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

Authors: Zhenyu Lu; Tingya Yang; Yanhan Hu; Junhao Hu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 Section2, 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 {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.Letr(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
(1)ℙ{r(t+δ)=j∣r(t)=i}={γijδ+o(δ)ifi≠j,1+rijδ+o(δ)ifi=j,
where δ>0. Here γij>0 is the transition rate from i to j if i≠j while
(2)γii=-∑j≠i‍γij.
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 fixedT>0, we consider the stochastic pantograph equation with Markovian switching of the form
(3)dX(t)=b(X(t),X(qt),r(t))dt+σ(X(t),X(qt),r(t))dW(t),t∈[t0,T],
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
(4)0<t0<t0qϵ<t0q2ϵ2<t0q3ϵ3<⋯<t0qn-1ϵn-1<t0qnϵn,n=[logϵ/qTt0]+1,0<Δi=t0qiϵi-t0qi-1ϵi-1=t0qiϵi(1-qϵ)<ϵqT(1-qϵ)=(ϵq-1)T<1.The integral version of (3) is given by the following:
(5)X(t)=ξ(θ)+∫t0t‍b(X(s),X(qs),r(s))ds+∫t0t‍σ(X(s),X(qs),r(s))dW(s).To guarantee the existence and uniqueness of the solutions of (3) we introduce the following conditions: (A1)
b:ℝn×ℝn×S→ℝn and there exists L1>0 such that
(6)|b(x1,y1,j)-b(x2,y2,j)|≤L1|x1-x2|+V1(y1,y2)|y1-y2|forxi,yi∈ℝn, i=1,2,j∈S;(A2)
σ:ℝn×ℝn×S→ℝn×m and there exists L2>0 such that
(7)∥σ(x1,y1,j)-σ(x2,y2,j)∥≤L2|x1-x2|+V2(y1,y2)|y1-y2|forxi,yi∈ℝn, i=1,2,j∈S,where Vi:ℝn×ℝn→ℝ+ such that
(8)Vi(x,y)≤Ki(1+|x|qi+|y|qi),i=1,2
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 initial dataξ∈Cℱ0b([qt0,t0];ℝn) 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
(9)𝔼(supt0≤t≤T|X(t)|p)≤C.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
(10)𝔼(supt0≤t≤T|X(t)|p)≤C,p≥2.
From (A1), (A2), and (8), we can obtain
(11)|b(x,y,i)|≤C(1+|x|+|y|+|y|q1+1),x,y∈ℝn,(12)∥σ(x,y,i)∥≤C(1+|x|+|y|+|y|q2+1),x,y∈ℝn.
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](13)𝔼(supt0≤s≤t|X(s)|p)≤3p-1{|∫t0s‍ξ(θ)|p+𝔼(supt0≤s≤t|∫t0s‍b(X(α),X(qα),r(α))dα|p)+𝔼(supt0≤s≤t|∫t0s‍σ(X(α),X(qα),r(α))dW(α)|p)}≤C{1+𝔼∫t0t‍(|b(X(s),X(qs),r(s))|p+∥σ(X(s),X(qs),r(s))∥p)ds∫t0t‍}≤C{1+𝔼∫t0t‍|X(s)|pds+𝔼∫t0t‍(|X(qs)|p(q1+1)+|X(qs)|p(q2+1))ds}.
Letβ:=(q1+1)∨(q2+1); then
(14)𝔼(supt0≤s≤t|X(s)|p)≤C{1+𝔼∫t0t‍|X(s)|pds+𝔼∫t0t‍|X(qs)|pβds}.
By virtue of the Gronwall inequality, we get(15)𝔼(supt0≤s≤t|X(s)|p)≤C{1+𝔼∫t0t‍|X(qs)|pβds}.
