Journal of Applied Mathematics

Volume 2013, Article ID 467948, 7 pages

http://dx.doi.org/10.1155/2013/467948

## The Ornstein-Uhlenbeck-Type Model with a Hybrid Dividend Strategy

School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

Received 3 August 2013; Revised 18 December 2013; Accepted 18 December 2013

Academic Editor: Mina Abd-El-Malek

Copyright © 2013 Dan Zhu and Chuancun Yin. 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.

#### Abstract

We consider the Ornstein-Uhlenbeck-type model. We first introduce the model and then find the ordinary differential equations and boundary conditions satisfied by the dividend functions; closed-form solutions for the dividend value functions are given. We also study the distribution of the time value of ruin. Furthermore, the moments and moment-generating functions of total discounted dividends until ruin are discussed.

#### 1. Introduction

In recent years, the dividend problem has gained a lot of attention in the actuarial literature. Dividend strategies for insurance risk model were first proposed by de Finetti [1], who considered a discrete time random walk with step size and found that the optimal dividend strategy must be a barrier strategy. From then on, the problem of optimal dividend strategy has been studied in continuous time, for example, Asmussen and Taksar [2], Albrecher et al. [3], Gao and Yin [4], Gerber and Shiu [5, 6], Wan [7] and so on. The optimal dividend problem in a compound Poisson model was studied by Gerber and Shiu [8]. Optimal dividend in an Ornstein-Uhlenbeck-type model with credit and debit interest was considered in Cai et al. [9]. For a class of compound Poisson process perturbed by diffusion with a threshold dividend strategy, the expected discounted penalty function has been studied by Wan [7]. The perturbed Sparre Andersen model with a threshold dividend strategy was settled by Gao and Yin [10].

Recently, the multilayer dividend strategy as an extension of the threshold dividend strategy has drawn many authors attention. For example, the perturbed Sparre Andersen and compound Poisson risk models with multilayer dividend strategy have been studied by Yang and Zhang [11, 12]. The integrodifferential equations for the expected discounted penalty function were derived and solved; when the claims are subexponentially distributed, the asymptotic formula for ruin probability is obtained. The Ornstein-Uhlenbeck-type model is a very important model in applied probability which has recently gained a lot of attention. See, for example, Cai et al. [9] and Fang and Wu [13]. More general, Wong and Zhao [14] consider the optimal dividends and bankruptcy in an Ornstein-Uhlenbeck process with the surplus-dependent credit/debit interest rate. Motivated by the above work, in this paper, we consider a hybrid dividend strategy which combined a barrier strategy with a threshold strategy in an Ornstein-Uhlenbeck-type model. For simplicity, we consider only one threshold and one barrier.

The remainder of the paper is organized as follows. In Section 2, we describe the model and discuss the dividend functions until ruin, in Section 3, we give the limit of dividends level, and in Section 4 we get the expression by Laplace transform of ruin time. The partial differential equation with boundary conditions satisfied by the moments and moment-generating function is proved in Section 5.

#### 2. The Model

Consider the following surplus process: where is the force of interest, is the drift coefficient, is the diffusion coefficient, and is the standard Brownian motion. We will assume that the company pays dividends according to the following strategy governed by parameters and . Whenever the modified surplus is below the level , no dividends are paid. However, when the modified surplus is above and below the , dividends are paid continuously at a constant rate . When the modified surplus is above , dividends are completely paid. For , let denote the aggregate dividend paid by time , where and are caused by the different parts of dividends, respectively. Thus, is the modified surplus at time . Let be the force of interest for valuation; in this paper, we assume that . Let be the indicator function of event and let denote the present value of all dividends until ruin where is the time of ruin, and

For , we use the symbol to denote the expectation of . That is,

Define the random times

with the convention .

Lemma 1. *Assume that is twice continuously differentiable on . For , satisfies the following ordinary differential equation:
**
for , satisfies the following ordinary differential equation:
**
for , satisfies the following equation:
**
with boundary conditions
*

*Proof. *By virtually the same arguments as in Yin and Wen [15], we can prove (9) and (10). The boundary conditions can be derived the same as in Gerber and Shiu [6] or Cai et al. [9].

The ordinary differential equation (9) has two positive independent solutions , such that is strictly decreasing and is strictly increasing (see e.g., [16]). Let , be such solution for the ordinary differential equation (10), where is strictly decreasing and is strictly increasing. In Cai et al. [9], the authors pointed out that these independent solutions are given by
where and are called the confluent hypergeometric functions of the first and second kind, respectively.

Denote

The expressions of the expected discounted dividend payments are given by Theorem 2.

Theorem 2. *(i) For ,
**(ii) For ,
*

*Proof. *When , by the strong Markov property of the process , we have

For , using Itô's formula or Dynkin's formula as in Li et al. [17], we find that satisfies the ordinary differential equation (9), with boundary conditions , . Assume that the solution of the equation is ; from the boundary conditions, we obtain

So we have

Similarly, when ,

For , using Itô's formula or Dynkin's formula, we have that satisfies the ordinary differential equation (10) with boundary conditions , . Assume that the solution of the equation is ; from the boundary conditions, we obtain
so we have

Similarly, satisfies (10) with boundary conditions , , so we get
where

For , dividends will be payable if the surplus process without ruin and reaches , so that

For , dividends are paid continuously at a constant rate , so we obtain

Using and , we get

With some careful calculations, we obtain and , so we get the results (18) and (19).

