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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 420648, 10 pages
Convergence Rate of Numerical Solutions for Nonlinear Stochastic Pantograph Equations with Markovian Switching and Jumps
1College of Electrical and Information Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, China
2Jiangsu Meteorological Observatory, Nanjing 210008, China
3College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China
Received 8 April 2013; Accepted 5 June 2013
Academic Editor: Zidong Wang
Copyright © 2013 Zhenyu Lu et al. 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.
The sufficient conditions of existence and uniqueness of the solutions for nonlinear stochastic pantograph equations with Markovian switching and jumps are given. It is proved that Euler-Maruyama scheme for nonlinear stochastic pantograph equations with Markovian switching and Brownian motion is of convergence with strong order 1/2. For nonlinear stochastic pantograph equations with Markovian switching and pure jumps, it is best to use the mean-square convergence, and the order of mean-square convergence is close to 1/2.
Stochastic modelling has been used with great success in a variety of application areas, including control theory, biology, epidemiology, mechanic, and neural networks, economics, and finance [1–5]. In general, stochastic different equations do not have explicit solutions. Therefore, approximate schemes for stochastic differential equations with Markovian switching and Poisson jumps have been investigated by many authors [3, 6, 7]. The convergence results of numerical solutions of stochastic differential equations with Markovian switching and Poisson jumps under the Lipschitz condition and the linear growth condition are obtained by using Euler-Maruyama scheme or semi-implicit Euler scheme. However, recently, more and more convergence results have been given under weaker conditions than the Lipschitz condition and the linear growth condition. Gyöngy and Rásonyi  revealed the convergence rate of Euler approximations for stochastic differential equations whose diffusion coefficient is not Lipschitz but only ()-Hölder continuous for some . Mao et al.  discussed and -convergence of the Euler-Maruyama scheme for stochastic differential equations with Markovian switching under non-Lipschitz coefficients. Wu et al.  proved existence of the nonnegative and the strong convergence of the Euler-Maruyama Scheme for the Cox-Ingersoll-Ross model with delay whose diffusion coefficient is nonlinear and non-Lipschitz continuous. Bao and Yuan  studied the convergence rate for stochastic differential delay equations whose coefficients may be highly nonlinear with respect to the delay variable.
So far, the research of the numerical solutions for stochastic pantograph equations has just begun [12–15]. Fan et al.  gave the strong convergence for stochastic pantograph equations under the Lipschitz condition and the linear growth condition. Ronghua et al.  proved that the Euler approximation solution converges to the analytic solution in probability under weaker conditions, but the convergence rate has not been given.
In this paper, we will study the convergence rate for nonlinear stochastic pantograph equations with Markovian switching and Poisson jump under weaker conditions than the Lipschitz condition and the linear growth condition. The rest of the paper is organized as follows. In Section 2, we will give the existence and uniqueness of the analytic solutions for Markovian switching and Brownian motion case and also reveal that the convergence order of Euler-Maruyama scheme is . In Section 3, we show that it is best to use the mean-square convergence for Markovian switching and the pure jump case and that the rate of mean-square convergence is close to .
2. Convergence Rate for Markovian Switching and Brownian Motion Case
Let be a complete probability space with a filtration satisfying the usual conditions. Let be an -dimensional Brownian motion defined on the probability space adapted to the filtration. For integer , let be the Euclidean space and the Hilbert-Schmidt norm for a matrix , where is its transpose. Throughout this paper, denotes a generic constant whose values may change from lines to lines.
Let be a right-continuous Markov chain on the probability space taking values in a finite state space with the generator given by where . Here is the transition rate from to if while We assume that the Markov chain is independent of the Brownian motion . It is well known that almost every sample path of is a right continuous step function with finite number of sample jumps in any finite subinterval of .
For fixed , we consider the stochastic pantograph equation with Markovian switching of the form with initial data , , , . is a Markov chain. On the time interval , let , and we define the partition
The integral version of (3) is given by the following:
To guarantee the existence and uniqueness of the solutions of (3) we introduce the following conditions: (A1) and there exists such that for , ; (A2) and there exists such that for , , where such that for some and arbitrary .
Remark 1. From (A1)-(A2), we know that the coefficients of (3) are much weaker than those of the Lipschitz condition and the linear growth condition. In many examples, and do not satisfy the Lipschitz condition or the linear growth condition but can be covered by (A1)-(A2).
