Convergence Rate of Numerical Solutions for Nonlinear Stochastic ...

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 420648, 10 pages http://dx.doi.org/10.1155/2013/420648

Research Article Convergence Rate of Numerical Solutions for Nonlinear Stochastic Pantograph Equations with Markovian Switching and Jumps Zhenyu Lu,1 Tingya Yang,2 Yanhan Hu,3 and Junhao Hu3 1

College of Electrical and Information Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, China 2 Jiangsu Meteorological Observatory, Nanjing 210008, China 3 College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China Correspondence should be addressed to Junhao Hu; [email protected] 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.

1. Introduction 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 EulerMaruyama 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. Gy¨ongy and R´asonyi [8] revealed the convergence rate of Euler approximations for stochastic differential equations whose diffusion coefficient is not Lipschitz but only (1/2 + 𝛼)-H¨older continuous for some 𝛼 > 0. Mao et al. [9] discussed 𝐿1 and 𝐿2 -convergence of the Euler-Maruyama scheme for stochastic differential

equations with Markovian switching under non-Lipschitz coefficients. Wu et al. [10] 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 [11] 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. [12] gave the strong convergence for stochastic pantograph equations under the Lipschitz condition and the linear growth condition. Ronghua et al. [14] 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

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Abstract and Applied Analysis

for Markovian switching and Brownian motion case and also reveal that the convergence order of Euler-Maruyama scheme is 1/2. In Section 3, we show that it is best to use the meansquare convergence for Markovian switching and the pure jump case and that the rate of mean-square convergence is close to 1/2.

The integral version of (3) is given by the following: 𝑡

𝑋 (𝑡) = 𝜉 (𝜃) + ∫ 𝑏 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠)) d𝑠 𝑡0

(5)

𝑡

+ ∫ 𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠)) d𝑊 (𝑠) . 𝑡0

2. Convergence Rate for Markovian Switching and Brownian Motion Case Let (Ω, F, P) be a complete probability space with a filtration {F𝑡 }𝑡≥0 satisfying the usual conditions. Let 𝑊(𝑡) be an 𝑚-dimensional Brownian motion defined on the probability space adapted to the filtration. For integer 𝑛 > 0, let (R𝑛 , ⟨⋅, ⋅⟩, | ⋅ |) be the Euclidean space and ‖𝐴‖ := √ trace (𝐴∗ 𝐴) the Hilbert-Schmidt norm for a matrix 𝐴, where 𝐴∗ is its transpose. Throughout this paper, 𝐶 > 0 denotes a generic constant whose values may change from lines to lines. Let 𝑟(𝑡), 𝑡 ≥ 0 be a right-continuous Markov chain on the probability space taking values in a finite state space 𝑆 = {1, 2, . . . , 𝑁} with the generator Γ = (𝛾𝑖𝑗 )𝑁×𝑁 given by P {𝑟 (𝑡 + 𝛿) = 𝑗 | 𝑟 (𝑡) = 𝑖} = {

𝛾𝑖𝑗 𝛿 + 𝑜 (𝛿) 1 + 𝑟𝑖𝑗 𝛿 + 𝑜 (𝛿)

if 𝑖 ≠ 𝑗, if 𝑖 = 𝑗, (1)

where 𝛿 > 0. Here 𝛾𝑖𝑗 > 0 is the transition rate from 𝑖 to 𝑗 if 𝑖 ≠ 𝑗 while 𝛾𝑖𝑖 = −∑ 𝛾𝑖𝑗 .

(2)

𝑗 ≠ 𝑖

We assume that the Markov chain 𝑟(⋅) is independent of the Brownian motion 𝑊(⋅). It is well known that almost every sample path of 𝑟(⋅) is a right continuous step function with finite number of sample jumps in any finite subinterval of R+ := [0, +∞). For fixed 𝑇 > 0, we consider the stochastic pantograph equation with Markovian switching of the form d𝑋 (𝑡) = 𝑏 (𝑋 (𝑡) , 𝑋 (𝑞𝑡) , 𝑟 (𝑡)) d𝑡 + 𝜎 (𝑋 (𝑡) , 𝑋 (𝑞𝑡) , 𝑟 (𝑡)) d𝑊 (𝑡) ,

𝑡 ∈ [𝑡0 , 𝑇] , (3)

with initial data 𝑋(𝜃) = 𝜉(𝜃), 𝑟(𝜃) = 𝑟0 , 𝜃 ∈ [𝑞𝑡0 , 𝑡0 ], 0 < 𝑡0 , 0 < 𝑞 < 1. 𝑟(𝑡) is a Markov chain. On the time interval [𝑡0 , 𝑇], let 𝑞 < 𝜖 < min{1, ((𝑇 + 1)/𝑇)𝑞}, and we define the partition 0 < 𝑡0
0 such that 󵄩󵄩󵄩𝜎 (𝑥1 , 𝑦1 , 𝑗) − 𝜎 (𝑥2 , 𝑦2 , 𝑗)󵄩󵄩󵄩 󵄩 󵄩 (7) 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ≤ 𝐿 2 󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨 + 𝑉2 (𝑦1 , 𝑦2 ) 󵄨󵄨󵄨𝑦1 − 𝑦2 󵄨󵄨󵄨 for 𝑥𝑖 , 𝑦𝑖 ∈ R𝑛 , 𝑖 = 1, 2, 𝑗 ∈ 𝑆, where 𝑉𝑖 : R𝑛 × R𝑛 → R+ such that 󵄨 󵄨𝑞 𝑉𝑖 (𝑥, 𝑦) ≤ 𝐾𝑖 (1 + |𝑥|𝑞𝑖 + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 𝑖 ) ,

(8)

for some 𝐾𝑖 > 0, 𝑞𝑖 ≥ 1 and arbitrary 𝑥, 𝑦 ∈ R𝑛 . 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, 𝑏 and 𝜎 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 𝑏 ([𝑞𝑡0 , 𝑡0 ]; R𝑛 ) and 𝑟(0) = 𝑟0 ∈ 𝑆, 𝑋(𝑡) is a initial data 𝜉 ∈ 𝐶F 0 unique global strong solution of (3). Moreover, for any 𝑝 ≥ 2 there exists 𝐶 > 0 such that E ( sup |𝑋 (𝑡)|𝑝 ) ≤ 𝐶.