Let
(16)pi:=([logϵ/qTt0]+2-i)pβ[logϵ/q(T/t0)]+1-i,i=1,2,…,[logϵ/qTt0]+1,
where [a] denotes the integer part of real number a; thus, for β≥1 and p≥2, we have
(17)pi+1β<pi,p[logϵ/q(T/t0)]+1=p,i=1,2,…,[logϵ/qTt0].
Together withξ∈Cℱ0b([qt0,t0];ℝn) and ϵ<1, we obtain that
(18)𝔼(supt0≤s≤(t0/q)ϵ|X(s)|p1)≤C{1+𝔼∫t0(t0/q)ϵ‍|X(qs)|p1βds}≤C{1+𝔼∫qt0t0‍|X(s)|p1βds}≤C.
In the similar way, combining (15) with the Hölder inequality further leads to
(19)𝔼(supt0≤s≤(t0/q2)ϵ2|X(s)|p2)≤C{1+𝔼∫t0(t0/q2)ϵ2‍|X(qs)|p2βds}≤C{1+∫t0(t0/q)ϵ‍(𝔼|X(s)|p1)p2β/p1ds}≤C.
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.
GivenΔ>0, then {r(kΔ),k=0,1,2,…} is a discrete Markov chain with the one-step transition probability matrix
(20)P(Δ)=(Pi,j(Δ))N×N=eΔΓ.
Given a fixed step sizeΔ>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(21)r(Δ)={i,i∈S-{N}suchthat∑j=1i-1‍Pr(0),j(Δ)≤ξ1<∑j=1i‍Pr(0),j(Δ),N,∑j=1N-1‍Pr(0),j(Δ)≤ξ1,
where we set ∑j=10Pr(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
(22)r((k+1)Δ)={i,i∈S-{N}suchthat∑j=1i-1‍Pr(kΔ),j(Δ)≤ξk+1<∑j=1i‍Pr(kΔ),j(Δ),N,∑j=1N-1‍Pr(kΔ),j(Δ)≤ξk+1.
The procedure can be carried out independently to obtain more trajectories.Define the Euler-Maruyama approximation for (3) by
(23)dY(t)=b(Y-(t),Y-(qt),r-(t))dt+σ(Y-(t),Y-(qt),r-(t))dW(t),t∈[t0,T],
where Y-(t):=Y(ti), r-(t):=r(ti) for t∈[ti,ti+1), i=0,1,…,[logϵ/q(T/t0)], which
(24)t0<t0qϵ=t1<t0q2ϵ2=t2<⋯<t0qn-1ϵn-1=tn-1<t0qnϵn=tn
and Y-(θ)=ξ(θ),r-(θ)=r0,θ∈[qt0,t0].By using the method of Lemma2, we obtain
(25)𝔼(supt0≤t≤T|Y(t)|p)≤C,(26)𝔼|Y(t)-Y-(t)|p≤CΔp/2,t∈[t0,T],
where Δ=max{Δ1,Δ2,…,Δn}:=(ϵ/q-1)T, Δ→0, when ϵ→q.Lemma 4.
If (A1) and (A2) hold, then(27)𝔼∫t0T‍|b(Y-(s),Y-(qs),r(s))-b(Y-(s),Y-(qs),r-(s))|pds≤CΔ,𝔼∫t0T‍∥σ(Y-(s),Y-(qs),r(s))-σ(Y-(s),Y-(qs),r-(s))∥pds≤CΔ.Proof.
Letn=[logϵ/q(T/t0)]+1, then
(28)𝔼∫t0T‍|b(Y-(s),Y-(qs),r(s))-b(Y-(s),Y-(qs),r-(s))|pds=∑i=0n-1‍𝔼∫titi+1‍|b(Y-(s),Y-(qs),r(s))-b(Y-(s),Y-(qs),r(ti))|pds.
