Convergence Rate of Numerical Solutions for Nonlinear Stochastic Pantograph Equations with Markovian Switching and Jumps

Abstract

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 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 [8] revealed the convergence rate of Euler approximations for stochastic differential equations whose diffusion coefficient is not Lipschitz but only (1/2+𝛼)-Hölder 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 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 mean-square convergence for Markovian switching and the pure jump case and that the rate of mean-square convergence is close to 1/2.

2. Convergence Rate for Markovian Switching and Brownian Motion Case

Let (Ω,ℱ,ℙ) be a complete probability space with a filtration {ℱ_𝑡}_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 (ℝ^𝑛,⟨⋅,⋅⟩,|⋅|) 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=(𝛾_𝑖𝑗)_𝑁×𝑁 given by ℙ𝑟()(𝑡)=𝛾𝑡+𝛿=𝑗∣𝑟=𝑖^𝑖𝑗𝛿+𝑜(𝛿)if𝑖≠𝑗,1+𝑟_𝑖𝑗𝛿+𝑜(𝛿)if𝑖=𝑗, (1) where 𝛿>0. Here 𝛾_𝑖𝑗>0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 while 𝛾_𝑖𝑖=−_𝑗≠𝑖𝛾_𝑖𝑗. (2) 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 (𝑡)𝑋(𝑡)(𝑡)𝑋(𝑡)(𝑡)(𝑡)𝑡d𝑋=𝑏,𝑋𝑞𝑡,𝑟d𝑡+𝜎,𝑋𝑞𝑡,𝑟d𝑊,𝑡∈₀,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<𝑡₀𝑞𝑡𝜖<⁰𝑞²𝜖²<𝑡₀𝑞³𝜖³𝑡<⋯<⁰𝑞^𝑛−1𝜖^𝑛−1<𝑡₀𝑞^𝑛𝜖^𝑛,𝑛=log_𝜖/𝑞𝑇𝑡₀+1,0<Δ_𝑖=𝑡₀𝑞^𝑖𝜖^𝑖−𝑡₀𝑞^𝑖−1𝜖^𝑖−1=𝑡₀𝑞^𝑖𝜖^𝑖𝑞1−𝜖<𝜖𝑞𝑇𝑞1−𝜖=𝜖𝑞−1𝑇<1. (4)

The integral version of (3) is given by the following: 𝑋(𝑡)=𝜉(𝜃)+^𝑡_𝑡0𝑏𝑋(𝑠)(𝑠)+,𝑋𝑞𝑠,𝑟d𝑠_𝑡𝑡0𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)d𝑊(𝑠). (5)

To guarantee the existence and uniqueness of the solutions of (3) we introduce the following conditions: (A1)𝑏∶ℝ^𝑛×ℝ^𝑛×𝑆→ℝ^𝑛 and there exists 𝐿₁>0 such that 𝑏𝑥₁,𝑦₁,𝑗−𝑏𝑥₂,𝑦₂,𝑗≤𝐿₁𝑥₁−𝑥₂+𝑉₁𝑦₁,𝑦₂𝑦₁−𝑦₂ (6) for 𝑥_𝑖,𝑦_𝑖∈ℝ^𝑛, 𝑖=1,2,𝑗∈𝑆; (A2)𝜎∶ℝ^𝑛×ℝ^𝑛×𝑆→ℝ^𝑛×𝑚 and there exists 𝐿₂>0 such that 𝜎𝑥₁,𝑦₁,𝑗−𝜎𝑥₂,𝑦₂,𝑗≤𝐿₂𝑥₁−𝑥₂+𝑉₂𝑦₁,𝑦₂𝑦₁−𝑦₂ (7) for 𝑥_𝑖,𝑦_𝑖∈ℝ^𝑛, 𝑖=1,2,𝑗∈𝑆, where 𝑉_𝑖∶ℝ^𝑛×ℝ^𝑛→ℝ₊ such that 𝑉_𝑖𝑥,𝑦≤𝐾_𝑖1+|𝑥|^𝑞𝑖+𝑦^𝑞𝑖,𝑖=1,2 (8) for some 𝐾_𝑖>0,𝑞_𝑖≥1 and arbitrary 𝑥,𝑦∈ℝ^𝑛.

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 initial data 𝜉∈𝐶^𝑏_ℱ0([𝑞𝑡₀,𝑡₀];ℝ^𝑛) and 𝑟(0)=𝑟₀∈𝑆, 𝑋(𝑡) is a unique global strong solution of (3). Moreover, for any 𝑝≥2 there exists 𝐶>0 such that 𝔼sup_{𝑡0≤𝑡≤𝑇}|𝑋(𝑡)|^𝑝≤𝐶. (9)

