# Approximate Solutions of Hybrid Stochastic Pantograph Equations with Levy Jumps

Authors: Wei Mao; Xuerong Mao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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 inL2 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.


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## Body

## 1. Introduction

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 L2 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 [21], we need to develop several new techniques to deal with Levy jumps. Some known results in Fan et al. [14], Milošević and Jovanović [16], and Marion et al. [21] are generalized to cover a class of more general SPEwMsLJs.The paper is organized as follows. In Section2, 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 L2 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(Ω,ℱ,P) be a complete probability space with a filtration (ℱt)t≥0 satisfying the usual condition; that is, the filtration (ℱt) is continuous on the right and (ℱ0) contains all P-null sets. Let {W(t), t≥0} be a d-dimensional Wiener process defined on the probability space (Ω,ℱ,P) adapted to the filtration (ℱt)t≥0. Let D([0,T],Rn) denote the family of function f from [0,T]→Rn that are right continuous and have limits on the left. Also D([0,T],Rn) is equipped with the norm ∥f∥=sup0≤t≤T|f(t)|, where |·| is the Euclidean norm in Rn; that is, |x|=x⊤x(x∈Rn). Let T>0, P≥2, and ℒP([0,T];Rn) denote the family of all Rn-valued measurable (ℱt)-adapted processes f={f(t)}0≤t≤T such that Esup0≤t≤T|f(t)|P<∞. Let (Rn,ℬ(Rn)) be a measurable space and π(du) a σ-finite measure on it. Let p=p(t), and let t∈Dp be a stationary ℱt-Poisson point process on Rn with characteristic measure π. Denote by N(dt,du) the Poisson counting measure associated with p; that is,
(1)N(t,A)=∑s∈Dp,s≤t‍IA(p(s)).
We refer to Mao [3] for the properties of a Wiener process and SDDEs and to Ikeda and Watanabe [22] for the details on Poisson point process.Letr(t), t≥0, be a right-continuous Markov chain on the probability space (Ω,ℱ,P) taking values in a finite state space S={1,2,…,N} with generator Γ=(γij)N×N given by
(2)P(r(t+Δ)=j∣r(t)=i)={γijΔ+∘(Δ),ifi≠j,1+γijΔ+∘(Δ),ifi=j,
where Δ>0. Here γij≥0 is the transition rate from i to j, i≠j, while
(3)γii=-∑j≠i‍γij.
We assume that Markov chain r(·) is independent of the Brownian motion W(·) and compensated Poisson random measure N~(·,·). It is known that almost every sample path of r(·) is right-continuous step function with a finite number of simple jumps in any finite subinterval of R+.In this paper, we study the following hybrid stochastic pantograph equations with Levy jump:(4)dX(t)=f(X(t),X(qt),r(t))dt+g(X(t),X(qt),r(t))dW(t)+∫Rnh(X(t),X(qt),r(t),u)N~(dt,du),X(0)=X0,
where 0<q<1 and
(5)f:R+×Rn×S→Rn,g:R+×Rn×S→Rn×m,h:R+×Rn×S×Rn→Rn,W(t) is a standard m-dimensional Brownian motion, and N~(dt,du) is the compensated Poisson random measure given by
(6)N~(dt,du)=N(dt,du)-π(du)dt.
Here π(du) is the Levy measure associated to N.LetC2(Rn×S,R+) denote the family of all nonnegative functions V(x,i) on Rn×S which are continuously twice differentiable in x. For each V∈C2(Rn×S,R+), define an operator LV from Rn×Rn×S to R by
(7)LV(x,y,i)≡Vx(x,i)f(x,y,i)+12trace[g⊤(x,y,i)Vxx(x,i)g(x,y,i)]+∫Rn‍[V(x+h(x,y,u),i)-V(x,i)+∫Rn‍-Vx(x,i)h(x,y,u)]π(du)+∑j=1N‍γijV(x,j),
where
(8)Vx(x,i)=(∂V(x,i)∂x1,…,∂V(x,i)∂xn),Vxx(x,i)=(∂2V(x,i)∂xi∂xj)n×n.In order to define the Euler approximate solution of (4), we need the property of embedded discrete Markov chain. The following lemma [23] describes this property.Lemma 1.
