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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.

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 [

Analysis of optimal dividends for Lévy risk processes is of particular interest which have undergone an intensive development. For example, Avram et al. [

The paper is organized as follows. In Section

To present the mathematical formulation of the problem of study, let us first introduce some notations and definitions. Let

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

We denote by

Following similar reasoning to Yuen and Yin [

In this section, we recap some basic facts about ladder processes and potential measure. Consider the dual process

We next introduce the notions of a special Bernstein function and complete Bernstein function and two useful results. Recall that a function

Let

Suppose that

Note that the tail of the Lévy measure

Define the probability of ruin by

For simplicity, we write the Lévy measure

For the Lévy process

(i) Suppose

(ii) Suppose

if

We first prove (i). Since

We now prove (ii). The log-convexity of

For

For a spectrally negative Lévy process, that is, in the case of

Parallel to the results of Loeffen [

(i) Suppose

(ii) Suppose

Since

Let

We now present the main results of the paper about the optimality of the barrier strategy

Suppose that

Suppose that

Suppose that

Suppose that

Before proving the main results, we give two lemmas which are similar to those in Loeffen [

Suppose that

As

Suppose that

If

For

Note that

For

We now present the proofs of Theorems

Define the jump measure of

Note that

It follows from (

If

If

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

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).