By (11), we compute
(29)𝔼∫titi+1‍|b(Y-(s),Y-(qs),r(s))-b(Y-(s),Y-(qs),r(ti))|pds≤2p-1𝔼∫titi+1‍(|b(Y-(s),Y-(qs),r(s))|p+|b(Y-(s),Y-(qs),r(ti))|p)×I{r(s)≠r(ti)}ds≤C𝔼∫titi+1‍(|Y-(qs)|p(q1+1)1+|Y-(s)|p+|Y-(qs)|p(q1+1))I{r(s)≠r(ti)}ds≤C∫titi+1‍𝔼[+|Y-(qs)|p(q1+1))I{r(s)≠r(ti)|r(ti)]]𝔼[|Y-(qs)|p(q1+1)(+|Y-(qs)|p(q1+1))1+|Y-(s)|p+|Y-(qs)|p(q1+1))I{r(s)≠r(ti)}|r(ti)]]ds=C∫titi+1‍𝔼[𝔼[𝔼[I{r(s)≠r(ti)|r(ti)]∫titi+1‍](+|Y-(qs)|p(q1+1)1+|Y-(s)|p+|Y-(qs)|p(q1+1))|r(ti)∫titi+1‍]×𝔼[I{r(s)≠r(ti)}|r(ti)]∫titi+1‍]ds.
By the Markov property, we have(30)𝔼[I{r(s)≠r(tk)}∣r(tk)]=∑i∈S‍I{r(tk)=i}P(r(s)≠i∣r(tk)=i)=∑i∈S‍I{r(tk)=i}∑j≠i‍(γij(s-tk)+o(s-tk))≤(max1≤i≤n(-γii)Δ+o(Δ))∑i∈S‍I{r(tk)=i}≤CΔ.
Substituting the above inequality into (29) yields
(2)𝔼∫titi+1‍|b(Y-(s),Y-(qs),r(s))-b(Y-(s),Y-(qs),r(ti))|pds(31)≤CΔ∫titi+1‍[1+|Y-(ti)|p+|Y-(qti)|p(q1+1)]ds≤CΔ.
So, (28) becomes
(32)𝔼∫t0T‍|b(Y-(s),Y-(qs),r(s))-b(Y-(s),Y-(qs),r-(s))|pds≤CΔ.
Similarly, we also obtain that(33)𝔼∫t0T‍∥σ(Y-(s),Y-(qs),r(s))-σ(Y-(s),Y-(qs),r-(s))∥pds≤CΔ.
The proof is complete.Theorem 5.
Under (A1) and (A2), for anyp≥2 there exits C>0 such that
(34)𝔼(supt0≤t≤T|X(t)-Y(t)|p)≤CΔp/2;
that is, the convergence order of Euler-Maruyama scheme (23) is 1/2.Proof.
Letδ>1 and ε>0; then∫ε/δε‍(1/x)dx=lnx|ε/δε=lnδ, and there is a continuous nonnegative function ψδε(x)(x≥0), which is zero outside [ε/δ,ε], such that(35)∫ε/δε‍ψδε(x)dx=1,ψδε(x)≤2xlnδ,x>0.
Define(36)ϕδε(x):=∫0x‍∫0y‍ψδε(z)dzdy,x>0,Vδε(x):=ϕδε(|x|),x∈ℝn.
For any t∈[t0,T], let
(37)Z(t):=X(t)-Y(t),Z-(t):=Y(t)-Y-(t),Z~(t):=(X(t),Y-(t))∈ℝ2n.
Using the Itô formula, we have(38)Vδε(Z(t))=∫t0t‍〈Y-(s),Y-(Vδε)x(Z(s)),b(X(s),X(qs),r(s))-b(Y-(s),Y-(qs),r-(s))〉ds+12∫t0t‍trace{(-σ(Y-(s),Y-(qs),r-(s)))σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r-(s)))*×(Vδϵ)xx(Z(s))×(-σ(Y-(s),Y-(qs),r-(s)))}σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r-(s)))}ds+∫t0t‍〈Y-(s),Y-(qs),r-(s)(Vδε)x(Z(s)),σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r-(s))〉dW(s):=I1(t)+I2(t)+I3(t).
