Convergence Analysis of Semi-Implicit Euler Methods for Solving Stochastic Age-Dependent Capital System with Variable Delays and Random Jump Magnitudes

Abstract

We consider semi-implicit Euler methods for stochastic age-dependent capital system with variable delays and random jump magnitudes, and investigate the convergence of the numerical approximation. It is proved that the numerical approximate solutions converge to the analytical solutions in the mean-square sense under given conditions.

1. Introduction

Stochastic differential equations have been widely used to model the phenomena arising in many branches of science and industry fields such as biology, economic, finance, and ecology [1–4]. Recently, the numerical construction of stochastic age-dependent capital system has received a great deal of attention [5–9]. In [8], exponential stability of numerical solutions for a class stochastic age-dependent capital system with Poisson jumps was studied by Zhang et al. in the case of deterministic magnitude. Zhang and Rathinasamy [9] studied convergence of numerical solutions for a class of stochastic age-dependent capital system with random jump magnitudes, which extended the analysis in [8] to the case where the jump magnitudes are random. However, in many real problems, the capital systems can be modeled by stochastic dynamical systems whose evolutions depend not only on the current states, but also on their historical states. So we need a stochastic delay model including an extra term, which is called time delay, to simulate them. In this paper, we consider stochastic age-dependent capital system with variable delays and random jump magnitudes: where , , and . The stock of capital goods of age at time is denoted by . This makes that total output produced at time defined as ; also is the age of the capital, the investment in the new capital. , + , + denote effects of the external environment for capital system, such as innovations in techniques, natural disasters, introduction of new products, and changes in laws and government policies, and so on. is a variable delay, and the is the appreciation (when ) or depreciation (when ) of the production capacity, and represents the volatility of the capital stock. is a standard Wiener process. is a Poisson process with mean and , , are independent, identically distributed random variables representing magnitudes for each jump. We assume that for some there is a constant such that ; that is, the 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 . Each sector of all the firms has an identical neoclassical technology and produces output using labor and capital. The production function is neoclassical, where is the total sum of capital at time and is the labor force.

In general most equations of stochastic age-dependent capital system with variable delays and random jump magnitudes do not have explicit solution. Thus, numerical approximation schemes are invaluable tools for exploring its properties. There is a significant amount of literature that has been published concerning approximate schemes for stochastic differential equations with jumps [9–11] or stochastic differential delay equations [12–14]. In [15], Chalmers and Higham gave the semi-implicit Euler approximate solutions and proved the convergence of semi-implicit Euler methods under the Lipschitz condition. However, in many situations, the coefficients , , and are only locally Lipschitz continuous. It is therefore useful to establish the strong convergence of the semi-implicit Euler method under the local Lipschitz condition. In this paper, we relax the global Lipschitz condition on the coefficients which was imposed in [15] and prove that the semi-implicit Euler approximate solutions converge to the exact solutions of (1) in the mean-square sense under the local Lipschitz condition.

In Section 2, we will collect some notation and hypotheses concerning (1), and the semi-implicit Euler method is used to produce numerical solutions. In Section 3, we give the useful lemmas which are essential to prove our main results, that is, Theorem 17.

2. Preliminaries and the Semi-Implicit Euler Approximation

Throughout this paper, it will be denoted by the space of functions that are square-integrable over the domain . Let

is the 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 complete probability space with covariance operator and taking its values in the separable Hilbert space : where is an orthonormal set of eigenvectors of that are mutually independent real Wiener processes with incremental covariance , , and ( denotes the trace of an operator). For an operator is the space of all bounded linear operators from into ; it is denoted by ; its denotes 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 with a left-hand side limit, and contains all -null sets). Let be a scalar Poisson process with intensity which is independent from Wiener process . Denote by the family of all bounded, -measurable, and -valued random variables. Let , , and denote the family of all -measurable, -valued variables which satisfies . In this paper, , , and are family of nonlinear operators, -measurable almost surely in .

The integral version of (1) is given by

For system (6), the discrete semi-implicit Euler approximation on is given by the iterative scheme with initial value ; ; and represents the integer part of . Here the time increment is , for some sufficiently large integer such that and , for , and and , , are the Wiener and Poisson increments, respectively.