Forh>0 and n≥0, then {rnh,n=0,1,2,…} is a discrete Markov chain with the one-step transition probability matrix
(9)P(h)=(Pij(h))N×N=ehΓ.Given a step sizeh>0, the discrete Markov chain {rnh,n=0,1,2,…} can be simulated as follows (see Mao and Yuan [24]). Let r(0)=i0 and generate a random number ζ1 which is uniformly distributed in [0,1]. If ζ1=1, then let r1h=i1=N or otherwise find the unique integer i1∈S for
(10)∑j=1i1-1‍Pi0,j(h)≤ζ1<∑j=1i1‍Pi0,j(h)
and let r1h=i1, where we set ∑i=10‍Pi0,j(h)=0 as usual. Generate independently a new random number ζ2 which is again uniformly distributed in [0,1]. If ζ2=1 then let r2h=i2=N or otherwise find the unique integer i2∈S for
(11)∑j=1i2-1‍Pi1,j(h)≤ζ2<∑j=1i2‍Pi1,j(h)
and let r2h=i2. Repeating this procedure, a trajectory of {rnh,n=0,1,2,…} 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 {rnh=r(nh),n=0,1,2,…}. For system (4), the discrete approximation is given by the iterative scheme
(12)Yn+1=Yn+f(Yn,Y[qn],rnh)h+g(Yn,Y[qn],rnh)ΔWn+∫Rn‍h(Yn,Y[qn],rnh,u)N~(h,du),
with initial value Y0=X(0), and [u] represents the integer part of u. Here tn=nh for n≥0. We have Yn≈X(tn), Y[qn]≈X(qtn), ΔWn=W(tn+1)-  W(tn), and N~(h,du)=N~(tn+1,du)-N~(tn,du).Let us introduce the following notations:(13)Z1(t)=Yn,Z2(t)=Y[qn],r¯(t)=rnh
for t∈[tn,tn+1). Then we define the continuous Euler approximate solution as follows:
(14)Y(t)=Y(0)+∫0t‍f(Z1(s),Z2(s),r¯(s))ds+∫0t‍g(Z1(s),Z2(s),r¯(s))dW(s)+∫0t‍∫Rn‍h(Z1(s),Z2(s),r¯(s),u)N~(ds,du),(Z1(s),Z2(s),r¯(s),u)N~(ds,du),0≤t≤T,
which interpolates the discrete approximation (7).In order to establish the strong convergence theorem, we suppose the following assumptions are satisfied.Assumption 2.
For eachi∈S and u∈Rn,
(15)|f(0,0,i)|2=|g(0,0,i)|2=∫Rn‍|h(0,0,i,u)|2π(du)=0.Assumption 3.
For everyd≥1, there exists a positive constant Kd such that for all x1,y1,x2,y2,u∈Rn and i∈S, |x1|∨|y1|∨|x2|∨|y2|≤d,
(16)|f(x1,y1,i)-f(x2,y2,i)|2∨|g(x1,y1,i)-g(x2,y2,i)|2∨∫Rn‍|h(x1,y1,i,u)-h(x2,y2,i,u)|2π(du)≤Kd(|x1-x2|2+|y1-y2|2).


## 3. Strong Convergence of Numerical Solutions

In this section, we will prove that the Euler approximate solutions converge to the true solutions inL2 sense under the local Lipschitz condition.Theorem 4.
If Assumptions2 and 3 hold, then the Euler approximate solutions converge to the true solutions of (4) in L2 sense with order 1/2; that is, there exists a positive constant Cd such that
(17)E[sup0≤t≤T∧τd|X(t)-Y(t)|2]≤Cd[h+∘(h)],
where θd=inf{t∈[0,T]:|X(t)|≥d} and ρd=inf{t∈[0,T]:|Y(t)|≥d}, and let τd=θd∧ρd.The proof of Theorem4 is very technical, so we present some useful lemmas.Lemma 5.
Under Assumptions2 and 3, for any t∈[0,T] and P≥2, there exists a positive constant C1(d) such that
(18)E∫0t‍|Y(s∧τd)-Z1(s∧τd)|Pds≤C1(d)hP/2T,
where C1(d) is a positive constant independent of the step size h.Proof.
For anyt∈[0,T∧τd], there exists an integer n such that t∈[nh,(n+1)h). Then
(19)Y(t)-Z1(t)=Y(t)-Yn=∫nht‍f(Z1(s),Z2(s),r¯(s))ds+∫nht‍g(Z1(s),Z2(s),r¯(s))dW(s)+∫nht‍∫Rn‍h(Z1(s),Z2(s),r¯(s),u)N~(ds,du).