By virtue of condition (A1), the Hölder inequality, and Lemma4, we deduce that
(39)𝔼(supt0≤s≤t|I1(s)|p)≤(t-t0)p-1𝔼∫t0t‍|〈Y-(s),Y-(qs),r-(Vδε)x(Z(s)),b(X(s),X(qs),r(s))-b(Y-(s),Y-(qs),r-(s))〉|pds≤(t-t0)p-1𝔼∫t0t‍|Y-(s),Y-(qs)b(X(s),X(qs),r(s))-b(Y-(s),Y-(qs),r-(s))|pds≤CΔp-1∫t0t‍𝔼(-b(Y-(s),Y-(qs),r-(s))|p|-b(Y-(s),Y-(qs),r-(s))|pb(X(s),X(qs),r(s))-b(Y-(s),Y-(qs),(s))|p+|b(Y-(s),Y-(qs),r(s))-b(Y-(s),Y-(qs),r-(s))|p)ds≤CΔp-1∫t0t‍{(𝔼V12p(Z~(qs)))12𝔼|Z(s)|p+(𝔼V12p(Z~(qs)))1/2×(𝔼|Z(qs)|2p)1/2+𝔼|Z-(s)|p+(𝔼V12p(Z~(qs)))1/2×(𝔼|Z-(qs)|2p)1/2(𝔼V12p(Z~(qs)))12}ds+CΔp.
By the Hölder inequality and (A2), we have(40)𝔼(supt0≤s≤t|I2(s)|p)≤12(t-t0)p-1𝔼×∫t0t‍|-σ(Y-(s),Y-(qs),r-(s)))}trace{)(-σ(Y-(s),Y-(qs),r-(s)))*σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r-(s)))*×(Vδε)xx(Z(s))×(Y-(s),Y-(qs),r-(s)σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r-(s)))-σ(Y-(s),Y-(qs),r-(s)))}|pds≤CΔp-1∫t0t‍𝔼{-σ(Y-(s),Y-(qs),r-(s))∥2∥(Vδϵ)xx(Z(s))∥×∥(Y-(s),Y-(qs),r-(s))σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r-(s))∥2}pds≤CΔp-1𝔼∫t0t‍1|Z(s)|p×{-σ(Y-(s),Y-(qs),r-(s))∥2p∥Y-(s),Y-(qs),r(s)σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r(s))∥2p+∥σ(Y-(s),Y-(qs),r(s))-σ(Y-(s),Y-(qs),r-(s))∥2p}≤CΔp-1∫t0t‍{×(𝔼|Z-(qs)|4p)1/2}𝔼|Z(s)|p+1εp(𝔼V24p(Z~(qs)))1/2(𝔼|Z(qs)|4p)1/2+1εp𝔼|Z-(s)|2p+1εp(𝔼V24p(Z~(qs)))1/2×(𝔼|Z-(qs)|4p)1/2}ds+CΔpεp.
Making use of the Burkhold-Davis-Gundy inequality yields(41)𝔼(supt0≤s≤t|I3(s)|p)≤CΔp/2-1𝔼∫t0t‍∥-σ(Y-(s),Y-(qs),r-(s))σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r-(s))∥pds≤CΔp/2-1𝔼∫t0t‍(∥-σ(Y-(s),Y-(qs),r-(s))-σ(Y-(s),Y-(qs),r(s)))∥pσ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r(s))∥p+∥-σ(Y-(s),Y-(qs),r(s)))σ(X(s),X(qs),r(s))-σ(Y-(s),Y-(qs),r-(s))-σ(Y-(s),Y-(qs),r(s)))∥p)ds≤CΔp/2-1∫t0t‍{𝔼|Z(s)|p+(𝔼V22p(Z~(qs)))1/2×(𝔼|Z(qs)|2p)1/2+𝔼|Z-(s)|p+(𝔼V22p(Z~(qs)))1/2×(𝔼|Z-(qs)|2p)1/2}ds+CΔp/2.