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 𝔼sup_{𝑡0≤𝑡≤𝑇}|𝑋(𝑡)|^𝑝≤𝐶,𝑝≥2. (10)
From (A1), (A2), and (8), we can obtain 𝑏𝑥,𝑦,𝑖≤𝐶1+|𝑥|+𝑦+𝑦^𝑞1+1,𝑥,𝑦∈ℝ^𝑛,𝜎𝑥,𝑦,𝑖≤𝐶1+|𝑥|+𝑦+𝑦^𝑞2+1,𝑥,𝑦∈ℝ^𝑛. (11)(12)
Substituting (11) and (12) into (5) and by the Hölder inequality and the Burkhold-Davis-Gundy inequality, we have that for any 𝑝≥2 and 𝑡∈[𝑡₀,𝑇]𝔼sup_{𝑡0≤𝑠≤𝑡}|𝑋(𝑠)|^𝑝≤3^𝑝−1𝜉(𝜃)^𝑝+𝔼sup_{𝑡0≤𝑠≤𝑡}^𝑠_𝑡0𝑏𝑋(𝛼),𝑋𝑞𝛼,𝑟(𝛼)d𝛼^𝑝+𝔼sup_{𝑡0≤𝑠≤𝑡}^𝑠_𝑡0𝜎𝑋(𝛼),𝑋𝑞𝛼,𝑟(𝛼)d𝑊(𝛼)^𝑝≤𝐶1+𝔼^𝑡_𝑡0𝑏𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)^𝑝+𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶1+𝔼^𝑡_𝑡0|𝑋(𝑠)|^𝑝d𝑠+𝔼^𝑡_𝑡0𝑋𝑞𝑠^{𝑝(𝑞1+1)}+𝑋𝑞𝑠^{𝑝(𝑞2+1)}d𝑠. (13)
Let 𝛽∶=(𝑞₁+1)∨(𝑞₂+1); then 𝔼sup_{𝑡0≤𝑠≤𝑡}|𝑋(𝑠)|^𝑝≤𝐶1+𝔼^𝑡_𝑡0|𝑋(𝑠)|^𝑝d𝑠+𝔼^𝑡_𝑡0𝑋𝑞𝑠^𝑝𝛽d𝑠. (14)
By virtue of the Gronwall inequality, we get𝔼sup_{𝑡0≤𝑠≤𝑡}|𝑋(𝑠)|^𝑝≤𝐶1+𝔼^𝑡_𝑡0𝑋𝑞𝑠^𝑝𝛽d𝑠. (15) Let 𝑝_𝑖∶=log_𝜖/𝑞𝑇𝑡₀+2−𝑖𝑝𝛽^{[log𝜖/𝑞(𝑇/𝑡0)]+1−𝑖},𝑖=1,2,…,log_𝜖/𝑞𝑇𝑡₀+1, (16) where [𝑎] denotes the integer part of real number 𝑎; thus, for 𝛽≥1 and 𝑝≥2, we have 𝑝_𝑖+1𝛽<𝑝_𝑖,𝑝_{[log𝜖/𝑞(𝑇/𝑡0)]+1}=𝑝,𝑖=1,2,…,log_𝜖/𝑞𝑇𝑡₀. (17)
Together with 𝜉∈𝐶^𝑏_ℱ0([𝑞𝑡₀,𝑡₀];ℝ^𝑛) and 𝜖<1, we obtain that 𝔼sup_{𝑡0≤𝑠≤(𝑡0/𝑞)𝜖}|𝑋(𝑠)|^𝑝1≤𝐶1+𝔼^(𝑡0_𝑡^/𝑞)𝜖₀𝑋𝑞𝑠^𝑝1𝛽d𝑠≤𝐶1+𝔼^𝑡0_𝑞𝑡0|𝑋(𝑠)|^𝑝1𝛽d𝑠≤𝐶. (18)
In the similar way, combining (15) with the Hölder inequality further leads to 𝔼sup_{𝑡0≤𝑠≤(𝑡0/𝑞2)𝜖2}|𝑋(𝑠)|^𝑝2≤𝐶1+𝔼^{(𝑡0/𝑞2)𝜖2}_𝑡0𝑋𝑞𝑠^𝑝2𝛽d𝑠≤𝐶1+^(𝑡0_𝑡^/𝑞)𝜖₀𝔼|𝑋(𝑠)|^{𝑝1𝑝2𝛽/𝑝1}d𝑠≤𝐶. (19)
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 Δ>0, then {𝑟(𝑘Δ),𝑘=0,1,2,…} is a discrete Markov chain with the one-step transition probability matrix 𝑃(Δ)=𝑃_𝑖,𝑗(Δ)_𝑁×𝑁=𝑒^ΔΓ. (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.

Define 𝑟(Δ)=𝑖,𝑖∈𝑆−{𝑁}suchthat∑^𝑖−1_𝑗=1𝑃_𝑟(0),𝑗(Δ)≤𝜉₁<∑^𝑖_𝑗=1𝑃_𝑟(0),𝑗(Δ),𝑁,∑^𝑁−1_𝑗=1𝑃_𝑟(0),𝑗(Δ)≤𝜉₁, (21) where we set ∑⁰_𝑗=1𝑃_𝑟(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 𝜉_𝑘+1 and setting 𝑟)Δ)={𝑁}((𝑘+1𝑖,𝑖∈𝑆−suchthat∑^𝑖−1_𝑗=1𝑃_{𝑟(𝑘Δ),𝑗}(Δ)≤𝜉_𝑘+1<∑^𝑖_𝑗=1𝑃_{𝑟(𝑘Δ),𝑗}(Δ),𝑁,∑^𝑁−1_𝑗=1𝑃_{𝑟(𝑘Δ),𝑗}(Δ)≤𝜉_𝑘+1. (22) The procedure can be carried out independently to obtain more trajectories.

