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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 718627, 15 pages
Approximate Solutions of Hybrid Stochastic Pantograph Equations with Levy Jumps
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.
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.
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 , we need to develop several new techniques to deal with Levy jumps. Some known results in Fan et al. , Milošević and Jovanović , and Marion et al.  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  for the properties of a Wiener process and SDDEs and to Ikeda and Watanabe  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
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 ). 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.
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.
The proof of this lemma is similar to that of Lemma 5.
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. , Xiao and Zhang , and Milošević and Jovanović . 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.
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