Abstract and Applied Analysis

Volume 2013 (2013), Article ID 718627, 15 pages

http://dx.doi.org/10.1155/2013/718627

Research Article

Approximate Solutions of Hybrid Stochastic Pantograph Equations with Levy Jumps

Wei Mao¹ and Xuerong Mao²

¹School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing 210013, China

²Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH, UK

Received 17 March 2013; Accepted 9 May 2013

Academic Editor: Svatoslav Staněk

Copyright © 2013 Wei Mao and Xuerong Mao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milošević and Jovanović (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory.

1. Introduction

Stochastic delay differential equations (SDDEs) have come to play an important role in many branches of science and industry. Such models have been used with great success in a variety of application areas, including biology, epidemiology, mechanics, economics and finance. Similar to SDEs, an explicit solution can rarely be obtained for SDDEs. It is necessary to develop numerical methods and to study the properties of these methods. There are many results for the numerical solutions of SDDEs [1–12].

Recently, as a special case of SDDEs, a class of stochastic pantograph delay equations (SPEs) has been received a great deal of attention and various studies have been carried out on the convergence of SPEs [13–16]. However, all equations of the above-mentioned works are driven by white noise perturbations with continuous initial data, and white noise perturbations are not always appropriate to interpret real data in a reasonable way. In real phenomena, the state of stochastic pantograph delay system may be perturbed by abrupt pulses or extreme events. A more natural mathematical framework for these phenomena takes into account other than purely Brownian perturbations. In particular, we incorporate the Levy perturbations with jumps into stochastic pantograph delay system to model abrupt changes.

The study of the convergence of the numerical solutions to SDDEs with jumps is in its infancy [17–20], and there is no research on the numerical solutions to SPEs with Markovian switching and Levy jumps (SPEwMsLJs). In this paper, we study the strong convergence of the Euler method for a class of SPEs with Markovian switching and Levy jumps (SPEwMsLJs). SPEwMsLJs may be regarded as an extension of SPEs with Markovian switching and SPEs with Levy jumps. The main aim is to prove that the Euler approximate solutions converges to the true solutions for SPEwMsLJs in sense. On the other hand, we study the convergence in probability of the Euler approximate solutions to the true solutions under local Lipschitz condition and some additional conditions in term of Lyapunov-type functions. It should be pointed out that the proof for SPEwMsLJs is certainly not a straightforward generalization of that for SPEs and SPEwMs without Levy jumps. Although the way of analysis follows the ideas of [21], we need to develop several new techniques to deal with Levy jumps. Some known results in Fan et al. [14], Milošević and Jovanović [16], and Marion et al. [21] are generalized to cover a class of more general SPEwMsLJs.

The paper is organized as follows. In Section 2, we introduce some notations and hypotheses concerning (4), and the Euler methods is used to produce a numerical solutions. In Section 3, we establish some useful lemmas and prove that the approximate solutions converge to the true solutions of SPEwMsLJs in sense. By applying Theorem 4, we study the convergence in probability of the approximate solutions to the true solutions in Section 4. Finally, we give an illustrative example in Section 5.

2. Preliminaries and the Approximate Solution

Let be a complete probability space with a filtration satisfying the usual condition; that is, the filtration is continuous on the right and contains all -null sets. Let , be a -dimensional Wiener process defined on the probability space adapted to the filtration . Let denote the family of function from that are right continuous and have limits on the left. Also is equipped with the norm , where is the Euclidean norm in ; that is, . Let , , and denote the family of all -valued measurable -adapted processes such that . Let be a measurable space and a -finite measure on it. Let , and let be a stationary -Poisson point process on with characteristic measure . Denote by the Poisson counting measure associated with ; that is,

We refer to Mao [3] for the properties of a Wiener process and SDDEs and to Ikeda and Watanabe [22] for the details on Poisson point process.

Let , , be a right-continuous Markov chain on the probability space taking values in a finite state space with generator given by

where . Here is the transition rate from to , , while

We assume that Markov chain is independent of the Brownian motion and compensated Poisson random measure . It is known that almost every sample path of is right-continuous step function with a finite number of simple jumps in any finite subinterval of .

In this paper, we study the following hybrid stochastic pantograph equations with Levy jump:

where and

is a standard -dimensional Brownian motion, and is the compensated Poisson random measure given by

Here is the Levy measure associated to .

Let denote the family of all nonnegative functions on which are continuously twice differentiable in . For each , define an operator from to by

where

In order to define the Euler approximate solution of (4), we need the property of embedded discrete Markov chain. The following lemma [23] describes this property.

Lemma 1. For and , then is a discrete Markov chain with the one-step transition probability matrix

Given a step size , the discrete Markov chain can be simulated as follows (see Mao and Yuan [24]). Let and generate a random number which is uniformly distributed in . If , then let or otherwise find the unique integer for

and let , where we set as usual. Generate independently a new random number which is again uniformly distributed in . If then let or otherwise find the unique integer for

and let . Repeating this procedure, a trajectory of can be generated. This procedure can be carried out independently to obtain more trajectories.

Now we define the Euler approximate solution to (4) with discrete Markov chain . For system (4), the discrete approximation is given by the iterative scheme

with initial value , and represents the integer part of . Here for . We have , ,   , and .

Let us introduce the following notations:

for . Then we define the continuous Euler approximate solution as follows:

which interpolates the discrete approximation (7).

In order to establish the strong convergence theorem, we suppose the following assumptions are satisfied.

Assumption 2. For each and ,

Assumption 3. For every , there exists a positive constant such that for all and , ,

3. Strong Convergence of Numerical Solutions

In this section, we will prove that the Euler approximate solutions converge to the true solutions in sense under the local Lipschitz condition.

