Research Article | Open Access
The Exit Time and the Dividend Value Function for One-Dimensional Diffusion Processes
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.
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 , 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 .
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.
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.
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 )).
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 )).
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 )). 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 , 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  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
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 , 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.
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.
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