#### 3. The Special Dividends Strategy

In this section, we consider the limit of dividends level. Let ; by the expressions of and , we have where

Substituting the above expressions into (18) and (19), and setting we obtain, for ,

Let we obtain, for ,

Then dividing numerator and denominator of (34) and (36) by , we get the expected discounted dividend payments for the threshold strategy which is (15) in Fang and Wu [13].

*Remark 3. *Similarly, when , we get the expected discounted dividend payments for the barrier strategy
which is (9) and (10) in Cai et al. [9].

#### 4. The Time Value of Ruin under a Hybrid Dividend Strategy

In this section, we focus on the Laplace transform of the time value of ruin. We assume that dividends are paid according to threshold strategy with parameters , and barrier strategy with parameter . Let denote the Laplace transform of the time value of ruin; for ,

Let

Theorem 4. *For , one has
**
and, for , one has
**
where
*

*Proof. *For , applying the strong Markov property, we obtain

Similarly, for ,

It can be verified that and , so we get

From (46) we obtain and , so we get the results (41) and (42).

*Remark 5. *(1) Let denote the expected discounted penalty at ruin; in this model, the penalty at ruin is a constant , so we get
Substitution of (41) and (42) into (47) yields an expression for .

(2) Let be the probability of ruin. Note that
then

From (14)–(18), we have

Taking limit in (41) and (42) yields .

#### 5. The Moment-Generating Function of

In this section, the moment-generating function of the hybrid dividend payments is discussed. We adopt a similar approach to that of Gao and Yin in Section 3 [10].

Let denote the moment-generating function of , and let denote the th moment function. The following theorem provides differential equations for the function .

Theorem 6. *The moment-generating function satisfies the partial differential equations
**
with boundary conditions
*

*Proof. *(i) We first provide the solution of . Consider the SDE
note that, in the case , the solution to the ODE is . To solve the SDE, consider the process

Let and , so we have ,
we get , so we obtain

Now the solution for is

(ii) Now we derive the integrodifferential equations for .

For , we consider a small time interval , the time such that the surplus will not reach before ; in view of the strong Markov property of the surplus process , we have

By Taylor expansion, we have

Subtracting from each side of the above equation, dividing by , and then letting , we achieve

Similarly, for , we consider a small time interval , the time such that the surplus will not reach ; we have

By Taylor expansion, we get

Subtracting from each side of (63), dividing by , and then letting , we achieve

The proof of boundary conditions is routine. This ends the proof of Theorem 6.

Theorem 7. *For , satisfies
**
and, for , satisfies
**
with the boundary conditions
*

*Proof. *Recall that , and ; using the representation
and equating the coefficients of in (53) yield the ordinary differential equations (66) and (67).

*Remark 8. *When , we get .

#### Conflict of Interests

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

#### Acknowledgments

The authors would like to thank the editor and the anonymous referees for their helpful comments, which have led to this improved version 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

- B. de Finetti, “Su un'impostazione alternativa della teoria collettiva del rischio,” in
*Proceedings of the transactions of the 15th International Congress of Actuaries*, vol. 2, pp. 433–443, 1957. - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Albrecher, R. Kainhofer, and R. F. Tichy, “Simulation methods in ruin models with non-linear dividend barriers,”
*Mathematics and Computers in Simulation*, vol. 62, pp. 277–287, 2003. View at Google Scholar - H. Gao and C. Yin, “A perturbed risk process compounded by a geometric Brownian motion with a dividend barrier strategy,”
*Applied Mathematics and Computation*, vol. 205, no. 1, pp. 454–464, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Wan, “Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion,”
*Insurance*, vol. 40, no. 3, pp. 509–523, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. U. Gerber and E. S. W. Shiu, “On optimal dividend strategies in the compound Poisson model,”
*North American Actuarial Journal*, vol. 10, no. 2, pp. 76–93, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - 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 · View at Google Scholar · View at MathSciNet - H. Gao and C. Yin, “The perturbed Sparre Andersen model with a threshold dividend strategy,”
*Journal of Computational and Applied Mathematics*, vol. 220, no. 1-2, pp. 394–408, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Yang and Z. Zhang, “On a perturbed Sparre Andersen risk model with multi-layer dividend strategy,”
*Journal of Computational and Applied Mathematics*, vol. 232, no. 2, pp. 612–624, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Yang and Z. Zhang, “The perturbed compound Poisson risk model with multi-layer dividend strategy,”
*Statistics & Probability Letters*, vol. 79, no. 1, pp. 70–78, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Y. Wong and J. Zhao, “Optimal dividends and bankruptcy procedures: analysis of the Ornstein-Uhlenbeck process,”
*Journal of Computational and Applied Mathematics*, vol. 236, no. 2, pp. 150–166, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Yin and Y. Wen, “An extension of Paulsen-Gjessing's risk model with stochastic return on investments,”
*Insurance*, vol. 52, no. 3, pp. 469–476, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - L. Breiman,
*Probability*, Addison-Wesley, Reading, Mass, USA, 1968. View at MathSciNet - P. Li, C. C. Yin, and M. Zhou, “The exit time and the dividend value function for one-dimensional diffusionn processes,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 675202, 9 pages, 2013. View at Publisher · View at Google Scholar