(9)

𝑡0 ≤𝑡≤𝑇

Proof. From (A1) and (A2), 𝑏 and 𝜎 are locally Lipschitzian. So, (3) has a unique local solution [3]. In order to verify that (3) has a unique global solution on time interval [𝑡0 , 𝑇], it is sufficient to show that E ( sup |𝑋 (𝑡)|𝑝 ) ≤ 𝐶, 𝑡0 ≤𝑡≤𝑇

(4)

𝑖 = 1, 2

𝑝 ≥ 2.

(10)

From (A1), (A2), and (8), we can obtain 󵄨 󵄨 󵄨 󵄨𝑞 +1 󵄨 󵄨󵄨 󵄨󵄨𝑏 (𝑥, 𝑦, 𝑖)󵄨󵄨󵄨 ≤ 𝐶 (1 + |𝑥| + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 1 ) , 󵄨 󵄨 󵄨 󵄨𝑞 +1 󵄩 󵄩󵄩 󵄩󵄩𝜎 (𝑥, 𝑦, 𝑖)󵄩󵄩󵄩 ≤ 𝐶 (1 + |𝑥| + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 2 ) ,

𝑥, 𝑦 ∈ R𝑛 , (11) 𝑥, 𝑦 ∈ R𝑛 . (12)

Abstract and Applied Analysis

3

Substituting (11) and (12) into (5) and by the H¨older inequality and the Burkhold-Davis-Gundy inequality, we have that for any 𝑝 ≥ 2 and 𝑡 ∈ [𝑡0 , 𝑇]

𝑏 ([𝑞𝑡0 , 𝑡0 ]; R𝑛 ) and 𝜖 < 1, we obtain Together with 𝜉 ∈ 𝐶F 0 that

E(

𝑝

E ( sup |𝑋 (𝑠)| )

sup

𝑡0 ≤𝑠≤(𝑡0 /𝑞)𝜖

|𝑋 (𝑠)|𝑝1 )

𝑡0 ≤𝑠≤𝑡

≤ 𝐶 {1 + E ∫

󵄨󵄨𝑝 󵄨󵄨 󵄨 󵄨 ≤ 3𝑝−1 {󵄨󵄨󵄨󵄨𝜉 (𝜃)󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨

(𝑡0 /𝑞)𝜖

𝑡0

󵄨󵄨𝑝 󵄨󵄨 𝑠 󵄨 󵄨 + E ( sup 󵄨󵄨󵄨󵄨∫ 𝑏 (𝑋 (𝛼) , 𝑋 (𝑞𝛼) , 𝑟 (𝛼)) d𝛼󵄨󵄨󵄨󵄨 ) 󵄨󵄨 𝑡0 ≤𝑠≤𝑡󵄨󵄨 𝑡0 󵄨󵄨𝑝 󵄨󵄨 𝑠 󵄨 󵄨 + E (sup 󵄨󵄨󵄨󵄨∫ 𝜎 (𝑋 (𝛼) , 𝑋 (𝑞𝛼) , 𝑟 (𝛼)) d𝑊 (𝛼)󵄨󵄨󵄨󵄨 )} 󵄨󵄨 𝑡0 ≤𝑠≤𝑡󵄨󵄨 𝑡0

󵄨𝑝 𝛽 󵄨󵄨 󵄨󵄨𝑋 (𝑞𝑠)󵄨󵄨󵄨 1 d𝑠}

(18)

𝑡0

≤ 𝐶 {1 + E ∫ |𝑋 (𝑠)|𝑝1 𝛽 d𝑠} 𝑞𝑡0

≤ 𝐶. In the similar way, combining (15) with the H¨older inequality further leads to

𝑡

󵄨𝑝 󵄨 ≤ 𝐶 {1 + E ∫ (󵄨󵄨󵄨𝑏 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠))󵄨󵄨󵄨 𝑡

E(

0

󵄩𝑝 󵄩 +󵄩󵄩󵄩𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠))󵄩󵄩󵄩 ) d𝑠}

sup 𝑡0 ≤𝑠≤(𝑡0

/𝑞2 )𝜖2

|𝑋 (𝑠)|𝑝2 )

≤ 𝐶 {1 + E ∫

𝑡0

𝑡

≤ 𝐶 {1 + E ∫ |𝑋 (𝑠)|𝑝 d𝑠 𝑡0

≤ 𝐶 {1 + ∫

𝑡

(13)

E ( sup |𝑋 (𝑠)|𝑝 ) 𝑡0 ≤𝑠≤𝑡

(14)

𝑡

󵄨𝑝𝛽 󵄨 ≤ 𝐶 {1 + E ∫ |𝑋 (𝑠)| d𝑠 + E ∫ 󵄨󵄨󵄨𝑋 (𝑞𝑠)󵄨󵄨󵄨 d𝑠} . 𝑡 𝑡 𝑝

0

By virtue of the Gronwall inequality, we get

In the following, we define the Euler-Maruyama based computational method. The method makes use of the following lemma.

𝑃 (Δ) = (𝑃𝑖,𝑗 (Δ))𝑁×𝑁 = 𝑒ΔΓ .

𝑡

󵄨𝑝𝛽 󵄨 E ( sup |𝑋 (𝑠)| ) ≤ 𝐶 {1 + E ∫ 󵄨󵄨󵄨𝑋 (𝑞𝑠)󵄨󵄨󵄨 d𝑠} . 𝑡0

(15)

𝑇 ] + 2 − 𝑖) 𝑝𝛽[log𝜖/𝑞 (𝑇/𝑡0 )]+1−𝑖 , 𝑡0 𝑇 𝑖 = 1, 2, . . . , [log𝜖/𝑞 ] + 1, 𝑡0

(16)

𝑝[log𝜖/𝑞 (𝑇/𝑡0 )]+1 = 𝑝, 𝑇 ]. 𝑡0

Define 𝑖−1

where [𝑎] denotes the integer part of real number 𝑎; thus, for 𝛽 ≥ 1 and 𝑝 ≥ 2, we have

𝑖 = 1, 2, . . . , [log𝜖/𝑞

(20)

Given a fixed step size Δ > 0 and the one-step transition probability matrix 𝑃(Δ) in (20), the discrete Markov chain {𝑟(𝑘Δ), 𝑘 = 0, 1, 2, . . .} can be simulated as follows: let 𝑟(0) = 𝑖0 , and compute a pseudorandom number 𝜉1 from the uniform (0, 1) distribution.