Using the inequality |a+b+c|P≤3P-1[|a|P+3|b|P+3|c|P], we get
(20)E[supnh≤t≤(n+1)h|Y(t)-Z1(t)|P]≤3P-1[E(supnh≤t≤(n+1)h|∫nht‍f(Z1(s),≤3P-1supnh≤t≤(n+1)h∫nht‍fZ2(s),r¯(s))ds{∫nht‍f(Z1(s),}|P{supnh≤t≤(n+1)h})+3P-1E×(supnh≤t≤(n+1)h|∫nht‍g(Z1(s),+3P-1Esupnh≤t≤(n+1)hZ2(s),r¯(s))dW(s){∫nht‍}|P{supnh≤t≤(n+1)h})+3P-1E(supnh≤t≤(n+1)h|∫nht‍∫Rn‍h(Z1(s),∫nht‍∫Rn‍+3Psupnh≤t≤(n+1)h-1EEZ2(s),r¯(s),u)+3P-1∫nht‍∫Rn‍supnh≤t≤(n+1)hE×N~(ds,du){∫nht‍∫Rn‍}|P{∫nht‍∫Rn‍supnh≤t≤(n+1)h})].
By the Hölder inequality and Assumptions 2 and 3, we have
(21)E(supnh≤t≤(n+1)h|∫nht‍f(Z1(s),Z2(s),r¯(s))ds|P)≤E(|∫nh(n+1)h‍f(Z1(s),Z2(s),r¯(s))ds|P)≤hP-1E∫nh(n+1)h‍|f(Z1(s),Z2(s),r¯(s))|Pds≤hP-1E∫nh(n+1)h‍[|f(Z1(s),Z2(s),r¯(s))|2]P/2ds≤hP-1E∫nh(n+1)h‍[2|f(Z1(s),Z2(s),r¯(s))≤hP-1E∫nh(n+1)h‍-f(0,0),r¯(s)|2≤hP-1E∫nh(n+1)h‍+2|f(0,0),r¯(s)|2]P/2ds≤hP-1E∫nh(n+1)h‍[2Kd(|Z1(s)|2+|Z2(s)|2)]P/2ds≤hP-12P-1KdP/2E∫nh(n+1)h(|Z1(s)|P+|Z2(s)|P)ds.
By the definition of τd, we have |Y(t)|<d, t∈[0,τd∧T]. so we get that |Z1(t)|P<dP, |Z2(t)|P<dP, and
(22)E(supnh≤t≤(n+1)h|∫nht‍f(Z1(s),Z2(s),r¯(s))ds|P)≤2PdPKdP/2hP.
By using the Burkholder-Davis-Gundy inequality and Assumptions 2 and 3, we have
(23)E(supnh≤t≤(n+1)h|∫nht‍g(Z1(s),Z2(s),r¯(s))dW(s)|P)≤CPE[∫nh(n+1)h‍|g(Z1(s),Z2(s),r¯(s))|2ds]P/2≤CPh(P/2)-1E[∫nh(n+1)h‍|g(Z1(s),Z2(s),r¯(s))|Pds]≤Cp2PdPKdP/2hP/2,(24)E(supnh≤t≤(n+1)h|∫nht‍∫Rn‍h(Z1(s),Z2(s),r¯(s),u)N~(ds,du)|P)≤CPE[∫nh(n+1)h‍∫Rn‍|h(Z1(s),Z2(s),r¯(s),u)|2π(du)ds]P/2≤CPh(P/2)-1E×∫nh(n+1)h‍[∫Rn‍|h(Z1(s),Z2(s),r¯(s),u)|2π(du)]P/2ds≤CPh(P/2)-1E×∫nh(n+1)h‍{∫Rn‍[{-h(0,0,r¯(s),u)|2}2|h(Z1(s),Z2(s),r¯(s),u)×∫Rn‍∫nh(n+1)h‍-h(0,0,r¯(s),u)|2×∫Rn‍∫nh(n+1)h‍+|h(0,0,r¯(s),u)|2]π(du){∫Rn‍[2|h(Z1(s),Z2(s),r¯(s),u)-h(0,0,r¯(s),u)|2}}P/2ds≤CPh(P/2)-1E∫nh(n+1)h‍[2Kd(|Z1(s)|2+|Z2(s)|2)]P/2ds≤Cp2PdPKdP/2hP/2.