Lemma 2. Assume that (A1) and (A2) hold. Then, for any initial data and , is a unique global strong solution of (3). Moreover, for any there exists such that
Proof. From (A1) and (A2), and are locally Lipschitzian. So, (3) has a unique local solution . In order to verify that (3) has a unique global solution on time interval , it is sufficient to show that
From (A1), (A2), and (8), we can obtain
Substituting (11) and (12) into (5) and by the Hölder inequality and the Burkhold-Davis-Gundy inequality, we have that for any and
Let ; then
By virtue of the Gronwall inequality, we get Let where denotes the integer part of real number ; thus, for and , we have
Together with and , we obtain that
In the similar way, combining (15) with the Hölder inequality further leads to
Repeating the previous procedures we then get (9). So the existence and uniqueness have been proved.
In the following, we define the Euler-Maruyama based computational method. The method makes use of the following lemma.
Lemma 3. Given , then is a discrete Markov chain with the one-step transition probability matrix
Given a fixed step size and the one-step transition probability matrix in (20), the discrete Markov chain can be simulated as follows: let , and compute a pseudorandom number from the uniform distribution.
Define where we set as usual. Having computed , we can compute by drawing a uniform pseudo-random number and setting The procedure can be carried out independently to obtain more trajectories.
Define the Euler-Maruyama approximation for (3) by where , for , , which and .
By using the method of Lemma 2, we obtain where , , when .
Lemma 4. If (A1) and (A2) hold, then
Theorem 5. Under (A1) and (A2), for any there exits such that that is, the convergence order of Euler-Maruyama scheme (23) is .
Proof. Let and ; then, and there is a continuous nonnegative function , which is zero outside , such that
Define For any , let
Using the Itô formula, we have
By virtue of condition (A1), the Hölder inequality, and Lemma 4, we deduce that
By the Hölder inequality and (A2), we have
Making use of the Burkhold-Davis-Gundy inequality yields
Moreover, by (8), (9), and (25), we have
Thus, combing (39), and (40) with (41), for any and , we get
Let , and using the Gronwall inequality, we have
Let by , it is easy to see that such that
Noting that for and substituting into (44) yields that Using (46) and the Hölder inequality, further gives that
Repeating the previous procedures, the desired result follows.
In this section, under general conditions, we reveal that the convergence order of Euler-Maruyama scheme for stochastic pantograph equations with Markovian switching and Brownian motion is . In Section 3, we will discuss the convergence rate for stochastic pantograph equation with Markovian switching and pure jumps.
3. Convergence Rate for Markovian Switching and Pure Jumps Case
Let be the Borel -algebra on , and a -finite measure defined on . Let , , be a stationary -Poisson point process on with characteristic measure . Denote by the Poisson counting measure associated with , that is, for . Let be the compensated Poisson measure associated with . In what follows, we further assume that for any .
In this section, we consider the following stochastic pantograph equation with Markovian switching and pure jumps on : with initial data and , .
We assume that (A1) and there exists such that for , , ; (A3) and there exists such that for , and , where such that for some and arbitrary .
From (A3), the jump coefficient may be also highly nonlinear. We define the Euler-Maruyama scheme associated with (49) by where , for , , and , for .
In order to state the main theorem, the following two lemmas are useful.
Lemma 6 (see ). Let and assume that
Then there exists such that
Lemma 7. Let and hold. Then (49) has a unique global solution . Moreover, for any there exists such that
Now we present the main theorem in this section.
Theorem 8. Let and hold. For any and arbitrary , there exists , independent of , such that
Proof. The proof is similar to that of Theorem 5. Set
Using the Itô formula and the Taylor expansion we have that for
By the property of , we deduce that
From (8), (52), and (56), we compute that for any
Applying Lemma 6, Lemma 4, (57), and the Hölder inequality, we obtain that
Together with the Gronwall inequality and taking , we get
For and any , let
It is easy to see that
Noting that for , from (65) we obtain
Then, together with (67) and the Hölder inequality, it further gives that
Repeating the previous procedures, we have
The proof is complete.
This work was supported by the National Natural Science Foundation of China (nos. 61104062 and 61174077), Jiangsu Qing Lan Project, and PAPD.
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