Define the Euler-Maruyama approximation for (3) by d𝑌(𝑡)=𝑏𝑌(𝑡),𝑌𝑞𝑡,𝑟(𝑡)d𝑡+𝜎𝑌(𝑡),𝑌𝑞𝑡,𝑟(𝑡)d𝑊(𝑡),𝑡∈𝑡₀,𝑇, (23) where 𝑌(𝑡)∶=𝑌(𝑡_𝑖), 𝑟(𝑡)∶=𝑟(𝑡_𝑖) for 𝑡∈[𝑡_𝑖,𝑡_𝑖+1), 𝑖=0,1,…,[log_𝜖/𝑞(𝑇/𝑡₀)], which 𝑡₀<𝑡₀𝑞𝜖=𝑡₁<𝑡₀𝑞²𝜖²=𝑡₂𝑡<⋯<⁰𝑞^𝑛−1𝜖^𝑛−1=𝑡_𝑛−1<𝑡₀𝑞^𝑛𝜖^𝑛=𝑡_𝑛 (24) and 𝑌(𝜃)=𝜉(𝜃),𝑟(𝜃)=𝑟₀,𝜃∈[𝑞𝑡₀,𝑡₀].

By using the method of Lemma 2, we obtain 𝔼sup_{𝑡0≤𝑡≤𝑇}|𝑌(𝑡)|^𝑝𝔼𝑌(𝑡)−≤𝐶,𝑌(𝑡)^𝑝≤𝐶Δ^𝑝/2,𝑡∈𝑡₀,𝑇, (25)(26) where Δ=max{Δ₁,Δ₂,…,Δ_𝑛}∶=(𝜖/𝑞−1)𝑇, Δ→0, when 𝜖→𝑞.

Lemma 4. If (A1) and (A2) hold, then 𝔼^𝑇_𝑡0𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝𝔼d𝑠≤𝐶Δ,_𝑇𝑡0𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶Δ. (27)

Proof. Let 𝑛=[log_𝜖/𝑞(𝑇/𝑡₀)]+1, then 𝔼^𝑇_𝑡0𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝=d𝑠_{𝑛−1𝑖=0}𝔼^𝑡𝑖+1_𝑡𝑖𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟𝑡_𝑖^𝑝d𝑠. (28)
By (11), we compute 𝔼^𝑡𝑖+1_𝑡𝑖𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟𝑡_𝑖^𝑝d𝑠≤2^𝑝−1𝔼^𝑡𝑖+1_𝑡𝑖𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝+𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟𝑡_𝑖^𝑝×𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑖)}}d𝑠≤𝐶𝔼^𝑡𝑖+1_𝑡𝑖1+𝑌(𝑠)^𝑝+𝑌𝑞𝑠^{𝑝(𝑞1+1)}𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑖)}}d𝑠≤𝐶^𝑡𝑖+1_𝑡𝑖𝔼𝔼1+𝑌(𝑠)^𝑝+𝑌𝑞𝑠^{𝑝(𝑞1+1)}𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑖)}}𝑟𝑡_𝑖d𝑠=𝐶^𝑡𝑖+1_𝑡𝑖𝔼𝔼1+𝑌(𝑠)^𝑝+𝑌𝑞𝑠^{𝑝(𝑞1+1)}𝑟𝑡_𝑖×𝔼𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑖)}}𝑟𝑡_𝑖d𝑠. (29)
By the Markov property, we have 𝔼𝐼_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑟𝑡_𝑘=_𝑖∈𝑆𝐼_{{𝑟(𝑡𝑘)=𝑖}}𝑃𝑟(𝑠)≠𝑖∣𝑟𝑡_𝑘==𝑖_𝑖∈𝑆𝐼_{{𝑟(𝑡𝑘)=𝑖}𝑗≠𝑖}𝛾_𝑖𝑗𝑠−𝑡_𝑘+𝑜𝑠−𝑡_𝑘≤max_{1≤𝑖≤𝑛}−𝛾_𝑖𝑖Δ+𝑜(Δ)_𝑖∈𝑆𝐼_{{𝑟(𝑡𝑘)=𝑖}}≤𝐶Δ. (30)
Substituting the above inequality into (29) yields 𝔼^𝑡𝑖+1_𝑡𝑖𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟𝑡_𝑖^𝑝d𝑠≤𝐶Δ^𝑡𝑖+1_𝑡𝑖1+𝑌𝑡_𝑖^𝑝+𝑌𝑞𝑡_𝑖^{𝑝(𝑞1+1)}d𝑠≤𝐶Δ. (2)(31)
So, (28) becomes 𝔼^𝑇_𝑡0𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶Δ. (32)
Similarly, we also obtain that 𝔼^𝑇_𝑡0𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶Δ. (33)
The proof is complete.

