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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 467948, 7 pages
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.
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.
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 , 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 , Albrecher et al. , Gao and Yin , Gerber and Shiu [5, 6], Wan  and so on. The optimal dividend problem in a compound Poisson model was studied by Gerber and Shiu . Optimal dividend in an Ornstein-Uhlenbeck-type model with credit and debit interest was considered in Cai et al. . 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 . The perturbed Sparre Andersen model with a threshold dividend strategy was settled by Gao and Yin .
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.  and Fang and Wu . More general, Wong and Zhao  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 , we can prove (9) and (10). The boundary conditions can be derived the same as in Gerber and Shiu  or Cai et al. .
The ordinary differential equation (9) has two positive independent solutions , such that is strictly decreasing and is strictly increasing (see e.g., ). Let , be such solution for the ordinary differential equation (10), where is strictly decreasing and is strictly increasing. In Cai et al. , 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.
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. , 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
Let we obtain, for ,
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 ,
Theorem 4. For , one has and, for , one has where
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
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
Remark 8. When , we get .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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.
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