Theorem 4. If Assumptions 2 and 3 hold, then the Euler approximate solutions converge to the true solutions of (4) in sense with order ; that is, there exists a positive constant such that

where and , and let .

The proof of Theorem 4 is very technical, so we present some useful lemmas.

Lemma 5. Under Assumptions 2 and 3, for any and , there exists a positive constant such that

where is a positive constant independent of the step size .

Proof. For any , there exists an integer such that . Then

Using the inequality , we get

By the Hölder inequality and Assumptions 2 and 3, we have

By the definition of , we have , . so we get that , , and

By using the Burkholder-Davis-Gundy inequality and Assumptions 2 and 3, we have

Combing (22)–(24) together, we have

where . The proof is complete.

Lemma 6. Under Assumptions 2 and 3, for any and , then

where is a positive constant independent of the stepsize .

The proof of this lemma is similar to that of Lemma 5.

Lemma 7. Under Assumptions 2 and 3, for any and , then

where is a positive constant independent of the stepsize .

The proof of this lemma is similar to that of [16, 24].

Proof of Theorem 4. Combining (4) and (14), one has

Then applying the generalized Itô’s formula, we can show that

Hence, for any , we get

By Assumption 3 and Lemmas 5–7, we have

Similarly, by Assumption 3 and Lemmas 5–7, we obtain

On the other hand, by the Burkholder-Davis-Gundy inequality, Young’s inequality, and Lemmas 5–7, we have for any

We have for any

where is some constant that may change from line to line. Similar, we have

Substituting (31)–(36) into (30), we obtain that

By choosing , sufficiently small and letting , we have

Therefore, we apply Gronwall's inequality to get

This completes the proof.

Remark 8. Under the local Lipschitz condition, Theorem 4 not only tells us the strong convergence of the approximate solutions to the true solutions but also tells us the rate of the convergence with order by (39).

Remark 9. When or , (4) reduces to which was studied by Fan et al. [14], Xiao and Zhang [15], and Milošević and Jovanović [16]. Our results in the present paper generalized and improved the results in [14–16].

4. Convergence of Numerical Solutions in Probability

In this section, by applying Theorem 4, we will show the convergence in probability of the approximate solutions to the true solutions under local Lipschitz condition. Before we give the convergence theorem, we need some additional conditions based on Lyapunov-type functions.

Assumption 10. For and , there exist a positive function , , and two constants such that

Assumption 11. There exists a positive constant such that, for all and with ,

Now, let us state our convergence theorem.

Theorem 12. Let the assumptions of Theorem 4 hold. Also assume that there exists a function satisfying (40)–(42). Then the Euler approximate solutions converges to the true solutions of (4) in the sense of the probability.

That is,

Proof. The proof is rather technical, and we divide it into three steps.

Step 1. We assume the existence of the nonnegative Lyapunov function satisfying (40). Applying the Itô’s formula, yields

Integrating from 0 to and taking expectations gives

By (41), we have

Thus, for any , it follows that

Using the Gronwall inequality, we obtain that

Let . By (40), we have . Noting that , as , we derive from (48) that

That is,

Recall that as . For a given , , and , it follows that

as . Let

Thus we have

Step 2. We will give the estimate of . By (14), applying the Itô’s formula to yields

By (7), we have

Integrating from 0 to , taking expectations, and by (41), we have

where

By (42) and Young’s inequality, we have

Let , the integer part of , and let be the indicator function of the set . Then

By setting and using the Markov property, we have

Inserting (60) into (58), we have, by Lemma 5,

Similarly, by Assumptions 2, 3, and 11, Lemma 5, and Markov property, we have

where . For the term in (56), by Assumptions 2, 3, and 11 and Lemma 5, we have

where . For the term in (56), by Assumptions 2, 3, and 11 and Lemma 5, we have

where

Inserting (65) into (64), we have

For the term , by Assumption 11, equation (56), and Lemma 5, we have

Combing these inequalities and (53), we derive that, for ,

where

For arbitrary , by the Gronwall inequality, we get

Noting that , as , we derive from (70) that

So we have

Step  3. Let be arbitrarily small. By setting

and using Theorem 4, we have

Combing (53) and (72) together, one gets

Hence on using (74), we conclude that

Recalling that as , we can choose sufficiently large for

and then choose sufficiently small for

to obtain

The proof of Theorem 12 is now complete.

5. An Example

In this section, we construct one example to demonstrate the effectiveness of this theory. Let be a right-continuous Markov chain taking values in with the generator

Let be a compensated Poisson random measures and is given by , where and

is the density function of a lognormal random variable. Of course and are assumed to be independent. Consider a linear stochastic pantograph delay equations with Markovian switching and pure jumps

Here , , , and . Then (82) can be regarded as the result of the two equations

switching among each other according to the movement of the Markov chain . Obviously, (82) satisfies Assumptions 2 and 3. Given the stepsize , we can have the Euler method

with . Let , , and , for . Then we define the continuous Euler approximate solution

Since the conditions of Theorem 4 are satisfied, then the approximate solution (85) will converge to the true solution of (82) in the sense of the -norm. To examine the convergence in the sense of the probability, we construct a function

Then

From the properties of the lognormal distribution, we have

If , then it follows that Assumptions 10 and 11 are satisfied. Consequently, the approximate solution (85) will converge to the true solution of (82) for any in the sense of Theorem 12.

Acknowledgment

This work is sponsored by Qing Lan project of Jiangsu Province (2012) and supported by the Grant of Jiangsu Second Normal University (Jsie2011zd04) and the Jiangsu Government Overseas Study Scholarship.

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