Theorem 5. Under (A1) and (A2), for any 𝑝≥2 there exits 𝐶>0 such that 𝔼sup_{𝑡0≤𝑡≤𝑇}|𝑋(𝑡)−𝑌(𝑡)|^𝑝≤𝐶Δ^𝑝/2; (34) that is, the convergence order of Euler-Maruyama scheme (23) is 1/2.

Proof. Let 𝛿>1 and 𝜀>0; then∫^𝜀_𝜀/𝛿(1/𝑥)d𝑥=ln𝑥|^𝜀_𝜀/𝛿=ln𝛿, and there is a continuous nonnegative function 𝜓_𝛿𝜀(𝑥)(𝑥≥0), which is zero outside [𝜀/𝛿,𝜀], such that^𝜀_𝜀/𝛿𝜓_𝛿𝜀(𝑥)d𝑥=1,𝜓_𝛿𝜀(𝑥)≤2𝑥ln𝛿,𝑥>0. (35)
Define 𝜙_𝛿𝜀(𝑥)∶=^𝑥₀^𝑦₀𝜓_𝛿𝜀(𝑧)𝑉d𝑧d𝑦,𝑥>0,_𝛿𝜀(𝑥)∶=𝜙_𝛿𝜀(|𝑥|),𝑥∈ℝ^𝑛. (36) For any 𝑡∈[𝑡₀,𝑇], let 𝑍(𝑡)∶=𝑋(𝑡)−𝑌(𝑡),𝑍(𝑡)∶=𝑌(𝑡)−𝑌(𝑡),𝑍(𝑡)∶=𝑋(𝑡),𝑌(𝑡)∈ℝ^2𝑛. (37)
Using the Itô formula, we have 𝑉_𝛿𝜀(𝑍(𝑡=))_𝑡𝑡0𝑉_𝛿𝜀𝑥(𝑍(𝑠)),𝑏𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)+1d𝑠2^𝑡_𝑡0trace𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^∗×𝑉_{𝛿𝜖𝑥𝑥}(𝑍(𝑠×𝜎𝑋(𝑠)(𝑠))),𝑋𝑞𝑠,𝑟−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)+d𝑠_𝑡𝑡0𝑉_𝛿𝜀𝑥(𝑍(𝑠)),𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)d𝑊(𝑠)∶=𝐼₁(𝑡)+𝐼₂(𝑡)+𝐼₃(𝑡). (38)
By virtue of condition (A1), the Hölder inequality, and Lemma 4, we deduce that 𝔼sup_{𝑡0≤𝑠≤𝑡}𝐼₁(𝑠)^𝑝≤𝑡−𝑡₀^𝑝−1𝔼^𝑡_𝑡0𝑉_𝛿𝜀𝑥(𝑍(𝑠)),𝑏𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝≤d𝑠𝑡−𝑡₀^𝑝−1𝔼^𝑡_𝑡0𝑏𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶Δ^𝑝−1𝑡_𝑡0𝔼𝑏𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,(𝑠)^𝑝+𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶Δ^𝑝−1𝑡_𝑡0𝔼|𝑍(𝑠)|^𝑝+𝔼𝑉₁^2𝑝𝑍𝑞𝑠^1/2×𝔼𝑍𝑞𝑠^2𝑝1/2+𝔼𝑍(𝑠)^𝑝+𝔼𝑉₁^2𝑝𝑍𝑞𝑠^1/2×𝔼𝑍𝑞𝑠^2𝑝1/2d𝑠+𝐶Δ^𝑝. (39)
By the Hölder inequality and (A2), we have 𝔼sup_{𝑡0≤𝑠≤𝑡}𝐼₂(𝑠)^𝑝≤12𝑡−𝑡₀^𝑝−1𝔼×^𝑡_𝑡0trace𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^∗×𝑉_{𝛿𝜀𝑥𝑥}(𝑍(𝑠×𝜎𝑋(𝑠)(𝑠)),𝑋𝑞𝑠,𝑟)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶Δ^𝑝−1𝑡_𝑡0𝔼𝑉_{𝛿𝜖𝑥𝑥}(𝑍(𝑠×𝜎𝑋(𝑠)(𝑠))),𝑋𝑞𝑠,𝑟−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^2𝑝d𝑠≤𝐶Δ^𝑝−1𝔼^𝑡_𝑡01|𝑍(𝑠)|^𝑝×𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^2𝑝+𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^2𝑝≤𝐶Δ^𝑝−1𝑡_𝑡0𝔼|𝑍(𝑠)|^𝑝+1𝜀^𝑝𝔼𝑉₂^4𝑝𝑍𝑞𝑠^1/2𝔼𝑍𝑞𝑠^4𝑝1/2+1𝜀^𝑝𝔼𝑍(𝑠)^2𝑝+1𝜀^𝑝𝔼𝑉₂^4𝑝𝑍𝑞𝑠^1/2×𝔼𝑍𝑞𝑠^4𝑝1/2d𝑠+𝐶Δ^𝑝𝜀^𝑝. (40)
