
is log-convex, then the barrier strategy at 
is an optimal strategy.
Proof of Theorem 4.1. If 
is discrete completely monotone and 
has a completely monotone density on
, then
has a completely monotone density on 
(cf. Chiu and Yin [18] ). The result follows from Lemma 5.1.
Proof of Theorem 4.2. It is well known that the property of DFR is preserved under the geometric sum (see Shanthikumar [19] , Corollary (3.6)), and since the sum of two log-convex functions is log-convex and the limit of a pointwise convergent sequence of log-convex functions is log-convex, it follows that
is also DFR. The result of Theorem 4.2 follows from Lemma 5.3.
Proof of Theorem 4.3. If 
is the exponential distribution function with mean
, then by (4.1) we have
Therefore,
Interchanging the order of summation yields
where
Note that 
and 
is increasing in
, it follows from Theorem 3.2 in Esary and Marshall [20] that 
has a density which is logarithmically convex on
. The result follows from Lemmas 5.2.
Remark 5.1. At the end of this paper, we give two conjectures. The first conjecture can be viewed as an extension of Theorem 4.3; The second conjecture can be viewed as an extension of Conjecture 1 and Theorem 4.2.
Conjecture 1. For model (2.1), if 
is a log-convex and 
has a density 
on 
that is log-convex, then the barrier strategy at 
is an optimal strategy for stochastic control problem (2.5).
Conjecture 2. For model (2.1), if 
is DFR and 
is DFR, then the barrier strategy at 
is an optimal strategy for stochastic control problem (2.5).
Acknowledgements
We would like to thank the anonymous referee who gave us many constructive suggestions and valuable comments on the previous version of this paper. The research was supported by the National Natural Science Foundation of China (No. 11171179), the Research Fund for the Doctoral Program of Higher Education of China (No. 20133705110002) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.
NOTES
*Corresponding author.