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

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.


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 ±1 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 momentgenerating function is proved in Section 5.

The Model
Consider the following surplus process: where  > 0 is the force of interest,  > 0 is the drift coefficient,  > 0 is the diffusion coefficient, and {  } ≥0 is the standard Brownian motion.We will assume that the company pays dividends according to the following strategy governed by parameters  2 >  1 > 0 and  > 0. Whenever the modified surplus is below the level  1 , no dividends are paid.However, when the modified surplus is above  1 and below the  2 , dividends are paid continuously at a constant rate .When the modified surplus is above  2 , dividends are completely paid.For  ≥ 0, let denote the aggregate dividend paid by time , where  1 () and  2 () are caused by the different parts of dividends, respectively.Thus, is the modified surplus at time .Let  > 0 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  ≥ 0, we use the symbol (;  1 ,  2 ) to denote the expectation of .That is, Define the random times with the convention inf 0 = ∞.
satisfies the following ordinary differential equation: for  1 <  ≤  2 , () satisfies the following ordinary differential equation: for  >  2 , () 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  1 ,  2 such that  1 is strictly decreasing and  2 is strictly increasing (see e.g., [16]).Let  3 ,  4 be such solution for the ordinary differential equation (10), where  3 is strictly decreasing and  4 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.

The Special Dividends Strategy
In this section, we consider the limit of dividends level.Let  2 → ∞; by the expressions of  and , we have where Substituting the above expressions into (18) and ( 19), and setting we obtain, for 0 <  ≤  1 , Let we obtain, for Then dividing numerator and denominator of ( 34) and (36) by  4 ( 2 )  4 ( 2 ), we get the expected discounted dividend payments for the threshold strategy which is (15) in Fang and Wu [13].

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 Proof.For 0 <  ≤  1 , applying the strong Markov property, we obtain Similarly, for It can be verified that   ( From ( 46) we obtain ( 1 ;  1 ,  2 ) and ( 2 ;  1 ,  2 ), so we get the results (41) and (42).