Making use of the Burkhold-Davis-Gundy inequality yields 𝔼sup_{𝑡0≤𝑠≤𝑡}𝐼₃(𝑠)^𝑝≤𝐶Δ^𝑝/2−1𝔼^𝑡_𝑡0𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶Δ^𝑝/2−1𝔼^𝑡_𝑡0𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝+𝜎𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝜎𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝d𝑠≤𝐶Δ^{𝑝/2−1𝑡}_𝑡0𝔼|𝑍(𝑠)|^𝑝+𝔼𝑉₂^2𝑝𝑍𝑞𝑠^1/2×𝔼𝑍𝑞𝑠^2𝑝1/2+𝔼𝑍(𝑠)^𝑝+𝔼𝑉₂^2𝑝𝑍𝑞𝑠^1/2×𝔼𝑍𝑞𝑠^2𝑝1/2d𝑠+𝐶Δ^𝑝/2. (41)
Moreover, by (8), (9), and (25), we have 𝔼𝑉₁^2𝑝𝑍𝑞𝑠∨𝔼𝑉₂^4𝑝𝑍𝔼𝑞𝑠≤𝐶,𝑍(𝑡)^𝑝≤𝐶Δ^𝑝/2. (42)
Thus, combing (39), and (40) with (41), for any 𝑡∈[𝑡₀,𝑇] and 𝑝≥2, we get 𝔼sup_{𝑡0≤𝑠≤𝑡}|𝑍(𝑠)|^𝑝≤2^𝑝−1𝜀^𝑝+𝔼sup_{𝑡0≤𝑠≤𝑡}𝑉^𝑝_𝛿𝜀(𝑍(𝑠𝜀))≤𝐶^𝑝+Δ^3𝑝/2−1+Δ^𝑝+Δ^2𝑝−1𝜀^𝑝+Δ^𝑝𝜀^𝑝+Δ^𝑝−1+Δ^𝑝/2+Δ^{𝑝/2−1𝑡}_𝑡0𝔼|𝑍(𝑠)|^𝑝d𝑠+^𝑡_𝑡0𝔼𝑍𝑞𝑠^2𝑝1/2+1d𝑠𝜀^𝑝𝑡_𝑡0𝔼𝑍𝑞𝑠^4𝑝1/2d𝑠. (43)
Let 𝜀=Δ^1/2, and using the Gronwall inequality, we have 𝔼sup_{𝑡0≤𝑠≤𝑡}|𝑍(𝑠)|^𝑝≤𝐶Δ^𝑝/2+Δ^{(𝑝/2)−1𝑡}_𝑡0𝔼𝑍𝑞𝑠^2𝑝1/2d𝑠+Δ^−1𝑡_𝑡0𝔼𝑍𝑞𝑠^4𝑝1/2d𝑠. (44)
Let 𝑝_𝑖∶=log_𝜖/𝑞𝑇𝑡₀+2−𝑖𝑝4^{[log𝜖/𝑞(𝑇/𝑡0)]+1−𝑖},𝑖=1,2,…,log_𝜖/𝑞𝑇𝑡₀+1; (45) by 𝑝≥2, it is easy to see that 𝑝_𝑖≥2 such that 4𝑝_𝑖+1<𝑝_𝑖,𝑝_{[log𝜖/𝑞(𝑇/𝑡0)]+1}=𝑝,𝑖=1,2,…,log_𝜖/𝑞𝑇𝑡₀. (46)
Noting that 𝑍(𝑠)=0 for 𝑠∈[𝑞𝑡₀,𝑡₀] and substituting 𝜖<1 into (44) yields that 𝔼sup_{𝑡0𝑡≤𝑠≤0𝜖/𝑞}|𝑍(𝑠)|^𝑝1≤𝐶Δ^𝑝1/2+Δ^{𝑝1/2−1(𝑡0}_𝑡^/𝑞)𝜖₀𝔼𝑍𝑞𝑠^2𝑝11/2d𝑠+Δ^−1(𝑡0_𝑡^/𝑞)𝜖₀𝔼𝑍𝑞𝑠^4𝑝11/2Δd𝑠≤𝐶^𝑝1/2+Δ^{𝑝1/2−1𝑡0}_𝑞𝑡0𝔼|𝑍(𝑠)|^2𝑝11/2d𝑠+Δ^−1𝑡0_𝑞𝑡0𝔼|𝑍(𝑠)|^4𝑝11/2d𝑠≤𝐶Δ^𝑝1/2. (47) Using (46) and the Hölder inequality, further gives that 𝔼sup_{𝑡0𝑡≤𝑠≤0/𝑞2𝜖2}|𝑍(𝑠)|^𝑝2≤𝐶Δ^𝑝2/2+Δ^{𝑝2/2−1(𝑡0/𝑞2)𝜖2}_𝑡0𝔼𝑍𝑞𝑠^2𝑝21/2d𝑠+Δ^{−1(𝑡0/𝑞2)𝜖2}_𝑡0𝔼𝑍𝑞𝑠^4𝑝21/2Δd𝑠≤𝐶^𝑝2/2+Δ^{𝑝2/2−1(𝑡0}_𝑡^/𝑞)𝜖₀𝔼|𝑍(𝑠)|^{𝑝1𝑝2/𝑝1}d𝑠+Δ^−1(𝑡0_𝑡^/𝑞)𝜖₀𝔼|𝑍(𝑠)|^{𝑝12𝑝2/𝑝1}d𝑠≤𝐶Δ^𝑝2/2. (48)
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 1/2. 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 ℬ(ℝ) be the Borel 𝜎-algebra on ℝ, and 𝜆(d𝑥) a 𝜎-finite measure defined on ℬ(ℝ). Let 𝑝=(𝑝(𝑡)), 𝑡∈𝐷_𝑝, be a stationary ℱ_𝑡-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)=∑_{𝑠∈𝐷𝑝,𝑠≤𝑡}𝐼_𝑈(𝑝(𝑠)) for 𝑈∈ℬ(ℝ). Let 𝑁(d𝑡,d𝑢)∶=𝑁(d𝑡,d𝑢)−d𝑡𝜆(d𝑢) be the compensated Poisson measure associated with 𝑁(d𝑡,d𝑢). In what follows, we further assume that ∫_𝑈|𝑢|^𝑝𝜆(𝑢)<∞ for any 𝑝≥2.

