Stochastic 𝜽-Methods for a Class of Jump-Diffusion Stochastic Pantograph Equations with Random Magnitude

Abstract

This paper is concerned with the convergence of stochastic -methods for stochastic pantograph equations with Poisson-driven jumps of random magnitude. The strong order of the convergence of the numerical method is given, and the convergence of the numerical method is obtained. Some earlier results are generalized and improved.

1. Introduction

Recently, the study of stochastic pantograph equations (SPEs) has many results [1–3]. SPEs have been extensively applied in many fields such as finance, control, and engineering. However, in general, SPEs have no explicit solutions, and the study of numerical solutions of SPEs has received a great deal of attention. Fan et al. [4] investigate the th moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Baker and Buckwar [5] gave strong approximations to the solution obtained by a continuous extension of the -Euler scheme and proved that the numerical solution produced by the continuous -method converges to the true solution with order 1/2. Fan et al. [6] investigated the existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations under the local Lipschitz condition and the linear growth condition. Li et al. [7] investigated the convergence of the Euler method of the stochastic pantograph equations with Markovian switching under the weaker conditions. Reference [8] studied convergence and stability of numerical methods of stochastic pantograph differential equations.

In practice, stochastic differential equations with jump and numerical methods are also discussed extensively. In [9–13] strong convergence and mean-square stability properties were analysed in the case of Poisson-driven jumps of deterministic magnitude. References [14, 15] discussed the numerical methods of stochastic differential equations with random jump magnitudes. Motivated by the papers above, in this paper, we focus on stochastic pantograph equations with random jump magnitudes. SPEs with random jump magnitudes may be regarded as an extension of stochastic pantograph equations. Jump models arise in many other application areas and have proved successful at describing unexpected, abrupt changes of state [16–18]. Typically, these models do not admit analytical solutions and hence must be simulated numerically. Similar to stochastic differential equations [19–21], explicit solutions can hardly be obtained for SPEs with random jump magnitudes. Thus, appropriate numerical approximation schemes such as the Euler (or Euler-Maruyama) are needed to apply them in practice or to study their properties.

The paper is organised as follows. In Section 2, we introduce the SPEs with random jump magnitudes and define stochastic -methods of (1). The main result of the paper is rather technical, so we present several lemmas in Section 3 and then complete the proof in Section 4.

2. Preliminaries

Throughout this paper, we 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). Let be the Euclidean norm in . Let be -measurable and right-continuous, and . Let be a -dimensional Brownian motion defined on the probability space.

Consider a class of jump-diffusion stochastic pantograph equations with random magnitude of the form on with the initial value and , where is a Poisson process with mean ; ; and , are independent, identically distributed random variables representing magnitudes for each jump.

Throughout, we assume that the jump magnitudes have bounded moments; that is, for some , there is a constant such that

We further employ the following assumptions.

Assumption 1. The functions , , and satisfy the global Lipschitz condition, that is, for each , there is a positive constant such that where .

Assumption 2 (linear growth condition). There is a positive constant such that for all for all .

In fact, the global Lipschitz condition (3) implies the linear growth condition (4). Under these conditions, it can be shown similarly as in [20] that (1) has a unique solution with all moments bounded.

We note for later reference that (1) involves the jump process where and , are the jump times.

One generalisation of stochastic -Euler methods [6, 21] to system (1) has the form and where . Here is a step size, which satisfies for some positive integer , . , , and are the Brownian and Poisson increments, respectively.

For , we define and denote Then, we define the continuous-time approximation which interpolates the discrete numerical approximation (6). So a convergence result for immediately provides a result for .

3. Lemmas

Throughout our analysis, , , denote generic constants, independent of . The main theorem of the paper is rather technical. We will present a number of useful lemmas in the section and then complete the proof in Section 4.

Lemma 3. Under Assumption 2, there exists such that for all ,

Proof. From (6), we have Note that . Now, using Assumption 2, Using and Assumption 2, we have
For the jump, we convert to the compensated Poisson increment with and and Assumption 2. We then obtain Combining (13), (14), and (15) with (12) yields where , is a constant dependent on and .
Now choosing sufficiently small such that and noting that (2) implies that each , we obtain The result then follows from an application of the discrete Gronwall inequality. The proof is complete.

Lemma 4. Under Assumption 2, there exists such that, for all ,

Proof. Consider . In this interval we have Thus, Thus, by virtue of (12)–(14) and Lemma 3, we have where .
In a similar way we obtain (18). The proof is complete.

Lemma 5. Under Assumption 2, there exists such that, for all ,

Proof. Consider . By (9), we have Thus, Therefore, in view of the Hölder inequality and , we have Then, applying Itô and martingale isometries and Assumption 2, we have Now, note that (2) implies that each , on , , , , , and . Hence, applying Lemma 3, we obtain where . In the following we consider : Let ; the proof is complete.

4. Main Results

We can now state and prove our main result of this paper.

Theorem 6. Under Assumption 1 for some and Assumptions 1–2, there exists and such that, for all ,

Proof. The analysis uses ideas from [15], where analogous results are derived in the stochastic differential equations. By construction, we have Now for any we have By Assumption 1 and Hölder inequality, we have
By Assumption 1, the Cauchy-Schwarz inequality, and the Doob inequality in the two martingale terms and the martingale isometry, We also have where is the smallest integer such that .
Now the number of nonzero terms in the summation in (34) is a random variable that is not independent of the summands. To obtain a useful bound, we recall the following Young’s inequality: where and .
Hence, Now we can apply the Hölder inequality as follows: Using (37) in (35), we have which follows from the Hölder inequality and (2); we yield Choosing , we have Substituting (40) into (34) yields as .
Now, substituting (31), (32), (33), and (41) into (30) yields From Lemmas 4 and 5, we have where .
By the Gronwall inequality, we have The proof is complete.

Remark 7. Theorem 6 shows that the order of convergence in mean square is close to 1. Moreover, stochastic -methods give strong convergence rate arbitrarily close to order 1/2 under appropriate moment bounds on the jump magnitude. This problem class is now widely used in mathematical finance.
By Theorem 6, we obtain the following corollaries.

Corollary 8. Under Assumption 1,

The convergent result can be extended to the case of nonlinear coefficients that are local Lipschitz [6, 7, 12] based on the style of analysis in [22].

Corollary 9. Under the local Lipschitz condition and Assumption 2,

Remark 10. Corollary 9 shows that the numerical solution converges to the true solution. However, the order of the convergence of the numerical method is not given under the local Lipschitz condition. If we remove jump and discuss the system without time lag, our results are reduced to the results derived in [6, 14]. In other words, our results are the generalization of paper [6, 14].

Conflict of Interests

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

Acknowledgments

The work is supported by the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China under Grant 61304067, 11271146 and 61304175 the Natural Science Foundation of Hubei Province of China under Grant 2013CFB443.