Research Article | Open Access

# The Exit Time and the Dividend Value Function for One-Dimensional Diffusion Processes

**Academic Editor:**Yong Zhou

#### Abstract

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.

#### 1. Introduction

Diffusion processes have extensive applications in economics, finance, queueing, mathematical biology, and electric engineering. See, for example, [1–4] and the references therein. The main tool for studying various properties of diffusion is the result on exit times from an interval. Motivated by Yin et al. in [5], who considered the exit problems for jump processes with applications to dividend problems. In this paper, we consider the Laplace transforms of some random variables involving the exit time for the general one-dimension diffusion processes with applications to dividend problems.

Let be a one-dimensional time-homogeneous diffusion process, which is defined by the following stochastic differential equation: where is a Brownian motion and are constants. It is well known that under certain conditions on the coefficients and , the SDE (1) has a unique strong solution for each starting point. The solution is a time-homogeneous strong Markov process with infinitesimal generator as follows: for any twice continuously differentiable function .

Define

For , we consider the following Laplace transforms:

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 2 studies the Laplace transforms of exit times and considers some popular diffusions. Some applications in the calculation of dividend value functions for the barrier strategy and the threshold strategy are considered in Section 3.

#### 2. Laplace Transform

In this section, we consider the Laplace transform of the exit time for the general diffusion process defined by (1).

Theorem 1. *The function defined by (5) satisfies the following differential equation:
**
with the boundary conditions , .*

*Proof. *We assume that is twice continuously differentiable and satisfies the following differential equation:
Applying Dynkin’s formula to , we obtain
Since is a stopping time, it follows from the optional sampling theorem that
and letting , we get
By the definitions of , we get
Substituting (9) and (10) into (13) and (14), we get
This completes the proof.

Theorem 2. *The function satisfies the following differential equation:
**
with the boundary conditions , . *

*Proof. *The proof of this theorem is similar to that of Theorem 1. We first assume that is twice continuously differentiable and satisfies the following differential equation:
Applying Dynkin’s formula to , and after the same discussion as of Theorem 1, we also can obtain (13) and (14). Substituting (17) into (13) and (14), we get
This completes the proof.

According to the definition of (7), we can lead to the following theorem from Theorems 1 and 2.

Theorem 3. *The function satisfies the following differential equation:
**
with the boundary conditions , . *

Now, we consider some examples.

*Example 4. *The Bessel process: , where is a real number. We assume that in this process.

First, we consider the following differential equation: It is well known that the increasing and decreasing solutions are, respectively, as follows: where , and and are the usual modified Bessel functions.

Then, from Theorem 1, we can give as follows: where the constants and are to be determined. From the boundary conditions (22), we can obtain the expression of the constants and as follows: So, we get

According to Theorem 2 and (21), we can give as follows: where the constants and are to be determined. From the boundary conditions (25), we can determine the constants and obtain the expression of as follows:

According to Theorem 3, the expression of can be obtained from solving the following differential equation: Furthermore, from the definition of , we also can get the expression of . The two methods can lead to the same results as follows:

*Example 5 (the square root process (see [6])). *

We assume that and consider the following differential equation:
We assume that is not an integer, the two linear independent solutions are
where and are the confluent hypergeometric functions of the first and second kinds, respectively. Then, as the way at used in Example 4, and from Theorems 1 and 2, we get that the expressions of and are as follows:
where
So, we can get

*Example 6 (the Ornstein-Uhlenbeck process (see [7])). *
The Ornstein-Uhlenbeck process above is the only process that is simultaneously Gaussian, Markov, and stationary, and has been discussed extensively, see, for example [2–4, 8].

We consider the following differential equation:

In the case of , , the two independent solutions to
are
where and are, respectively, the Hermite and parabolic functions. We obtain the expressions of and as follows:
where
By the definition of , we get

For the general and , the two independent solutions of (36) are, respectively, as follows:
Then, we obtain the expressions of and as follows:
where
Finally, we get

*Example 7 (the Gompertz Brownian motion process (see [9])). *
We assume that .

Now, consider the differential equation It is well known that the increasing and decreasing solutions are, respectively, as follows: where and , as in Example 5, are the first and second Kummer’s function, respectively. From the boundary conditions and , we get where From the boundary conditions and , we get Then, we obtain

#### 3. Applications to Dividend Value Function

##### 3.1. Barrier Strategy

In this subsection, we consider the barrier strategy for dividend payments which are discussed in various model, see, for example, [10–13]. More specifically, we assume that the company pays dividends according to the following strategy governed by parameter . Whenever the surplus is above the level , the excess will be paid as dividends, and when the surplus is below nothing is paid out. We denote the aggregate dividends paid in the time interval by , the modified risk process by , the ruin time by , and the present value of all dividends until ruin by , here, is the discount factor, and the expectation of by

Now, we want to derive the dividend value function by the Laplace transform of exit time. We denote where is defined by (3). Let in the function be defined by (6), we get the definition of . So, we get the following lemma from Theorem 2.

Lemma 8. *The function defined by (56) satisfies the following differential equation:
**
with the boundary conditions , . *

Then, we have the following theorem.

Theorem 9. *For , one has
**
where is defined by (56).*

*Proof. *The one-dimensional diffusion model defined by (1) is a time-homogeneous strong Markov process. Then, when , we have
where is the shift operator. By the definition of , we get
From (59) and (60), we obtain
where can be determined from Lemma 8. From [11], for the barrier strategy, we have the following boundary condition:
Then, we have
So, we get the result. This completes the proof.

Now, we consider the examples discussed in Section 2.