Let

𝑝𝑖+1 𝛽 < 𝑝𝑖 ,

d𝑠}

Lemma 3. Given Δ > 0, then {𝑟(𝑘Δ), 𝑘 = 0, 1, 2, . . .} is a discrete Markov chain with the one-step transition probability matrix

0

𝑝𝑖 := ([log𝜖/𝑞

𝑝2 𝛽/𝑝1

(E|𝑋 (𝑠)|𝑝1 )

(19)

Repeating the previous procedures we then get (9). So the existence and uniqueness have been proved.

Let 𝛽 := (𝑞1 + 1) ∨ (𝑞2 + 1); then

𝑡0 ≤𝑠≤𝑡

󵄨𝑝 𝛽 󵄨󵄨 󵄨󵄨𝑋 (𝑞𝑠)󵄨󵄨󵄨 2 d𝑠}

≤ 𝐶.

𝑡0

𝑝

(𝑡0 /𝑞)𝜖

𝑡0

󵄨𝑝(𝑞 +1) 󵄨 󵄨𝑝(𝑞 +1) 󵄨 +E ∫ (󵄨󵄨󵄨𝑋 (𝑞𝑠)󵄨󵄨󵄨 1 + 󵄨󵄨󵄨𝑋 (𝑞𝑠)󵄨󵄨󵄨 2 ) d𝑠} .

𝑡

(𝑡0 /𝑞2 )𝜖2

(17)

{ { 𝑖, 𝑖 ∈ 𝑆 − {𝑁} such that∑𝑃𝑟(0),𝑗 (Δ) ≤ 𝜉1 { { { 𝑗=1 { { { 𝑖 { { < ∑𝑃𝑟(0),𝑗 (Δ) , 𝑟 (Δ) = { { { 𝑗=1 { { { 𝑁−1 { { { { 𝑁, ∑ 𝑃𝑟(0),𝑗 (Δ) ≤ 𝜉1 , 𝑗=1 { (21)

4

Abstract and Applied Analysis

where we set ∑0𝑗=1 𝑃𝑟(0),𝑗 (Δ) = 0 as usual. Having computed 𝑟(0), 𝑟(Δ), . . . , 𝑟(𝑘Δ), we can compute 𝑟((𝑘 + 1)Δ) by drawing a uniform (0, 1) pseudo-random number 𝜉𝑘+1 and setting

Proof. Let 𝑛 = [log𝜖/𝑞 (𝑇/𝑡0 )] + 1, then 𝑇 󵄨 E ∫ 󵄨󵄨󵄨󵄨𝑏 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠)) 𝑡 0

𝑖−1

{ { 𝑖, 𝑖 ∈ 𝑆 − {𝑁} such that ∑𝑃𝑟(𝑘Δ),𝑗 (Δ) { { { 𝑗=1 { { { 𝑖 { { ≤ 𝜉𝑘+1 0; then∫𝜀/𝛿 (1/𝑥)d𝑥 = ln 𝑥|𝜀𝜀/𝛿 = ln 𝛿, and there is a continuous nonnegative function 𝜓𝛿𝜀 (𝑥)(𝑥 ≥ 0), which is zero outside [𝜀/𝛿, 𝜀], such that 𝜀

2 , 𝜓𝛿𝜀 (𝑥) ≤ 𝑥 ln 𝛿

∫ 𝜓𝛿𝜀 (𝑥) d𝑥 = 1, 𝜀/𝛿

𝑥 > 0.

(35)

𝑥

𝑦

0

0

𝑉𝛿𝜀 (𝑥) := 𝜙𝛿𝜀 (|𝑥|) ,

𝑥 > 0,

(36)

𝑥 ∈ R𝑛 .

𝑍 (𝑡) := 𝑋 (𝑡) − 𝑌 (𝑡) ,

̃ (𝑡) := (𝑋 (𝑡) , 𝑌 (𝑡)) ∈ R2𝑛 . 𝑍

0

󵄨𝑝 −𝑏 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))󵄨󵄨󵄨󵄨 d𝑠

󵄨𝑝 −𝑏 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , (𝑠))󵄨󵄨󵄨󵄨

󵄨 + 󵄨󵄨󵄨󵄨𝑏 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠)) 󵄨𝑝 −𝑏 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))󵄨󵄨󵄨󵄨 ) d𝑠

For any 𝑡 ∈ [𝑡0 , 𝑇], let

𝑍 (𝑡) := 𝑌 (𝑡) − 𝑌 (𝑡) ,

𝑡 󵄨 E ∫ 󵄨󵄨󵄨󵄨𝑏 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (s)) 𝑡

𝑡 󵄨󵄨 ≤ 𝐶Δ𝑝−1 ∫ E ( 󵄨󵄨󵄨𝑏 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠)) 󵄨 𝑡0

Define 𝜙𝛿𝜀 (𝑥) := ∫ ∫ 𝜓𝛿𝜀 (𝑧) d𝑧d𝑦,

𝑝−1

≤ (𝑡 − 𝑡0 )

(37)

𝑡 { 1/2 2𝑝 ̃ ≤ 𝐶Δ𝑝−1 ∫ {E|𝑍 (𝑠)|𝑝 + (E𝑉1 (𝑍 (𝑞𝑠))) 𝑡0 {

󵄨𝑝 󵄨 󵄨2𝑝 1/2 󵄨 × (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 ) + E󵄨󵄨󵄨󵄨𝑍 (𝑠)󵄨󵄨󵄨󵄨

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Abstract and Applied Analysis 1/2

