Abstract
We develop a new split-step (SS) method for stochastic age-dependent capital system with random jump magnitudes. The main aim of this paper is to investigate the convergence of the SS method for a class of stochastic age-dependent capital system with random jump magnitudes. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory.
1. Introduction
Stochastic partial differential equations are becoming increasingly used to model real-world phenomena in different fields, such as economics, biology, and physics. Recently, the study of the stochastic age-dependent (vintage) capital system has received a great deal of attention. For example, Wang studied stability of solutions for stochastic investment system [1]. Zhang et al. studied the convergence and exponential stability of numerical solutions to the stochastic age-dependent capital system [2, 3].
In the stochastic age-dependent capital system, due to the effects of external environment for capital system, such as innovations in technique, introduction of new products, natural disasters, and changes in laws and government policies, the size of the capital systems increases or decreases drastically. So Poisson jumps with deterministic jump magnitude have been used in stochastic age-dependent population equations. For example, Li et al. [4] studied the Euler numerical method for stochastic age-dependent population equations with Poisson jumps. L. Wang and X. Wang [5] analysed the convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps. Rathinasamy et al. [6] developed the numerical method for stochastic age-dependent population equations with Poisson jump and phase semi-Markovian switching. However, the random jump magnitude is now commonly seen in financial models [7ā9]. In this paper, we will consider the following stochastic age-dependent capital system with random jump magnitudes as shown in [10]: where denotes the stock of capital goods of age at time , , , and . is defined as total output produced in year ; also is the age of the capital; the investment in the new capital. The is the appreciation (when ) or depreciation (when ) of the production capacity, and represents the volatility of the capital stock. The value of is the actual jump and is the underlying random variables of the magnitudes, and often it is called āmark" of the jump.
Also is a standard Wiener process. is a scalar Poisson process with intensity . It is assumed that for some there is a constant such that ; that is, the th moment of the jump magnitude is bounded. The maximum physical lifetime of capital , the planning interval of calendar time , the depreciation rate of capital, and the capital density (the initial distribution of capital over age) are given. The denotes the accumulative rate of capital at the moment of , , and is the technical progress at the moment of . This makes that total output produced in year be defined as . In each sector all the firms have an identical neoclassical technology and produce output using labor and capital. The production function is neoclassical, where is the total sum of capital at time and is the labor force.
The integral version of (1) is given by the equation where for fixed .
Since the system (1) does not have closed form solutions, it is necessary to develop numerical methods for (1). Recently, Zhang and Rathinasamy [10] first derived the numerical solutions for stochastic age-dependent capital system with random jump magnitudes. However, their method belongs to the classic explicit Euler method and has a lower accuracy in [10] if we do not consider the appropriate step sizes.
Higham and Kloeden [11] first constructed the split-step backward Euler (SSBE) method for nonlinear stochastic differential equations with Poisson jumps. Tan and Wang [12] studied the convergence and stability of the SSBE method for linear stochastic delay integrodifferential equations. Ding et al. [13] developed the split-step method for solving the stochastic differential equations. Rathinasamy [14] investigated the split-step methods for stochastic age-dependent population equations with Markovian switching. Thus, we can construct the SS method for stochastic age-dependent population equations with random Poisson jumps.
In this paper, we will investigate the convergence of the SS method for system (1). The outline of the paper is as follows. In Section 2, we will introduce some preliminary results which are essential for our analysis. Section 3 will show us the SS method for solving stochastic age-dependent population equations with random Poisson jumps. In Section 4, several lemmas which are useful for our main result are proved. We give the main result that the numerical solutions converge to the true solutions with strong order 1/2 in Section 5. At last, a numerical example is given to verify the results obtained from the theory.