Moreover, by (8), (9), and (25), we have
(42)𝔼V12p(Z~(qs))∨𝔼V24p(Z~(qs))≤C,𝔼|Z-(t)|p≤CΔp/2.
Thus, combing (39), and (40) with (41), for any t∈[t0,T] and p≥2, we get
(43)𝔼(supt0≤s≤t|Z(s)|p)≤2p-1{εp+𝔼(supt0≤s≤tVδεp(Z(s)))}≤C{+1εp∫t0t‍(𝔼|Z(qs)|4p)1/2ds}}εp+Δ3p/2-1+Δp+Δ2p-1εp+Δpεp+Δp-1+Δp/2+Δp/2-1{∫t0t‍𝔼|Z(s)|pds+∫t0t‍(𝔼|Z(qs)|2p)1/2ds+1εp∫t0t‍(𝔼|Z(qs)|4p)1/2ds}}.
Letε=Δ1/2, and using the Gronwall inequality, we have
(44)𝔼(supt0≤s≤t|Z(s)|p)≤C{Δp/2+Δ(p/2)-1∫t0t‍(𝔼|Z(qs)|2p)1/2ds+Δ-1∫t0t‍(𝔼|Z(qs)|4p)1/2ds}.
Let(45)pi:=([logϵ/qTt0]+2-i)p4[logϵ/q(T/t0)]+1-i,i=1,2,…,[logϵ/qTt0]+1;
by p≥2, it is easy to see that pi≥2 such that
(46)4pi+1<pi,p[logϵ/q(T/t0)]+1=p,i=1,2,…,[logϵ/qTt0].
Noting thatZ(s)=0 for s∈[qt0,t0] and substituting ϵ<1 into (44) yields that
(47)𝔼(supt0≤s≤(t0/q)ϵ|Z(s)|p1)≤C{Δp1/2+Δp1/2-1∫t0(t0/q)ϵ‍(𝔼|Z(qs)|2p1)1/2ds+Δ-1∫t0(t0/q)ϵ‍(𝔼|Z(qs)|4p1)1/2ds}≤C{Δp1/2+Δp1/2-1∫qt0t0‍(𝔼|Z(s)|2p1)1/2ds+Δ-1∫qt0t0‍(𝔼|Z(s)|4p1)1/2ds}≤CΔp1/2.
Using (46) and the Hölder inequality, further gives that
(48)𝔼(supt0≤s≤(t0/q2)ϵ2|Z(s)|p2)≤C{Δp2/2+Δp2/2-1∫t0(t0/q2)ϵ2‍(𝔼|Z(qs)|2p2)1/2ds+Δ-1∫t0(t0/q2)ϵ2‍(𝔼|Z(qs)|4p2)1/2ds}≤C{Δp2/2+Δp2/2-1∫t0(t0/q)ϵ‍(𝔼|Z(s)|p1)p2/p1ds+Δ-1∫t0(t0/q)ϵ‍(𝔼|Z(s)|p1)2p2/p1ds}≤CΔp2/2.
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 is1/2. 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

Letℬ(ℝ) be the Borel σ-algebra on ℝ, and λ(dx) a σ-finite 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)=∑s∈Dp,s≤t‍IU(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 onℝn:
(49)dX(t)=b(X(t),X(qt),r(t))dt+∫U‍h(X(t),X(qt),u)N~(dt,du),t∈[t0,T]
with initial data X(θ)=ξ(θ) and r(θ)=r0, θ∈[qt0,t0].We assume that(A1)
b:ℝn×ℝn×S→ℝn and there exists L1>0 such that
(50)|b(x1,y1,j)-b(x2,y2,j)|≤L1|x1-x2|+V1(y1,y2)|y1-y2|
for xi,yi∈ℝn, i=1,2, j∈S;(A3)
h:ℝn×ℝn×U→ℝn and there exists L3>0 such that
(51)|h(x1,y1,u)-h(x2,y2,u)|≤(L3|x1-x2|+V3(y1,y2)|y1-y2|)|u|
for xi,yi∈ℝn,i=1,2, and u∈U, where V3:ℝn×ℝn→ℝ+ such that
(52)V3(x,y)≤K3(1+|x|q3+|y|q3)
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
(53)dY(t)=b(Y-(t),Y-(qt),r-(t))dt+∫U‍h(Y-(t),Y-(qt),u)N~(dt,du),
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]).