̃ (𝑞𝑠))) + (E𝑉1 (𝑍 2𝑝

Making use of the Burkhold-Davis-Gundy inequality yields

󵄨 󵄨2𝑝 1/2 } ×(E󵄨󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨󵄨 ) } d𝑠 + 𝐶Δ𝑝 . }

󵄨𝑝 󵄨 E ( sup 󵄨󵄨󵄨𝐼3 (𝑠)󵄨󵄨󵄨 ) (39)

𝑡0 ≤𝑠≤𝑡

𝑡 󵄩 ≤ 𝐶Δ𝑝/2−1 E ∫ 󵄩󵄩󵄩󵄩𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠)) 𝑡 0

By the H¨older inequality and (A2), we have

󵄩𝑝 −𝜎 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))󵄩󵄩󵄩󵄩 d𝑠 𝑡 󵄩 󵄩 ≤ 𝐶Δ𝑝/2−1 E ∫ (󵄩󵄩󵄩𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠)) 𝑡0 󵄩

󵄨𝑝 󵄨 E ( sup 󵄨󵄨󵄨𝐼2 (𝑠)󵄨󵄨󵄨 ) 𝑡0 ≤𝑠≤𝑡

󵄩𝑝 −𝜎 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))󵄩󵄩󵄩󵄩

1 𝑝−1 ≤ (𝑡 − 𝑡0 ) E 2

󵄩 + 󵄩󵄩󵄩󵄩𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠) )

𝑡

󵄨 × ∫ 󵄨󵄨󵄨󵄨trace { (𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠)) 𝑡

󵄩𝑝 −𝜎 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠)) 󵄩󵄩󵄩󵄩 ) d𝑠

0

∗

−𝜎 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠)))

𝑡

̃ (𝑞𝑠))) ≤ 𝐶Δ𝑝/2−1 ∫ {E|𝑍 (𝑠)|𝑝 + (E𝑉2 (𝑍 2𝑝

1/2

𝑡0

× (𝑉𝛿𝜀 )𝑥𝑥 (𝑍 (𝑠))

󵄨2𝑝 1/2 󵄨 × (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 )

× (𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠))

1/2 󵄨𝑝 󵄨 2𝑝 ̃ + E󵄨󵄨󵄨󵄨𝑍 (𝑠)󵄨󵄨󵄨󵄨 + (E𝑉2 (𝑍 (𝑞𝑠)))

󵄨𝑝 −𝜎 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))) }󵄨󵄨󵄨󵄨 d𝑠

󵄨 󵄨2𝑝 ×(E󵄨󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨󵄨 )

𝑡

󵄩 󵄩 ≤ 𝐶Δ𝑝−1 ∫ E { 󵄩󵄩󵄩(𝑉𝛿𝜖 )𝑥𝑥 (𝑍 (𝑠))󵄩󵄩󵄩 𝑡0

1/2

} d𝑠 + 𝐶Δ𝑝/2 .

󵄩 × 󵄩󵄩󵄩󵄩𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠))

(41)

󵄩2 −𝜎 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))󵄩󵄩󵄩󵄩 } d𝑠

Moreover, by (8), (9), and (25), we have

𝑝

≤ 𝐶Δ𝑝−1 E ∫

𝑡

𝑡0

̃ (𝑞𝑠)) ∨ E𝑉 (𝑍 ̃ (𝑞𝑠)) ≤ 𝐶, E𝑉1 (𝑍 2 2𝑝

1 |𝑍 (𝑠)|𝑝

Thus, combing (39), and (40) with (41), for any 𝑡 ∈ [𝑡0 , 𝑇] and 𝑝 ≥ 2, we get

󵄩2𝑝 −𝜎 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))󵄩󵄩󵄩󵄩 󵄩 + 󵄩󵄩󵄩󵄩𝜎(𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))

E ( sup |𝑍 (𝑠)|𝑝 ) 𝑡0 ≤𝑠≤𝑡

󵄩2𝑝 −𝜎 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))󵄩󵄩󵄩󵄩 } ≤ 𝐶Δ

𝑡

(42)

󵄨𝑝 󵄨 E󵄨󵄨󵄨󵄨𝑍 (𝑡)󵄨󵄨󵄨󵄨 ≤ 𝐶Δ𝑝/2 .

󵄩 × { 󵄩󵄩󵄩󵄩𝜎 (𝑋 (𝑠) , 𝑋 (𝑞𝑠) , 𝑟 (𝑠))

𝑝−1

4𝑝

𝑝

≤ 2𝑝−1 {𝜀𝑝 + E ( sup 𝑉𝛿𝜀 (𝑍 (𝑠)))}

𝑝

𝑡0 ≤𝑠≤𝑡

∫ {E|𝑍 (𝑠)| 𝑡0

1/2 1 4𝑝 ̃ 󵄨4𝑝 1/2 󵄨 + 𝑝 (E𝑉2 (𝑍 (𝑞𝑠))) (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 ) 𝜀 1/2 1 󵄨 󵄨2𝑝 1 4𝑝 ̃ + 𝑝 E󵄨󵄨󵄨󵄨𝑍 (𝑠)󵄨󵄨󵄨󵄨 + 𝑝 (E𝑉2 (𝑍 (𝑞𝑠))) 𝜀 𝜀

𝐶Δ𝑝 󵄨4𝑝 1/2 󵄨 × (E󵄨󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨󵄨 ) } d𝑠 + 𝑝 . 𝜀

≤ 𝐶 {𝜀𝑝 + Δ3𝑝/2−1 + Δ𝑝 + 𝑡

𝑡

0

0

󵄨2𝑝 󵄨 + Δ𝑝/2−1 {∫ E|𝑍 (𝑠)|𝑝 d𝑠 + ∫ (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 ) 𝑡 𝑡 +

(40)

Δ2𝑝−1 Δ𝑝 + 𝑝 + Δ𝑝−1 + Δ𝑝/2 𝜀𝑝 𝜀 1/2

d𝑠

1 𝑡 󵄨󵄨 󵄨4𝑝 1/2 ∫ (E󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 ) d𝑠}} . 𝑝 𝜀 𝑡0 (43)