2. Preliminaries
Throughout this paper, it will be denoted by the space of functions that are square-integrable over the domain . Let where is generalized partial derivative with respect to age and is a Sobolev space. such that . is the dual space of . We denote by , and the norms in , , and , respectively, by the scalar product in . , the duality product between and , is defined by
Let be a Wiener process defined on and taking its values in the separable Hilbert space , where is an orthonormal set of eigenvectors of , are mutually independent real Wiener processes with incremental covariance , , and (tr denotes the trace of an operator). For an operator to be the space of all bounded linear operators from into , it is denoted by the Hilbert-Schmidt norm; that is,
Let be the space of all continuous function from into with sup-norm =, , and .
Let be a complete probability space with a filtration , satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets).
Definition 1. Let be the stochastic basis and a Wiener process. Suppose that is a random variable such that . A stochastic process for fixed is said to be a solution on to the stochastic age-structured capital system for if the following conditions are satisfied:(D1) is a -measurable random variable;(D2), , , where denotes the space of all -valued processes (one will write for short) measurable (from into ), and satisfying . Here denotes the space of all continuous functions from to ;(D3)it satisfies the following equation: for all , , a.e. , where the stochastic integrals are understood in the ItƓ sense.
The parameter is the maximal age of the capital, so , .
As the standing hypotheses we always assume that the following conditions are satisfied.(1), and , .(2)(Lipschitz condition) there exists a positive constant such that and ,(i),(ii).(3) is nonnegative measurable in such that and is nonnegative continuous in such that (4), , , , where is a positive constant.
In an analogous way to the corresponding proof presented in [15], the following existence and uniqueness of solutions is established: under the conditions , (1) has a unique continuous solution on .
We note for the following jump process: where and is the indicator function for the set ; that is, .
Then the following lemma can be found in [16, 17].
Lemma 2. There exists a constant for any and for some such that
3. The Split-Step -Method
Let denote the th jump of occurrence time. Suppose, for example, that jumps arrive at distinct, ordered times ; let be a deterministic grid point of . We construct approximate solutions to models of the form (1) at a discrete set of times . This set is the superposition of the random jump times of a Poisson process on [0,] and a deterministic grid and satisfies max. Let , , and denote the increments of the time, Brownian motion, and the Poisson processes, respectively.
For system (1) the split-step approximate solution is defined by the iterative schemewith initial values , , ; is the numerical approximation of with ; the time increment is . Where , if we give , the SS method becomes the SSBE method in [11]. If , the SS method is an explicit method.
To answer the question of the existence of numerical solution, we will give the following lemma.
Lemma 3. Let conditions and hold, and let and ; then the implicit equation (12a) can be solved uniquely for , with probability 1.
Proof. Writing (12a) as , and using condition (2) and (3), we have Then the result follows from the classical Banach contraction mapping theorem [18].
When Lemma 3 followed, we find it is convenient to use continuous-time approximation solution in our strong convergence analysis; hence for , we can define the following step functions: where is the largest number such that .
When , Lemma 3 ensures the existence of by (12a); then we define with initial value , , .
It is easy to verify that ; that is, and coincide with the discrete solutions at the grid points. Hence we refer to as a continuous-time extension of the discrete approximation . So our plan is to prove a strong convergence result for .
4. Several Lemmas
In this section, we will give several lemmas which are useful for the following main result.
Lemma 4. Under the conditions , there are constants and such that
The proof is similar to that in [3].
Next lemma shows the relationship between and .
Lemma 5. Under conditions , let , , and ; then there exist two positive constants and such that where and are produced by (12a) and (12b).
Proof. Squaring both sides of (12a) and using the elementary inequality , we find Using the elementary inequality , we obtain Due to and conditions , we can get Taking mathematical expectation for both sides, we can obtain Since , thus ; then by , we have where and . The proof is completed.
The next lemma shows that the continuous numerical solutions have bounded second moments.
Lemma 6. Under conditions , let , ; then there exists a constant such that where , , and .