Combing (22)–(24) together, we have
(25)E|Y(t)-Z1(t)|P≤3P-1[2PdPKdP/2hP+2Cp2PdPKdP/2hP/2],E∫0t∧τd‍|Y(s)-Z1(s)|Pds≤C1(d)hP/2T,
where C1(d)=3P-1[2PdPKdP/2+2Cp2PdPKdP/2]. The proof is complete.Lemma 6.
Under Assumptions2 and 3, for any t∈[0,T] and P≥2, then
(26)E∫0t‍|Y(q(s∧τd))-Z2(s∧τd)|Pds≤C2(d)hP/2T,
where C2(d) is a positive constant independent of the stepsize h.The proof of this lemma is similar to that of Lemma5.Lemma 7.
Under Assumptions2 and 3, for any t∈[0,T] and P≥2, then
(27)E∫0t‍|f(Z1(s∧τd),Z2(s∧τd),r(s∧τd))E∫0t‍-f(Z1(s∧τd),Z2(s∧τd),r¯(s∧τd))|Pds∨E∫0t‍|g(Z1(s∧τd),Z2(s∧τd),r(s∧τd))-g(Z1(s∧τd),Z2(s∧τd),r¯(s∧τd))|Pds∨E∫0t‍∫Rn‍|h(Z1(s∧τd),Z2(s∧τd),r(s∧τd),u)∫0t‍∫Rn‍-h(Z1(s∧τd),Z2(s∧τd),r¯(s∧τd),u)|P∫0t‍∫Rn‍×π(du)ds≤C3(d)[hP/2+∘(hP/2)],
where C3(d) is a positive constant independent of the stepsize h.The proof of this lemma is similar to that of [16, 24].Proof of Theorem4.
Combining (4) and (14), one has
(28)X(t)-Y(t)=∫0t‍[f(X(s),X(qs),r(s))-f(Z1(s),Z2(s),r¯(s))]ds+∫0t‍[g(X(s),X(qs),r(s))-g(Z1(s),Z2(s),r¯(s))]dW(s)+∫0t‍∫Rn‍[h(X(s),X(qs),r(s),u)+∫0t‍∫Rn‍-h(Z1(s),Z2(s),r¯(s),u)]N~(ds,du).
Then applying the generalized Itô’s formula, we can show that
(29)|X(t)-Y(t)|2=2∫0t‍|X(s)-Y(s)|=22∫0t‍×|f(X(s),X(qs),r(s))=2∫0t‍-f(Z1(s),Z2(s),r-(s))|ds+∫0t‍|g(X(s),X(qs),r(s))-g(Z1(s),Z2(s),r-(s))|2ds+∫0t‍∫Rn‍|h(X(s),X(qs),r(s),u)+∫0t‍∫Rn‍-h(Z1(s),Z2(s),r-(s),u)|2π(du)ds+2∫0t‍|X(s)-Y(s)|+2∫0t‍×|g(X(s),X(qs),r(s))+2∫0t‍-g(Z1(s),Z2(s),r-(s))|dW(s)+∫0t‍∫Rn‍[|h(X(s),X(qs),r(s),u)2|X(s)-Y(s)|+∫0t‍∫Rn‍×|h(X(s),X(qs),r(s),u)+∫0t‍∫Rn‍-h(Z1(s),Z2(s),r-(s),u)||h(X(s),X(qs),r(s),u)]N~(ds,du)+∫0t‍∫Rn‍|h(X(s),X(qs),r(s),u)+∫0t‍∫Rn‍-h(Z1(s),Z2(s),r-(s),u)|2N~(ds,du).