LetΦ:ℝ+×U→ℝn and assume that
(54)∫t0t‍∫U‍𝔼|Φ(s,u)|pλ(du)ds<∞,t0>0,p≥2.
Then there existsD(p)>0 such that
(55)𝔼(supt0≤s≤t|∫t0s‍∫U‍Φ(r,u)N~(du,ds)|p)≤D(p){𝔼(∫t0t‍∫U‍|Φ(s,u)|2λ(du)ds)p/2+𝔼∫t0t‍∫U‍|Φ(s,u)|pλ(du)ds{𝔼(∫t0t‍∫U‍|Φ(s,u)|2λ(du)ds)p/2}.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
(56)𝔼(supt0≤t≤T|X(t)|p)∨𝔼(supt0≤t≤T|Y(t)|p)≤C,(57)𝔼|Y(t)-Y-(t)|p≤CΔ.Proof.
The proof is very similar to that of Lemma2 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
(58)𝔼(supt0≤t≤T|X(t)-Y(t)|p)≤CΔ(1+θ)1/[logϵ/q(T/t0)](1+α).Proof.
The proof is similar to that of Theorem5. Set (59)Z(t):=X(t)-Y(t),Z-(t):=Y(t)-Y-(t),Z~(t):=(X(t),Y-(t))∈ℝ2n,t∈[t0,T].
Define
(60)Γ1(t):=b(X(t),X(qt),r(t))-b(Y-(t),Y-(qt),r-(t)),Γ2(t,u):=h(X(t),X(qt),u)-h(Y-(t),Y-(qt),u).
Using the Itô formula and the Taylor expansion we have that for t∈[t0,T](61)Vδε(Z(t))=∫t0t‍〈(Vδε)x(Z(s)),Γ1(s)〉ds+∫t0t‍∫U‍{Vδε(Z(s)+Γ2(s,u))-Vδε(Z(s))-〈(Vδε)x(Z(s)),Γ2(s,u)〉}×λ(du)ds+∫t0t‍∫U‍{Vδε(Z(s)+Γ2(s,u))-Vδε(Z(s))}N~(du,ds)=∫t0t‍〈(Vδε)x(Z(s)),Γ1(s)〉ds+∫t0t‍∫U‍{∫01‍〈(Vδε)x(Z(s)+θΓ2(s,u))-(Vδε)x(Z(s)),Γ2(s,u)〉dθ∫01‍}×λ(du)ds+∫t0t‍∫U‍{∫01‍〈(Vδε)x(Z(s)+θΓ2(s,u)),Γ2(s,u)〉dθ∫01‍}N~(du,ds).
By the property ofVδε(x), we deduce that
(62)|Z(t)|≤ε+Vδε(Z(t))≤ε+∫t0t‍|Γ1(s)|ds+2∫t0t‍∫U‍|Γ2(s,u)|λ(du)ds+∫t0t‍∫U‍{∫01‍〈(Vδε)x(Z(s)+θΓ2(s,u)),Γ2(s,u)〉dθ∫01‍}N~(du,ds),t∈[t0,T].
From (8), (52), and (56), we compute that for any p≥2(63)𝔼(supt0≤t≤TV1p(Z~(qs)))∨𝔼(supt0≤t≤TV3p(Z~(qs)))≤C.