Proof. From (17), we can get Applying the It formula [19] to it yields Using conditions and the compensated Poisson process , we have Note where . Consider Taking (29)-(30) into (28), we compute that for some positive constants , , , Now, it follows that for any and by Lemma 5 By Burkholder-Davis-Gundy inequality, we have for some positive constants , , . Substituting (33) into (32) and then by Lemma 5, we can obtain that where Applying the continuous Gronwall inequality, we can easy get with .
The next lemma shows that the continuous-time approximation in (17) remains close to the step functions and in the mean square sense.
Lemma 7. Under conditions , , and let , ; then there exist two positive constants and that are independent of , such that where , , and are defined by (14), (15), and (17), respectively.
Proof. Consider ; we have Squaring both sides and using the element inequality , we have Now, the Cauchy-Schwarz inequality, condition (), and the compensated Poisson process give Taking mathematical expectation, by element inequality , and using the martingale isometry, we have By conditions () and (), we get Since and on , we have Then by Lemmas 5 and 6, we can derive where Thus we can prove (37), and a similar analysis gives the proof of (38).
Lemma 8. Under conditions , for any , there exists a positive constant such that
Proof . The proof is similar to that of Lemma 4.
5. Main Result
Now we use the above lemmas to prove a strong convergent result.
Theorem 9. Under condition , let , ; then there exists a constant such that
Proof. From (2) and (17), we have Using the generalized It formula it yields By Cauchy-Schwartz inequality and condition (), we have Hence for any Now using the Burkholder-Davis-Gundy inequality and Lemma 2, we have where are positive constants. Therefore, inserting (53) into (52) it has Then using Lemma 7, we have That is, where Using the Gronwall inequality, we have Since for any , thus we have The proof is completed.
Theorem 10. Under conditions , let , ; then there exists a constant such that
Proof. Let ; it is easy to see that Recall the Young inequality: for , we have Let ; we have Note By Lemma 4, then similarly, the result can be derived for so that Then By Theorem 9, (61) becomes Let ; then where . The proof is completed.
Theorem 11. Under the conditions there exists a constant such that for some , the numerical approximate solution (1) will converge to the exact solution to (1) in the sense
Proof. The proof is easily deduced from Theorem 10.
6. Numerical Example
In this section, we consider the following stochastic age-dependent capital system with random jump magnitudes age-dependent in [10] where is a scalar Brownian motion and is a scalar Poisson process with intensity . Let be defined in (16). Take , , and to be a lognormal random variable such that and .
It is easy to verify that the conditions are satisfied. Then the approximate solution will converge to the true solution of (1) for any in the sense of Theorem 11. Obviously, in (72) cannot be solved explicitly. It is necessary to know numerical approximation of .
Now we fix step sizes and and change the parameter in all figures. Then we compare the expectation of the numerical solution, where .
In Figure 1, we show the expectation of numerical solution of the system (72) by Euler method in [10] and split-step -method with = 0.2, respectively. From the figure, we can see that the split-step -method also reveals the age-dependent capital system tendency.
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In Figures 2 and 3, the four pictures are simulation for numerical solution of the system (72) by split-step -method with = 0.3, 0.4, 0.5, and 0.6, respectively.
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7. Conclusion
Due to the complexity of this model, the comparison between the Euler approximation and split-step methods is not possible in this paper. Similarly, error analysis is also not obtained in this paper due to unavailability of closed form solutions. They will be discussed in the future researches. But the visualization of Figures 1, 2, and 3 provides the information about the split-step methods which coincide with the Euler approximation. The comparison between Euler approximation and split-step methods in the stochastic differential equations in [11ā13] shows that the split-step methods are better than the Euler approximation. Similarly, we believe that this new approximation of the system (1) in this paper is a good approximation when compared with the Euler approximation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported with funds provided by the National Natural Science Foundation of China (nos. 11226321, 11272229, and 11102132). The second author is working under the research project named as UGC Research Award 2012ā2014 of University Grants Commission, Bahadur Shah Zafar Marg, New Delhi-110002 with award number F.30-1/2013(SA-II)/RA-2012-14-OB-TAM-1324.