In this section, we consider the following stochastic pantograph equation with Markovian switching and pure jumps on ℝ^𝑛: (𝑡)𝑋(𝑡)(𝑡)+d𝑋=𝑏,𝑋𝑞𝑡,𝑟d𝑡_𝑈ℎ𝑋(𝑡),𝑋𝑞𝑡,𝑢𝑁(d𝑡,d𝑢),𝑡∈𝑡₀,𝑇 (49) with initial data 𝑋(𝜃)=𝜉(𝜃) and 𝑟(𝜃)=𝑟₀, 𝜃∈[𝑞𝑡₀,𝑡₀].

We assume that (A1)𝑏∶ℝ^𝑛×ℝ^𝑛×𝑆→ℝ^𝑛 and there exists 𝐿₁>0 such that 𝑏𝑥₁,𝑦₁,𝑗−𝑏𝑥₂,𝑦₂,𝑗≤𝐿₁𝑥₁−𝑥₂+𝑉₁𝑦₁,𝑦₂𝑦₁−𝑦₂ (50) for 𝑥_𝑖,𝑦_𝑖∈ℝ^𝑛, 𝑖=1,2, 𝑗∈𝑆; (A3)ℎ∶ℝ^𝑛×ℝ^𝑛×𝑈→ℝ^𝑛 and there exists 𝐿₃>0 such that ℎ𝑥₁,𝑦₁,𝑢−ℎ𝑥₂,𝑦₂≤𝐿,𝑢₃𝑥₁−𝑥₂+𝑉₃𝑦₁,𝑦₂𝑦₁−𝑦₂|𝑢| (51) for 𝑥_𝑖,𝑦_𝑖∈ℝ^𝑛,𝑖=1,2, and 𝑢∈𝑈, where 𝑉₃∶ℝ^𝑛×ℝ^𝑛→ℝ₊ such that 𝑉₃𝑥,𝑦≤𝐾₃1+|𝑥|^𝑞3+𝑦^𝑞3 (52) for some 𝐾₃>0,𝑞₃≥1 and arbitrary 𝑥,𝑦∈ℝ^𝑛.

From (A3), the jump coefficient may be also highly nonlinear. We define the Euler-Maruyama scheme associated with (49) by d𝑌(𝑡)=𝑏𝑌(𝑡),𝑌𝑞𝑡,𝑟(𝑡+)d𝑡_𝑈ℎ𝑌(𝑡),𝑌𝑞𝑡,𝑢𝑁(d𝑡,d𝑢), (53) where 𝑌(𝑡)∶=𝑌(𝑡_𝑖), 𝑟(𝑡)∶=𝑟(𝑡_𝑖) for 𝑡∈[𝑡_𝑖,𝑡_𝑖+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→ℝ^𝑛 and assume that ^𝑡_𝑡0𝑈𝔼|Φ(𝑠,𝑢)|^𝑝𝜆(d𝑢)d𝑠<∞,𝑡₀>0,𝑝≥2. (54)
Then there exists 𝐷(𝑝)>0 such that 𝔼sup_{𝑡0≤𝑠≤𝑡}^𝑠_𝑡0𝑈Φ(𝑟,𝑢)𝑁(d𝑢,d𝑠)^𝑝≤𝐷𝑝𝔼^𝑡_𝑡0𝑈|Φ(𝑠,𝑢)|²𝜆(d𝑢)d𝑠^𝑝/2+𝔼^𝑡_𝑡0𝑈|Φ(𝑠,𝑢)|^𝑝𝜆(d𝑢)d𝑠. (55)