Applying Lemma6, Lemma 4, (57), and the Hölder inequality, we obtain that
(64)𝔼(supt0≤s≤t|Z(s)|p)≤2p-1{εp+𝔼(supt0≤s≤tVδεp(Z(s)))}≤C{+𝔼(∫t0t‍∫U‍|Γ2(s,u)|2λ(du)ds)p/2}εp+∫t0t‍𝔼|Γ1(s)|pds+∫t0t‍∫U‍𝔼|Γ2(s,u)|pλ(du)ds+𝔼(∫t0t‍∫U‍|Γ2(s,u)|2λ(du)ds)p/2}≤C{εp+∫t0t‍𝔼|Γ1(s)|pds+∫t0t‍∫U‍𝔼|Γ2(s,u)|pλ(du)ds}≤C{εp+∫t0t‍𝔼(|X(s)-Y-(s)|+V1(Z~(qs))|X(qs)-Y-(qs)|)pds+∫t0t‍𝔼|b(Y-(s),Y-(qs),r(s))-b(Y-(s),Y-(qs),r-(s))|pds+∫t0t‍𝔼(|X(s)-Y-(s)|+V3(Z~(qs))|X(qs)-Y~(qs)|)pds∫t0t‍}≤C{εp+Δ+∫t0t‍{(∫t0t‍V3p(Z~(qs))|Z-(qs)|p)𝔼|Z(s)|p+𝔼(V1p(Z~(qs))|Z(qs)|p)+𝔼(V1p(Z~(qs))|Z-(qs)|p)+𝔼(V3p(Z~(qs))|Z(qs)|p)+𝔼(V3p(Z~(qs))|Z-(qs)|p)∫t0t‍}ds∫t0t‍}≤C{εp+Δ+∫t0t‍𝔼|Z(s)|pds+∫t0t‍𝔼|Z(qs)|pds}.
Together with the Gronwall inequality and takingε=Δ1/p, we get(65)𝔼(supt0≤s≤t|Z(s)|p)≤C{Δ+∫t0t‍𝔼|Z(qs)|pds}.
Forθ∈(0,1) and any α∈(0,1), let
(66)pi:=p(1+θ)([logϵ/q(T/t0)]+1-i)(1+α),i=1,2,…,[logϵ/qTt0]+1.
It is easy to see that(67)(1+θ)pi+1<pi,p[logϵ/q(T/t0)]+1=p,i=1,2,…,[logϵ/qTt0].
Noting thatZ(t)=Z-(t)=0 for t∈[qt0,t0], from (65) we obtain
(68)𝔼(supt0≤s≤(t0/q)ϵ|Z(s)|p1)≤C{Δ+∫qt0t0‍𝔼|Z(s)|p1ds}≤CΔ.
Then, together with (67) and the Hölder inequality,  it  further gives that
(69)𝔼(supt0≤s≤(t0/q2)ϵ2|Z(s)|p2)≤C{Δ+∫t0(t0/q)ϵ‍(𝔼|Z(s)|p2(1+θ))1/(1+θ)ds}≤C{Δ+∫t0(t0/q)ε‍(𝔼|Z(s)|p1)p2/p1ds}≤CΔp2/p1.
Similarly,(70)𝔼(supt0≤s≤(t0/q3)ϵ3|Z(s)|p3)≤C{Δ+∫t0(t0/q2)ϵ2‍(𝔼|Z(s)|p3(1+θ))1/(1+θ)ds}≤C{Δ+∫t0(t0/q2)ϵ2‍(𝔼|Z(s)|p2)p3/p2ds}≤CΔp3/p1.
Repeating the previous procedures, we have(71)𝔼(supt0≤s≤T|Z(s)|p)≤CΔ(1+θ)1/[logϵ/q(T/t0)](1+α).
The proof is complete.



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*Source: 420648-2013-06-26.xml*