Define the step functions: where is the indicator function for set . Then we define the continuous semi-implicit Euler approximation: with , for fixed .

Remark 1. If the parameter in (7), then the semi-implicit Euler methods become the Euler methods, which have been studied in [10, 12, 14].

In order to establish the convergence theorem, we propose the following assumptions.

Assumption 2. The function is the time delay which satisfies For , there exists a constant such that

Assumption 3 (local Lipschitz condition). There exist two positive constants , such that, for all and ,

Assumption 4. is a nonnegative measurable function in such that and is a nonnegative continuous function in such that ; is a nonnegative constant in .

Assumption 5. The initial function is Holder-continuous with exponent ; that is, there exists a positive constant such that, for ,

Assumption 6. Consider , , and , .

Assumption 7. Consider , , , and , where is a positive constant.

Remark 8. If the local Lipschitz condition holds, then there exists a positive constant such that, for ,

3. Convergence of the Semi-Implicit Euler Approximate Solution

In this section, several lemmas which are useful for the following main result are given.

Lemma 9. Under Assumptions 2–7, there are constants and such that

The proof is similar to that of in [16].

Let and . Define the stopping time .

Lemma 10. Under Assumptions 2–7, there exists a constant such that

Proof. From (9), applying Itô formula to yields where is a compensated Poisson process. Note where .
Therefore, by Assumption 4, (15), and a variant of Cauchy-Schwarz inequality for any , along with (18), we get that Let , ; we have By Burkholder-Davis-Gundy inequality, we have where , , and are positive constants. Substituting (22) into (21) yields, again for a possibly different and , for every . Now, Gronwall lemma obviously implies the required result. The proof is complete.

Lemma 11. Under Assumptions 2–7, there exist constants and such that

The proof is similar to that of Lemma 9.

Lemma 12. Under Assumption 3 and , for any , where , are positive constants dependent only on , , and and independent of .

Proof. For any , there exists an integer such that ; then Applying the basic inequality to 5 terms on the above right-hand side, we have Now, the Cauchy-Schwarz inequality and Assumptions 2–7 give where is a constant. Because the differential operator is a bounded linear operator, we obtain Similarly, the second part of (25) can be obtained.

Lemma 13. Under Assumptions 2–7, for any , where , are positive constants dependent only on , , and and independent of .

Proof. For any , there exists an integer such that ; then . To show the estimate , let us consider the following five possible cases.(1)If , then + .Using the Cauchy-Schwarz inequality, martingale isometries, and (15), we have Hence, (2)If , then − .Using the Cauchy-Schwarz inequality, martingale isometries, and (15), we have Hence, (3)If or , then . So we get, by Assumption 5 on , (4)If , then and . We have, by Assumption 5 on , Hence, (5)If , then − and . We have, by Assumption 5 on , Hence, Combining these different cases, we get Similarly, we have

Lemma 14. There exists a constant for any and such that

Proof. The proof is basically similar to that of Theorem 3.4 in [15], and we thus omit it here.

We are now in a position to prove the main convergence results.

Theorem 15. If Assumptions 2–7 hold, then the semi-implicit Euler approximate solutions converge to the exact solutions of (1) in the mean-square sense; that is, where is a positive constant dependent only on , , and and and independent of .

Proof. Subtraction of (6) and (9) gives Therefore using the generalized Itô formula, along with the Cauchy-Schwarz inequality and Assumptions 2–7, yields By Burkholder-Davis-Gundy inequality, we have where , , , , and are positive constants. Let ; inserting (46) into (45) we obtain Hence, Applying Gronwall’s inequality, we obtain a bound of the form where .

Theorem 16. Under Assumptions 2–7 and let , then

Proof. Let , it is easy to see that By Young’s inequality , , for any , , , , and , we have Let , ; we have Note Then By Theorem 15, (51) becomes Let ; then The proof is completed.

Theorem 17. Under Assumptions 2–7 and there is a constant such that for some , the numerical approximate solution (7) will converge to the exact solution to (1) in the sense

Proof. The proof is easily deduced from Theorem 16.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was partially supported by The National Natural Science Foundation (61374040), Key Discipline of Shanghai (S30501), Scientific Innovation Program (13ZZ115), and Graduate Innovation Program of Shanghai (54-13-302-102, JWCXSL1301).