^{1}

^{1}

^{2}

^{1}

^{2}

We investigate the exit times from an interval for a general one-dimensional time-homogeneous diffusion process and their applications to the dividend problem in risk theory. Specifically, we first use Dynkin’s formula to derive the ordinary differential equations satisfied by the Laplace transform of the exit times. Then, as some examples, we solve the closed-form expression of the Laplace transform of the exit times for several popular diffusions, which are commonly used in modelling of finance and insurance market. Most interestingly, as the applications of the exit times, we create the connect between the dividend value function and the Laplace transform of the exit times. Both the barrier and threshold dividend value function are clearly expressed in terms of the Laplace transform of the exit times.

Diffusion processes have extensive applications in economics, finance, queueing, mathematical biology, and electric engineering. See, for example, [

Let

Define

For

We study the differential equations satisfied by the Laplace transforms and some applications of the popular dividend strategy in risk theory.

The rest of the paper is organized as follows. Section

In this section, we consider the Laplace transform of the exit time for the general diffusion process

The function

We assume that

The function

The proof of this theorem is similar to that of Theorem

According to the definition of (

The function

Now, we consider some examples.

The Bessel process:

First, we consider the following differential equation:

Then, from Theorem

According to Theorem

According to Theorem

We assume that

We consider the following differential equation:

In the case of

For the general

Now, consider the differential equation

In this subsection, we consider the barrier strategy for dividend payments which are discussed in various model, see, for example, [

Now, we want to derive the dividend value function by the Laplace transform of exit time. We denote

The function

Then, we have the following theorem.

For

The one-dimensional diffusion model defined by (

Now, we consider the examples discussed in Section

The Bessel process discussed in Example

We consider the square root process discussed in Example

We consider the Ornstein-Uhlenbeck process considered in Example

In the case of

For the general

We consider the Gompertz Brownian motion process discussed in Example

We consider the company pays dividends according to the threshold dividend strategy; that is, dividends are paid at a constant rate

We can mimic the discussion of Theorem

The function

We have the following theorem.

For

When

We just consider the square root process.

We consider the square root process discussed in Example

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

The authors are grateful to the anonymous referee’s careful reading and detailed helpful comments and constructive suggestions, which have led to a significant improvement of the 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.