Abstract and Applied Analysis

7

Let 𝜀 = Δ1/2 , and using the Gronwall inequality, we have

≤ 𝐶 {Δ𝑝2 /2 + Δ𝑝2 /2−1 ∫

(𝑡0 /𝑞)𝜖

𝑡0

E ( sup |𝑍 (𝑠)|𝑝 )

(𝑡0 /𝑞)𝜖

+Δ−1 ∫

𝑡0 ≤𝑠≤𝑡

𝑡0

𝑡

󵄨2𝑝 1/2 󵄨 ≤ 𝐶 {Δ𝑝/2 + Δ(𝑝/2)−1 ∫ (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 ) d𝑠 𝑡

(44)

0

(E|𝑍 (𝑠)|𝑝1 ) 2𝑝2 /𝑝1

(E|𝑍 (𝑠)|𝑝1 )

𝑝2 /𝑝1

d𝑠

d𝑠}

≤ 𝐶Δ𝑝2 /2 . (48)

𝑡

󵄨4𝑝 1/2 󵄨 +Δ ∫ (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 ) d𝑠} . −1

Repeating the previous procedures, the desired result follows.

𝑡0

Let 𝑝𝑖 := ([log𝜖/𝑞

𝑇 ] + 2 − 𝑖) 𝑝4[log𝜖/𝑞 (𝑇/𝑡0 ) ]+1−𝑖 , 𝑡0

(45)

𝑇 𝑖 = 1, 2, . . . , [log𝜖/𝑞 ] + 1; 𝑡0 by 𝑝 ≥ 2, it is easy to see that 𝑝𝑖 ≥ 2 such that 4𝑝𝑖+1 < 𝑝𝑖 ,

3. Convergence Rate for Markovian Switching and Pure Jumps Case

𝑝[log𝜖/𝑞 (𝑇/𝑡0 )]+1 = 𝑝,

𝑖 = 1, 2, . . . , [log𝜖/𝑞

(46)

𝑇 ]. 𝑡0

Noting that 𝑍(𝑠) = 0 for 𝑠 ∈ [𝑞𝑡0 , 𝑡0 ] and substituting 𝜖 < 1 into (44) yields that E(

sup |𝑍 (𝑠)|𝑝1 ) 𝑡0 ≤𝑠≤(𝑡0 /𝑞)𝜖

≤ 𝐶 {Δ𝑝1 /2 + Δ𝑝1 /2−1 ∫

(𝑡0 /𝑞)𝜖

𝑡0

+Δ−1 ∫

(𝑡0 /𝑞)𝜖

𝑡0

≤ 𝐶 {Δ

𝑝1 /2

+Δ

󵄨2𝑝 󵄨 (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 1 )

1/2

d𝑠

𝑝1 /2−1

∫ (E|𝑍 (𝑠)|

)

𝑞𝑡0

𝑡0

+Δ−1 ∫ (E|𝑍 (𝑠)|4𝑝1 ) 𝑞𝑡0

1/2

(47)

1/2

2𝑝1

d𝑠

𝑈

𝑡 ∈ [𝑡0 , 𝑇] (49)

(A1) 𝑏 : R𝑛 × R𝑛 × 𝑆 → R𝑛 and there exists 𝐿 1 > 0 such that

Using (46) and the H¨older inequality, further gives that sup |𝑍 (𝑠)|𝑝2 ) 2 2 𝑡0 ≤𝑠≤(𝑡0 /𝑞 )𝜖

E(

≤ 𝐶 {Δ

̃ (d𝑡, d𝑢) , + ∫ ℎ (𝑋 (𝑡) , 𝑋 (𝑞𝑡) , 𝑢) 𝑁

with initial data 𝑋(𝜃) = 𝜉(𝜃) and 𝑟(𝜃) = 𝑟0 , 𝜃 ∈ [𝑞𝑡0 , 𝑡0 ]. We assume that

d𝑠}

≤ 𝐶Δ𝑝1 /2 .

𝑝2 /2

Let B(R) be the Borel 𝜎-algebra on R, and 𝜆(d𝑥) a 𝜎finite measure defined on B(R). Let 𝑝 = (𝑝(𝑡)), 𝑡 ∈ 𝐷𝑝 , be a stationary F𝑡 -Poisson point process on R with characteristic measure 𝜆(⋅). Denote by 𝑁(d𝑡, d𝑢) the Poisson counting measure associated with 𝑝, that is, 𝑁(𝑡, 𝑈) = ̃ d𝑢) := 𝑁(d𝑡, d𝑢)− ∑𝑠∈𝐷𝑝 ,𝑠≤𝑡 𝐼𝑈(𝑝(𝑠)) for 𝑈 ∈ B(R). Let 𝑁(d𝑡, d𝑡𝜆(d𝑢) be the compensated Poisson measure associated with 𝑁(d𝑡, d𝑢). In what follows, we further assume that ∫𝑈 |𝑢|𝑝 𝜆(𝑢) < ∞ for any 𝑝 ≥ 2. In this section, we consider the following stochastic pantograph equation with Markovian switching and pure jumps on R𝑛 : d𝑋 (𝑡) = 𝑏 (𝑋 (𝑡) , 𝑋 (𝑞𝑡) , 𝑟 (𝑡)) d𝑡

󵄨4𝑝 1/2 󵄨 (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 1 ) d𝑠} 𝑡0

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 1/2. In Section 3, we will discuss the convergence rate for stochastic pantograph equation with Markovian switching and pure jumps.