Hence, for any t∈[0,T], we get
(30)Esup0≤t≤T∧τd|X(t)-Y(t)|2≤E∫0T∧τd‍|X(s)-Y(s)|2ds+E∫0τd∧T‍|f(X(s),X(qs),r(s))+E∫0τd∧T‍-f(Z1(s),Z2(s),r¯(s))|2ds+E∫0τd∧T‍|g(X(s),X(qs),r(s))+2E∫0τd∧T‍-g(Z1(s),Z2(s),r¯(s))|2ds+E∫0τd∧T‍∫Rn‍|h(X(s),X(qs),r(s),u)+E∫0τd∧T‍∫Rn‍-h(Z1(s),Z2(s),r¯(s),u)|2π(du)ds+2Esup0≤t≤τd∧T∫0t‍|X(s∧τd)-Y(s∧τd)|+2Esup0≤t≤τd∧T∫0t‍×|g(X(s∧τd),X(q(s∧τd)),r(s))+2Esup0≤t≤τd∧T∫0t‍-g(Z1(s),Z2(s),r¯(s))|dW(s)+Esup0≤t≤τd∧T∫0t‍∫Rn‍[-h(Z1(s),Z2(s),r¯(s),u)|2|X(s)-Y(s)|+Esup0≤t≤τd∧T∫0t‍∫Rn‍×|h(X(s),X(qs),r(s),u)+Esup0≤t≤τd∧T∫0t‍∫Rn‍-h(Z1(s),Z2(s),r¯(s),u)|]+Esup0≤t≤τd∧T∫0t‍∫Rn‍×N~(ds,du)+Esup0≤t≤τd∧T∫0t‍∫Rn‍|h(X(s),X(qs),r(s),u)+Esup0≤t≤τd∧T∫0t‍∫Rn‍-h(Z1(s),Z2(s),r¯(s),u)|2+Esup0≤t≤τd∧T∫0t‍∫Rn‍×N~(ds,du).
By Assumption 3 and Lemmas 5–7, we have
(31)E∫0τd∧T‍|f(X(s),X(qs),r(s))E∫0τd∧T‍-f(Z1(s),Z2(s),r¯(s))|2ds≤2E∫0τd∧T‍|f(X(s),X(qs),r(s))≤2E∫0τd∧T‍-f(Z1(s),Z2(s),r(s))|2ds+2E∫0τd∧T‍|f(Z1(s),Z2(s),r(s))+2E∫0τd∧T‍-f(Z1(s),Z2(s),r¯(s))|2ds≤2KdE∫0τd∧T‍(|X(s)-Z1(s)|2≤2KdE∫0τd∧T‍+|X(qs)-Z2(s)|2)ds+2C3(d)h≤2KdE×∫0τd∧T‍(2|X(s)-Y(s)|2+|Y(s)-Z1(s)|2)+(2|X(qs)-Y(qs)|2+2|Y(qs)-Z2(s)|2)ds+2C3(d)[h+∘(h)]≤4Kd[C1(d)+C2(d)]hT+2C3(d)[h+∘(h)]+8Kd∫0τd∧T‍Esup0≤u≤s|X(u)-Y(u)|2ds.
Similarly, by Assumption 3 and Lemmas 5–7, we obtain
(32)E∫0τd∧T‍|g(X(s),X(qs),r(s))E∫0τd∧T‍-g(Z1(s),Z2(s),r¯(s))|2ds≤4Kd[C1(d)+C2(d)]hT+2C3(d)[h+∘(h)]+8Kd∫0τd∧T‍Esup0≤u≤s|X(u)-Y(u)|2ds,(33)E∫0τd∧T‍∫Rn‍|h(X(s),X(qs),r(s),u)E∫0τd∧T‍∫Rn‍-h(Z1(s),Z2(s),r¯(s),u)|2π(du)ds≤4Kd[C1(d)+C2(d)]hT+2C3(d)[h+∘(h)]+8Kd∫0τd∧T‍Esup0≤u≤s|X(u)-Y(u)|2ds.
On the other hand, by the Burkholder-Davis-Gundy inequality, Young’s inequality, and Lemmas 5–7, we have for any ɛ1>0(34)Esup0≤t≤τd∧T∫0t‍|X(s)-Y(s)|Esup0≤t≤τd∧T∫0t‍×|g(X(s),X(qs),r(s))Esup0≤t≤τd∧T∫0t‍-g(Z1(s),Z2(s),r¯(s))|dW(s)≤3E[sup0≤t≤τd∧T|X(t)-Y(t)|≤3E×(∫0τd∧T‍|g(X(t),X(qt),r(t))≤3E×∫0τd∧T‍-g(Z1(t),Z2(t),r¯(t))|2dt{∫0τd∧T‍}){sup0≤t≤τd∧T}]1/2≤3[ɛ1Esup0≤t≤τd∧T|X(t)-Y(t)|2]1/2×[1ɛ1E(∫0τd∧T‍|g(X(t),X(qt),r(t))1ɛ1E∫0τd∧T‍-g(Z1(t),Z2(t),r¯(t))|2dt{∫0τd∧T‍})]1/2≤6ɛ1Esup0≤t≤τd∧T|X(t)-Y(t)|2+6ɛ1E∫0τd∧T‍|g(X(t),X(qt),r(t))+6ɛ1E∫0τd∧T‍-g(Z1(t),Z2(t),r¯(t))|2dt≤6ɛ1Esup0≤t≤τd∧T|X(t)-Y(t)|2+24ɛ1Kd[C1(d)+C2(d)]hT+12ɛ1C3(d)[h+∘(h)]+48ɛ1Kd∫0τd∧T‍Esup0≤u≤s|X(u)-Y(u)|2ds.