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 𝔼sup_{𝑡0≤𝑡≤𝑇}|𝑋(𝑡)|^𝑝∨𝔼sup_{𝑡0≤𝑡≤𝑇}|𝑌(𝑡)|^𝑝𝔼𝑌(𝑡)−≤𝐶,𝑌(𝑡)^𝑝≤𝐶Δ. (56)(57)

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

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 𝔼sup_{𝑡0≤𝑡≤𝑇}|𝑋(𝑡)−𝑌(𝑡)|^𝑝≤𝐶Δ^{(1+𝜃)1/[log𝜖/𝑞(𝑇/𝑡0)](1+𝛼)}. (58)

Proof. The proof is similar to that of Theorem 5. Set 𝑍(𝑡)∶=𝑋(𝑡)−𝑌(𝑡),𝑍(𝑡)∶=𝑌(𝑡)−𝑌(𝑡),𝑍(𝑡)∶=𝑋(𝑡),𝑌(𝑡)∈ℝ^2𝑛,𝑡∈𝑡₀,𝑇. (59) Define Γ₁(𝑡)∶=𝑏𝑋(𝑡),𝑋𝑞𝑡,𝑟(𝑡)−𝑏𝑌(𝑡),𝑌𝑞𝑡,𝑟(𝑡,Γ₂)(𝑡,𝑢)∶=ℎ𝑋(𝑡),𝑋𝑞𝑡,𝑢−ℎ𝑌(𝑡),𝑌𝑞𝑡,𝑢. (60) Using the Itô formula and the Taylor expansion we have that for 𝑡∈[𝑡₀,𝑇]𝑉_𝛿𝜀(𝑍(𝑡))=^𝑡_𝑡0𝑉_𝛿𝜀𝑥(𝑍(𝑠)),Γ₁(𝑠+)d𝑠_{𝑡𝑡0𝑈}𝑉_𝛿𝜀𝑍(𝑠)+Γ₂(𝑠,𝑢)−𝑉_𝛿𝜀(𝑍(𝑠−𝑉))_𝛿𝜀𝑥(𝑍(𝑠)),Γ₂(()+𝑠,𝑢)×𝜆d𝑢d𝑠_{𝑡𝑡0𝑈}𝑉_𝛿𝜀𝑍(𝑠)+Γ₂(𝑠,𝑢)−𝑉_𝛿𝜀(𝑍(𝑠𝑁()=))d𝑢,d𝑠_𝑡𝑡0𝑉_𝛿𝜀𝑥(𝑍(𝑠)),Γ₁(𝑠+)d𝑠_{𝑡𝑡0𝑈10}𝑉_𝛿𝜀𝑥𝑍(𝑠)+𝜃Γ₂()−𝑉𝑠,𝑢_𝛿𝜀𝑥(𝑍(𝑠)),Γ₂(()+𝑠,𝑢)𝑑𝜃×𝜆d𝑢d𝑠_{𝑡𝑡0𝑈10}𝑉_𝛿𝜀𝑥(𝑍(𝑠)+𝜃Γ₂(),Γ𝑠,𝑢₂(𝑠,𝑢)d𝜃𝑁(d𝑢,d𝑠). (61)
By the property of 𝑉_𝛿𝜀(𝑥), we deduce that |𝑍(𝑡)|≤𝜀+𝑉_𝛿𝜀(𝑍(𝑡))≤𝜀+^𝑡_𝑡0Γ₁(𝑠)d𝑠+2^𝑡_𝑡0𝑈Γ₂(𝜆()+𝑠,𝑢)d𝑢d𝑠_{𝑡𝑡0𝑈10}𝑉_𝛿𝜀𝑥𝑍(𝑠)+𝜃Γ₂(),Γ𝑠,𝑢₂(𝑁(),𝑡𝑠,𝑢)d𝜃d𝑢,d𝑠𝑡∈₀,𝑇. (62)
From (8), (52), and (56), we compute that for any 𝑝≥2𝔼sup_{𝑡0≤𝑡≤𝑇}𝑉^𝑝₁𝑍𝑞𝑠∨𝔼sup_{𝑡0≤𝑡≤𝑇}𝑉^𝑝₃𝑍𝑞𝑠≤𝐶. (63)