+Δ

𝑝2 /2−1

(𝑡0 /𝑞2 )𝜖2

+Δ−1 ∫

𝑡0

󵄨 󵄨󵄨 󵄨󵄨𝑏 (𝑥1 , 𝑦1 , 𝑗) − 𝑏 (𝑥2 , 𝑦2 , 𝑗)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 𝐿 1 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨 + 𝑉1 (𝑦1 , 𝑦2 ) 󵄨󵄨󵄨𝑦1 − 𝑦2 󵄨󵄨󵄨

(50)

for 𝑥𝑖 , 𝑦𝑖 ∈ R𝑛 , 𝑖 = 1, 2, 𝑗 ∈ 𝑆;

∫

(𝑡0 /𝑞2 )𝜖2

𝑡0

󵄨2𝑝 1/2 󵄨 (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 2 ) d𝑠

󵄨4𝑝 󵄨 (E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 2 )

1/2

d𝑠}

(A3) ℎ : R𝑛 × R𝑛 × 𝑈 → R𝑛 and there exists 𝐿 3 > 0 such that 󵄨 󵄨󵄨 󵄨󵄨ℎ (𝑥1 , 𝑦1 , 𝑢) − ℎ (𝑥2 , 𝑦2 , 𝑢)󵄨󵄨󵄨

󵄨 󵄨 󵄨 󵄨 ≤ (𝐿 3 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨 + 𝑉3 (𝑦1 , 𝑦2 ) 󵄨󵄨󵄨𝑦1 − 𝑦2 󵄨󵄨󵄨) |𝑢|

(51)

8

Abstract and Applied Analysis for 𝑥𝑖 , 𝑦𝑖 ∈ R𝑛 , 𝑖 = 1, 2, and 𝑢 ∈ 𝑈, where 𝑉3 : R𝑛 × R𝑛 → R+ such that 󵄨 󵄨𝑞 𝑉3 (𝑥, 𝑦) ≤ 𝐾3 (1 + |𝑥|𝑞3 + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 3 )

Proof. The proof is similar to that of Theorem 5. Set 𝑍 (𝑡) := 𝑋 (𝑡) − 𝑌 (𝑡) ,

(52)

𝑍 (𝑡) := 𝑌 (𝑡) − 𝑌 (𝑡) ,

for some 𝐾3 > 0, 𝑞3 ≥ 1 and arbitrary 𝑥, 𝑦 ∈ R𝑛 . From (A3), the jump coefficient may be also highly nonlinear. We define the Euler-Maruyama scheme associated with (49) by

𝑡 ∈ [𝑡0 , 𝑇] .

Define Γ1 (𝑡) := 𝑏 (𝑋 (𝑡) , 𝑋 (𝑞𝑡) , 𝑟 (𝑡)) − 𝑏 (𝑌 (𝑡) , 𝑌 (𝑞𝑡) , 𝑟 (𝑡)) ,

d𝑌 (𝑡) = 𝑏 (𝑌 (𝑡) , 𝑌 (𝑞𝑡) , 𝑟 (𝑡)) d𝑡 ̃ (d𝑡, d𝑢) , + ∫ ℎ (𝑌 (𝑡) , 𝑌 (𝑞𝑡) , 𝑢) 𝑁

(53)

𝑈

where 𝑌(𝑡) := 𝑌(𝑡𝑖 ), 𝑟(𝑡) := 𝑟(𝑡𝑖 ) for 𝑡 ∈ [𝑡𝑖 , 𝑡𝑖+1 ), 𝑖 = 0, 1, . . . , 𝑛 − 1, and 𝑌(𝜃) = 𝜉(𝜃), 𝑟(𝜃) = 𝑟0 , for 𝜃 ∈ [𝑞𝑡0 , 𝑡0 ]. In order to state the main theorem, the following two lemmas are useful. Lemma 6 (see [16]). Let Φ : R+ × 𝑈 → R𝑛 and assume that

Γ2 (𝑡, 𝑢) := ℎ (𝑋 (𝑡) , 𝑋 (𝑞𝑡) , 𝑢) − ℎ (𝑌 (𝑡) , 𝑌 (𝑞𝑡) , 𝑢) . (60) Using the Itˆo formula and the Taylor expansion we have that for 𝑡 ∈ [𝑡0 , 𝑇] 𝑡

𝑉𝛿𝜀 (𝑍 (𝑡)) = ∫ ⟨(𝑉𝛿𝜀 )𝑥 (𝑍 (𝑠)) , Γ1 (𝑠)⟩ d𝑠 𝑡0

𝑡

𝑡

∫ ∫ E|Φ (𝑠, 𝑢)|𝑝 𝜆 (d𝑢) d𝑠 < ∞, 𝑡0

̃ (𝑡) := (𝑋 (𝑡) , 𝑌 (𝑡)) ∈ R2𝑛 , 𝑍

(59)

𝑡0 > 0, 𝑝 ≥ 2.

𝑈

(54)

+ ∫ ∫ {𝑉𝛿𝜀 (𝑍 (𝑠) + Γ2 (𝑠, 𝑢)) 𝑡0

𝑈

− 𝑉𝛿𝜀 (𝑍 (𝑠))

Then there exists 𝐷(𝑝) > 0 such that

− ⟨(𝑉𝛿𝜀 )𝑥 (𝑍 (𝑠)) , Γ2 (𝑠, 𝑢)⟩}

󵄨󵄨 𝑠 󵄨󵄨𝑝 󵄨 ̃ (d𝑢, d𝑠)󵄨󵄨󵄨󵄨 ) E ( sup 󵄨󵄨󵄨󵄨∫ ∫ Φ (𝑟, 𝑢) 𝑁 󵄨󵄨 𝑡0 ≤𝑠≤𝑡󵄨󵄨 𝑡0 𝑈 󵄨 𝑡

× 𝜆 (d𝑢) d𝑠

≤ 𝐷 (𝑝) {E(∫ ∫ |Φ (𝑠, 𝑢)| 𝜆 (d𝑢) d𝑠) 𝑡0

𝑡

𝑝/2

2

𝑈

(55)

+ ∫ ∫ {𝑉𝛿𝜀 (𝑍 (𝑠) + Γ2 (𝑠, 𝑢)) 𝑡0

𝑈

̃ (d𝑢, d𝑠) −𝑉𝛿𝜀 (𝑍 (𝑠))} 𝑁

𝑡

+E ∫ ∫ |Φ (𝑠, u)|𝑝 𝜆 (d𝑢) d𝑠} . 𝑡0

Lemma 7. Let (𝐴1) and (𝐴3) hold. Then (49) has a unique global solution (𝑋(𝑡))𝑡∈[𝑡0 ,𝑇] . Moreover, for any 𝑝 ≥ 2 there exists 𝐶 > 0 such that E ( sup |𝑋 (𝑡)|𝑝 ) ∨ E ( sup |𝑌 (𝑡)|𝑝 ) ≤ 𝐶,

(56)

󵄨𝑝 󵄨 E󵄨󵄨󵄨󵄨𝑌 (𝑡) − 𝑌 (𝑡)󵄨󵄨󵄨󵄨 ≤ CΔ.

