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

and Applied Analysis 3 where the constants C 1 and C 2 are to be determined. From the boundary conditions (22), we can obtain the expression of the constants C 1 and C 2 as follows: C 1 = a V KV (√2δb) IV (2δa)KV (√2δb) − IV (2δb)KV (√2δa) , C 2 = a V IV (√2δb) IV (2δb)KV (√2δa) − IV (2δa)KV (√2δb) .


Introduction
Diffusion processes have extensive applications in economics, finance, queueing, mathematical biology, and electric engineering.See, for example, [1][2][3][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 onedimension diffusion processes with applications to dividend problems.
We study the differential equations satisfied by the Laplace transforms and some applications of the popular dividend strategy in risk theory.

Laplace Transform
In this section, we consider the Laplace transform of the exit time for the general diffusion process {  ;  ≥ 0} defined by (1).
Proof.We assume that () is twice continuously differentiable and satisfies the following differential equation: Applying Dynkin's formula to ℎ(,   ) =  − (  ), 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.
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. Example 4. The Bessel process:   = (( − 1)/2  ) +   , where  > 1 is a real number.We assume that  >  > 0 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 V = (−2)/2, and  V (⋅) and  V (⋅) are the usual modified Bessel functions.Then, from Theorem 1, we can give  1 () as follows: where the constants  1 and  2 are to be determined.From the boundary conditions ( 22), we can obtain the expression of the constants  1 and  2 as follows: So, we get According to Theorem 2 and (21), we can give  2 () as follows: where the constants  3 and  4 are to be determined.From the boundary conditions (25), we can determine the constants and obtain the expression of  2 () as follows: According to Theorem 3, the expression of () can be obtained from solving the following differential equation: Furthermore, from the definition of () =  1 () +  2 (), 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  >  > 0 and consider the following differential equation: We assume that (2]/ 2 ) 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  1 () and  2 () 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][3][4]8].We consider the following differential equation: In the case of  = 0,  = 1, the two independent solutions to are where  V (⋅) and  V (⋅) are, respectively, the Hermite and parabolic functions.We obtain the expressions of  1 () and  2 () 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  1 () and  2 () as follows: where Finally, we get Example 7 (the Gompertz Brownian motion process (see [9])).

Barrier Strategy.
In this subsection, we consider the barrier strategy for dividend payments which are discussed in various model, see, for example, [10][11][12][13].More specifically, we assume that the company pays dividends according to the following strategy governed by parameter  > 0. 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 [0, ] by   (), the modified risk process by (53) Now, we want to derive the dividend value function by the Laplace transform of exit time.We denote where  +  is defined by (3).Let  = 0 in the function  2 () be defined by ( 6), we get the definition of  2 ().So, we get the following lemma from Theorem 2. Lemma 8.The function  2 () defined by (56) satisfies the following differential equation: with the boundary conditions  2 (0) = 0,  2 () = 1.
Then, we have the following theorem.
Proof.The one-dimensional diffusion model defined by ( 1) is a time-homogeneous strong Markov process.Then, when 0 <  < , we have where  0 is the shift operator.By the definition of  0 , we get From ( 59) and (60), we obtain where  2 () 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  = 0 in Example 4, we get  2 () from  2 () as follows: Using the following formula: we have Then, from Theorem 9, and substituting  2 () and   2 () into (58), we obtain Example 11.We consider the square root process discussed in Example 5. Let  = 0 in Example 5, according to (, , 0) = 1, 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  = 0 in Example 6.From [14], we have the following formulas: where Γ() is the gamma function.
In the case of  = 0,  = 1, 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  = 0 in Example 7, and using (81), we lead to the expression of  2 () as follows: Using (71), we get Then, from Theorem )] . (84) 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   () =  ∫  0 (  () > ).Let   denote the present value of all dividends until ruin as follows: where   = inf{ ≥ 0 :   () = 0}.We denote by   (, ) the expected discounted value of dividend payments; that is, We can mimic the discussion of Theorem 1 to give the differential equation and the boundary conditions satisfied by  3 ().Lemma 14.The function  3 () defined by (88) satisfies the following differential equation: We have the following theorem.