*Example 10. *The Bessel process discussed in Example 4. Reference [14] gives the following helpful formulas:
Letting in Example 4, we get from as follows:
Using the following formula:
we have
Then, from Theorem 9, and substituting and into (58), we obtain

*Example 11. *We consider the square root process discussed in Example 5. Let in Example 5, according to , we obtain
where
Using the following helpful formulas (see [14]):
we get
Finally, we have

*Example 12. *We consider the Ornstein-Uhlenbeck process considered in Example 6, and let in Example 6. From [14], we have the following formulas:
where is the gamma function.

In the case of , , we have Then, we get

For the general , , we have where We can get Finally, we obtain

*Example 13. *We consider the Gompertz Brownian motion process discussed in Example 7. In [15], the authors point out as ,
Letting in Example 7, and using (81), we lead to the expression of as follows:
Using (71), we get
Then, from Theorem 9, we have

##### 3.2. Threshold Strategy

We consider the company pays dividends according to the threshold dividend strategy; that is, dividends are paid at a constant rate whenever the modified surplus is above the threshold , and no dividends are paid whenever the modified surplus is below . For recent publications on threshold strategy, see, for example, [3, 16, 17]. We define the modified risk process by where . Let denote the present value of all dividends until ruin as follows: where . We denote by the expected discounted value of dividend payments; that is, We denote

We can mimic the discussion of Theorem 1 to give the differential equation and the boundary conditions satisfied by .

Lemma 14. *The function defined by (88) satisfies the following differential equation:
**
with the boundary conditions , . *

We have the following theorem.

Theorem 15. *For , one has
**
and, for , one has
*

*Proof. *When , in view of the strong Markov property, we obtain
When , since is a stopping time, it follows from the strong Markov property of that
Using the continuity of the function at , we get
So, we get
Substituting the above expression into (92) and (93), we can get the results (90) and (91). This completes the proof.

We just consider the square root process.

*Example 16. *We consider the square root process discussed in Example 11. We first solve the differential equation satisfied by as follows
We assume that is not an integer, the two linear independent solutions are
Using (81), and from the boundary conditions , , we get
From (71), we have
Furthermore, and have been given in Example 11. From Theorem 15, we obtain
and for , we get
where is defined in Example 11.

#### Conflict of Interests

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

#### Acknowledgments

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.

#### References

- S. Asmussen and M. Taksar, “Controlled diffusion models for optimal dividend pay-out,”
*Insurance*, vol. 20, no. 1, pp. 1–15, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Cai, H. U. Gerber, and H. Yang, “Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest,”
*North American Actuarial Journal*, vol. 10, no. 2, pp. 94–119, 2006. View at: Publisher Site | Google Scholar - Y. Fang and R. Wu, “Optimal dividends in the Brownian motion risk model with interest,”
*Journal of Computational and Applied Mathematics*, vol. 229, no. 1, pp. 145–151, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. C. Yin and H. Q. Wang, “The first passage time and the dividend value function for one-dimension diffusion processes between two reflecting barriers,”
*International Journal of Stochastic Analysis*, vol. 2012, Article ID 971212, 15 pages, 2012. View at: Publisher Site | Google Scholar - C. C. Yin, Y. Shen, and Y. Z. Wen, “Exit problems for jump processes with applications to dividend problems,”
*Journal of Computational and Applied Mathematics*, vol. 245, pp. 30–52, 2013. View at: Publisher Site | Google Scholar - S. Ditlevsen, “A result on the first-passage time of an Ornstein-Uhlenbeck process,”
*Statistics and Probability Letters*, vol. 77, no. 18, pp. 1744–1749, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. C. Cox, J. E. Ingersoll Jr., and S. A. Ross, “A theory of the term structure of interest rates,”
*Econometrica*, vol. 53, no. 2, pp. 385–407, 1985. View at: Publisher Site | Google Scholar - Y. Madec and C. Japhet, “First passage time problem for a drifted Ornstein-Uhlenbeck process,”
*Mathematical Biosciences*, vol. 189, no. 2, pp. 131–140, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Wang and K. Pötzelberger, “Crossing probabilities for diffusion processes with piecewise continuous boundaries,”
*Methodology and Computing in Applied Probability*, vol. 9, no. 1, pp. 21–40, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. U. Gerber and E. S. W. Shiu, “On the time value of ruin,”
*North American Actuarial Journal*, vol. 2, no. 1, pp. 48–78, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. U. Gerber and E. S. W. Shiu, “Optimal dividends: analysis with Brownian motion,”
*North American Actuarial Journal*, vol. 8, no. 1, pp. 1–20, 2004. View at: Google Scholar - C. W. Wang, C. C. Yin, and E. Q. Li, “On the classical risk model with credit and debit interests under absolute ruin,”
*Statistics and Probability Letters*, vol. 80, no. 5-6, pp. 427–436, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. C. Yuen and C. Yin, “On optimality of the barrier strategy for a general Lévy risk process,”
*Mathematical and Computer Modelling*, vol. 53, no. 9-10, pp. 1700–1707, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. J. Zhang and J. M. Jin,
*Computation of Special Functions*, Nanjing University Press, Nanjing, China, 2011. - Abramowitz and M. Stegun,
*Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables*, United States Department of Commerce, US Government Printing Office, Washington, DC, USA, 1972. - Y. Chi and X. S. Lin, “On the threshold dividend strategy for a generalized jump-diffusion risk model,”
*Insurance*, vol. 48, no. 3, pp. 326–337, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. C. Y. Ng, “On a dual model with a dividend threshold,”
*Insurance*, vol. 44, no. 2, pp. 315–324, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2013 Peng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.