(57)

𝑡0 ≤𝑡≤𝑇

𝑡0 ≤𝑡≤𝑇

(61)

𝑡

= ∫ ⟨(𝑉𝛿𝜀 )𝑥 (𝑍 (𝑠)) , Γ1 (𝑠)⟩ d𝑠

𝑈

Proof. The proof is very similar to that of Lemma 2 and (25).

𝑡0

𝑡

1

+ ∫ ∫ {∫ ⟨(𝑉𝛿𝜀 )𝑥 (𝑍 (𝑠) + 𝜃Γ2 (𝑠, 𝑢)) 𝑡0

0

𝑈

−(𝑉𝛿𝜀 )𝑥 (𝑍 (𝑠)) , Γ2 (𝑠, 𝑢)⟩ 𝑑𝜃} × 𝜆 (d𝑢) d𝑠 𝑡

1

+ ∫ ∫ {∫ ⟨(𝑉𝛿𝜀 )𝑥 (𝑍 (𝑠) + 𝜃Γ2 (𝑠, 𝑢)) , 𝑡0

𝑈

0

̃ (d𝑢, d𝑠) . Γ2 (𝑠, 𝑢)⟩ d𝜃} 𝑁

Now we present the main theorem in this section. Theorem 8. Let (𝐴1) and (𝐴3) hold. For any 𝑝 ≥ 2 and arbitrary 𝜃, 𝛼 ∈ (0, 1), there exists 𝐶 > 0, independent of Δ, such that E ( sup |𝑋 (𝑡) − 𝑌 (𝑡)|𝑝 ) ≤ 𝐶Δ(1+𝜃) 𝑡0 ≤𝑡≤𝑇

1/[log𝜖/𝑞 (𝑇/𝑡0 )](1+𝛼)

. (58)

By the property of 𝑉𝛿𝜀 (𝑥), we deduce that |𝑍 (𝑡)| ≤ 𝜀 + 𝑉𝛿𝜀 (𝑍 (𝑡)) 𝑡

𝑡

0

0

󵄨 󵄨 󵄨 󵄨 ≤ 𝜀 + ∫ 󵄨󵄨󵄨Γ1 (𝑠)󵄨󵄨󵄨 d𝑠 + 2 ∫ ∫ 󵄨󵄨󵄨Γ2 (𝑠, 𝑢)󵄨󵄨󵄨 𝜆 (d𝑢) d𝑠 𝑡 𝑡 𝑈

Abstract and Applied Analysis 𝑡

9 𝑝 ̃ 󵄨𝑝 󵄨 (𝑞𝑠)) 󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 ) + E (𝑉3 (𝑍

1

+ ∫ ∫ {∫ ⟨(𝑉𝛿𝜀 )𝑥 (𝑍 (𝑠) +𝜃Γ2 (𝑠, 𝑢)) , 𝑡0

𝑈

0

󵄨𝑝 󵄨 𝑝 ̃ +E (𝑉3 (𝑍 (𝑞𝑠)) 󵄨󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨󵄨 ) } d𝑠}

̃ (d𝑢, d𝑠) , Γ2 (𝑠, 𝑢)⟩ d𝜃} 𝑁 𝑡 ∈ [𝑡0 , 𝑇] . (62)

𝑡

𝑡

0

0

󵄨𝑝 󵄨 ≤ 𝐶 {𝜀𝑝 + Δ + ∫ E|𝑍 (𝑠)|𝑝 d𝑠 + ∫ E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 d𝑠} . 𝑡 𝑡 (64)

From (8), (52), and (56), we compute that for any 𝑝 ≥ 2 ̃ (𝑞𝑠))) ∨ E ( sup 𝑉 (𝑍 ̃ (𝑞𝑠))) ≤ 𝐶. (63) E ( sup 𝑉1 (𝑍 3 𝑝

𝑝

𝑡0 ≤𝑡≤𝑇

𝑡0 ≤𝑡≤𝑇

Applying Lemma 6, Lemma 4, (57), and the H¨older inequality, we obtain that

Together with the Gronwall inequality and taking 𝜀 = Δ1/𝑝 , we get 𝑡

󵄨𝑝 󵄨 E ( sup |𝑍 (𝑠)|𝑝 ) ≤ 𝐶 {Δ + ∫ E󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 d𝑠} . 𝑡 𝑡0 ≤𝑠≤𝑡

𝑝

E ( sup |𝑍 (𝑠)| )

(65)

0

𝑡0 ≤𝑠≤𝑡

For 𝜃 ∈ (0, 1) and any 𝛼 ∈ (0, 1), let

𝑝

≤ 2𝑝−1 {𝜀𝑝 + E ( sup 𝑉𝛿𝜀 (𝑍 (𝑠)))} 𝑡0 ≤𝑠≤𝑡

𝑡

𝑝𝑖 := 𝑝 (1 + 𝜃)([log𝜖/𝑞 (𝑇/𝑡0 )]+1−𝑖)(1+𝛼) ,

󵄨󵄨𝑝

󵄨 ≤ 𝐶 {𝜀𝑝 + ∫ E󵄨󵄨󵄨Γ1 (𝑠)󵄨󵄨 d𝑠 𝑡

𝑖 = 1, 2, . . . , [log𝜖/𝑞

0

𝑡

󵄨𝑝 󵄨 + ∫ ∫ E󵄨󵄨󵄨Γ2 (𝑠, 𝑢)󵄨󵄨󵄨 𝜆 (d𝑢) d𝑠 𝑡0

𝑈

𝑇 ] + 1. 𝑡0

(66)