Applying Lemma 6, Lemma 4, (57), and the Hölder inequality, we obtain that 𝔼sup_{𝑡0≤𝑠≤𝑡}|𝑍(𝑠)|^𝑝≤2^𝑝−1𝜀^𝑝+𝔼sup_{𝑡0≤𝑠≤𝑡}𝑉^𝑝_𝛿𝜀(𝑍(𝑠𝜀))≤𝐶^𝑝+^𝑡_𝑡0𝔼Γ₁(𝑠)^𝑝+d𝑠_{𝑡𝑡0𝑈}𝔼Γ₂(𝑠,𝑢)^𝑝𝜆(d𝑢)d𝑠+𝔼^𝑡_𝑡0𝑈Γ₂(𝑠,𝑢)²𝜆(d𝑢)d𝑠^𝑝/2≤𝐶𝜀^𝑝+^𝑡_𝑡0𝔼Γ₁(𝑠)^𝑝+d𝑠_{𝑡𝑡0𝑈}𝔼Γ₂(𝑠,𝑢)^𝑝𝜆()𝜀d𝑢d𝑠≤𝐶^𝑝+^𝑡_𝑡0𝔼𝑋(𝑠)−𝑌(𝑠)+𝑉₁𝑍𝑞𝑠𝑋𝑞𝑠−𝑌𝑞𝑠^𝑝+d𝑠_𝑡𝑡0𝔼𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)−𝑏𝑌(𝑠),𝑌𝑞𝑠,𝑟(𝑠)^𝑝+d𝑠_𝑡𝑡0𝔼𝑋(𝑠)−𝑌(𝑠)+𝑉₃𝑍𝑞𝑠𝑋𝑞𝑠−𝑌𝑞𝑠^𝑝𝜀d𝑠≤𝐶^𝑝+Δ+^𝑡_𝑡0𝔼|𝑍(𝑠)|^𝑝+𝔼𝑉^𝑝₁𝑍𝑞𝑠𝑍𝑞𝑠^𝑝+𝔼𝑉^𝑝₁𝑍𝑞𝑠𝑍𝑞𝑠^𝑝+𝔼𝑉^𝑝₃𝑍𝑞𝑠𝑍𝑞𝑠^𝑝+𝔼𝑉^𝑝₃𝑍𝑞𝑠𝑍𝑞𝑠^𝑝𝜀d𝑠≤𝐶^𝑝+Δ+^𝑡_𝑡0𝔼|𝑍(𝑠)|^𝑝d𝑠+^𝑡_𝑡0𝔼𝑍𝑞𝑠^𝑝d𝑠. (64)
Together with the Gronwall inequality and taking 𝜀=Δ^1/𝑝, we get𝔼sup_{𝑡0≤𝑠≤𝑡}|𝑍(𝑠)|^𝑝≤𝐶Δ+^𝑡_𝑡0𝔼𝑍𝑞𝑠^𝑝d𝑠. (65)
For 𝜃∈(0,1) and any 𝛼∈(0,1), let 𝑝_𝑖∶=𝑝(1+𝜃)^{([log𝜖/𝑞(𝑇/𝑡0)]+1−𝑖)(1+𝛼)},𝑖=1,2,…,log_𝜖/𝑞𝑇𝑡₀+1. (66)
It is easy to see that (1+𝜃)𝑝_𝑖+1<𝑝_𝑖,𝑝_{[log𝜖/𝑞(𝑇/𝑡0)]+1}=𝑝,𝑖=1,2,…,log_𝜖/𝑞𝑇𝑡₀. (67)
Noting that 𝑍(𝑡)=𝑍(𝑡)=0 for 𝑡∈[𝑞𝑡₀,𝑡₀], from (65) we obtain 𝔼sup_{𝑡0𝑡≤𝑠≤0𝜖/𝑞}|𝑍(𝑠)|^𝑝1≤𝐶Δ+^𝑡0_𝑞𝑡0𝔼|𝑍(𝑠)|^𝑝1d𝑠≤𝐶Δ. (68)
Then, together with (67) and the Hölder inequality, it further gives that 𝔼sup_{𝑡0𝑡≤𝑠≤0/𝑞2𝜖2}|𝑍(𝑠)|^𝑝2≤𝐶Δ+^(𝑡0_𝑡^/𝑞)𝜖₀𝔼|𝑍(𝑠)|^{𝑝2(1+𝜃)1/(1+𝜃)}d𝑠≤𝐶Δ+^(𝑡0_𝑡^/𝑞)𝜀₀𝔼|𝑍(𝑠)|^{𝑝1𝑝2/𝑝1}d𝑠≤𝐶Δ^𝑝2/𝑝1. (69)
Similarly, 𝔼sup_{𝑡0𝑡≤𝑠≤0/𝑞3𝜖3}|𝑍(𝑠)|^𝑝3≤𝐶Δ+^{(𝑡0/𝑞2)𝜖2}_𝑡0𝔼|𝑍(𝑠)|^{𝑝3(1+𝜃)1/(1+𝜃)}d𝑠≤𝐶Δ+^{(𝑡0/𝑞2)𝜖2}_𝑡0𝔼|𝑍(𝑠)|^{𝑝2𝑝3/𝑝2}d𝑠≤𝐶Δ^𝑝3/𝑝1. (70)
Repeating the previous procedures, we have 𝔼sup_{𝑡0≤𝑠≤𝑇}|𝑍(𝑠)|^𝑝≤𝐶Δ^{(1+𝜃)01/[log𝜖/𝑞(𝑇/𝑡)](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.