It is easy to see that

𝑡

󵄨2 󵄨 +E(∫ ∫ 󵄨󵄨󵄨Γ2 (𝑠, 𝑢)󵄨󵄨󵄨 𝜆 (d𝑢) d𝑠) 𝑡 𝑈

𝑝/2

}

(1 + 𝜃) 𝑝𝑖+1 < 𝑝𝑖 ,

0

𝑝[log𝜖/𝑞 (𝑇/𝑡0 )]+1 = 𝑝,

𝑡

󵄨𝑝 󵄨 ≤ 𝐶 {𝜀𝑝 + ∫ E󵄨󵄨󵄨Γ1 (𝑠)󵄨󵄨󵄨 d𝑠 𝑡

𝑖 = 1, 2, . . . , [log𝜖/𝑞

0

𝑇 ]. 𝑡0

(67)

𝑡

󵄨𝑝 󵄨 + ∫ ∫ E󵄨󵄨󵄨Γ2 (𝑠, 𝑢)󵄨󵄨󵄨 𝜆 (d𝑢) d𝑠} 𝑡0

𝑈

𝑡

󵄨 󵄨 ≤ 𝐶 {𝜀𝑝 + ∫ E (󵄨󵄨󵄨󵄨𝑋 (𝑠) − 𝑌 (𝑠)󵄨󵄨󵄨󵄨 𝑡

Noting that 𝑍(𝑡) = 𝑍(𝑡) = 0 for 𝑡 ∈ [𝑞𝑡0 , 𝑡0 ], from (65) we obtain

0

̃ (𝑞𝑠)) 󵄨󵄨󵄨󵄨𝑋 (𝑞𝑠) − 𝑌 (𝑞𝑠)󵄨󵄨󵄨󵄨)𝑝 d𝑠 +𝑉1 (𝑍 󵄨 󵄨 𝑡

󵄨 + ∫ E 󵄨󵄨󵄨󵄨𝑏 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠)) 𝑡

E(

𝑡0

sup |𝑍 (𝑠)|𝑝1 ) ≤ 𝐶 {Δ + ∫ E|𝑍 (𝑠)|𝑝1 d𝑠} ≤ 𝐶Δ. 𝑞𝑡0 𝑡0 ≤𝑠≤(𝑡0 /𝑞)𝜖 (68)

0

󵄨𝑝 −𝑏 (𝑌 (𝑠) , 𝑌 (𝑞𝑠) , 𝑟 (𝑠))󵄨󵄨󵄨󵄨 d𝑠 𝑡 󵄨 󵄨 + ∫ E (󵄨󵄨󵄨󵄨𝑋 (𝑠) − 𝑌 (𝑠)󵄨󵄨󵄨󵄨 𝑡 0

̃ (𝑞𝑠)) 󵄨󵄨󵄨󵄨𝑋 (𝑞𝑠) − 𝑌 ̃ (𝑞𝑠)󵄨󵄨󵄨󵄨)𝑝 d𝑠} +𝑉3 (𝑍 󵄨 󵄨 𝑝

𝑡

𝑝

≤ 𝐶 {𝜀 + Δ + ∫ {E|𝑍 (𝑠)|

Then, together with (67) and the H¨older inequality, it further gives that

E(

sup |𝑍 (𝑠)|𝑝2 ) 2 2 𝑡0 ≤𝑠≤(𝑡0 /𝑞 ) 𝜖

≤ 𝐶 {Δ + ∫

𝑡0

𝑡0

𝑝 ̃ 󵄨𝑝 󵄨 + E (𝑉1 (𝑍 (𝑞𝑠)) 󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨 ) 󵄨 󵄨𝑝 𝑝 ̃ + E (𝑉1 (𝑍 (𝑞𝑠)) 󵄨󵄨󵄨󵄨𝑍 (𝑞𝑠)󵄨󵄨󵄨󵄨 )

(𝑡0 /𝑞)𝜖

≤ 𝐶 {Δ + ∫

(𝑡0 /𝑞)𝜀

𝑡0

≤ 𝐶Δ𝑝2 /𝑝1 .

1/(1+𝜃)

(E|𝑍 (𝑠)|𝑝2 (1+𝜃) ) 𝑝2 /𝑝1

(E|𝑍 (𝑠)|𝑝1 )

d𝑠}

d𝑠}

(69)

10

Abstract and Applied Analysis Similarly, E(

sup |𝑍 (𝑠)|𝑝3 ) 3 3 𝑡0 ≤𝑠≤(𝑡0 /𝑞 )𝜖 (𝑡0 /𝑞2 )𝜖2

≤ 𝐶 {Δ + ∫

𝑡0 2

(E|𝑍 (𝑠)|𝑝3 (1+𝜃) )

2

(𝑡0 /𝑞 )𝜖

≤ 𝐶 {Δ + ∫

𝑡0

(E|𝑍 (𝑠)|𝑝2 )

𝑝3 /𝑝2

1/(1+𝜃)

d𝑠}

(70)

d𝑠}

≤ 𝐶Δ𝑝3 /𝑝1 . Repeating the previous procedures, we have E ( sup |𝑍 (𝑠)|𝑝 ) ≤ 𝐶Δ(1+𝜃) 𝑡0 ≤𝑠≤𝑇

1/[log𝜖/𝑞 (𝑇/𝑡0 )](1+𝛼)

.

(71)

The proof is complete.

Acknowledgments 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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[9] X. Mao, C. Yuan, and G. Yin, “Approximations of EulerMaruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 936–948, 2007. [10] F. Wu, X. Mao, and K. Chen, “The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2641–2658, 2009. [11] J. Bao and C. Yuan, “Convergence rate of EM scheme for SDDEs,” http://arxiv.org/abs/1111.4130. [12] Z. Fan, M. Liu, and W. Cao, “Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1142–1159, 2007. [13] Z. Fan, M. Song, and M. Liu, “The 𝛼th moment stability for the stochastic pantograph equation,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 109–120, 2009. [14] L. Ronghua, L. Min, and P. Wan-kai, “Convergence of numerical solutions to stochastic pantograph equations with Markovian switching,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 414–422, 2009. [15] Y. Xiao and H. Y. Zhang, “A note on convergence of semiimplicit Euler methods for stochastic pantograph equations,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1419–1424, 2010. [16] C. Marinelli, C. Pr´evˆot, and M. R¨ockner, “Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise,” Journal of Functional Analysis, vol. 258, no. 2, pp. 616–649, 2010.

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