We have for any ɛ2>0(35)Esup0≤t≤τd∧T∫0t‍∫Rn‍[2|X(s)-Y(s)||h(X(s),X(qs),r(s),u)Esup0≤t≤τd∧T∫0t‍∫Rn‍-h(Z1(s),Z2(s),r¯(s),u)|]N~(ds,du)≤CE(∑t∈Dp,t≤τ∧T‍|X(t)-Y(t)|2≤CE∑t∈Dp,t≤τ∧T‍×|h(X(t),X(qt),r(t),pt)≤CE∑t∈Dp,t≤τ∧T‍-h(Z1(t),Z2(t),r¯(t),pt)|2{∑t∈Dp,t≤τ∧T‍})1/2≤CEsup0≤t≤τd∧T|X(t)-Y(t)|E×(∑t∈Dp,t≤τd∧T‍|h(X(t),X(qt),r(t),pt)∑t∈Dp,t≤τd∧T‍-h(Z1(t),Z2(t),r¯(t),pt)|2{∑t∈Dp,t≤τd∧T‍})1/2≤CE[ɛ2sup0≤t≤τd∧T|X(t)-Y(t)|2]1/2×E[1ɛ2∑t∈Dp,t≤τd∧T‍|h(X(t),X(qt),r(t),pt)×E1ɛ2(∑t∈Dp,t≤τd∧T‍-h(Z1(t),Z2(t),r¯(t),pt)|2{∑t∈Dp,t≤τd∧T‍}]1/2≤Cɛ2Esup0≤t≤τd∧T|X(t)-Y(t)|2+Cɛ2E(∑t∈Dp,t≤τd∧T‍|h(X(t),X(qt),r(t),pt)+Cɛ2E∑t∈Dp,t≤τd∧T‍-h(Z1(t),Z2(t),r¯(t),pt)|2∑t∈Dp,t≤τd∧T‍|h(X(t),X(qt),r(t),pt))≤Cɛ2Esup0≤t≤τd∧T|X(t)-Y(t)|2+Cɛ2E∫0τd∧T‍∫Rn‍|h(X(t),X(qt),r(t),u)+Cɛ2E∫0τd∧T‍∫Rn‍-h(Z1(t),Z2(t),r¯(t),u)|2π(du)dt≤Cɛ2Esup0≤t≤τd∧T|X(t)-Y(t)|2+4Cɛ2Kd[C1(d)+C2(d)]hT+2Cɛ2C3(d)[h+∘(h)]+8Cɛ2Kd∫0τd∧T‍Esup0≤u≤s|X(u)-Y(u)|2ds,
where C is some constant that may change from line to line. Similar, we have
(36)Esup0≤t≤τd∧T∫0t‍∫Rn‍|h(X(s),X(qs),r(s),u)Esup0≤t≤τd∧T∫0t‍∫Rn‍-h(Z1(s),Z2(s),r¯(s),u)|2N~(ds,du)≤CE(∑t∈Dp,t≤τd∧T‍|h(X(t),X(qt),r(t),pt)≤CE∑t∈Dp,t≤τd∧T‍-h(Z1(t),Z2(t),r¯(t),pt)|4{∑t∈Dp,t≤τd∧T‍})1/2≤CE(∑s∈Dp,s≤τd∧T‍|h(X(t),X(qt),r(t),pt)≤CE∑s∈Dp,s≤τd∧T‍-h(Z1(t),Z2(t),r¯(t),pt)|2{∑s∈Dp,s≤τd∧T‍})≤CE∫0τd∧T‍∫Rn‍|h(X(t),X(qt),r(t),u)≤CE∫0τd∧T‍∫Rn‍-h(Z1(t),Z2(t),r¯(t),u)|2π(du)dt≤4CKd[C1(d)+C2(d)]hT+2CC3(d)[h+∘(h)]+8CKd∫0τd∧T‍Esup0≤u≤s|X(u)-Y(u)|2ds.
Substituting (31)–(36) into (30), we obtain that
(37)E[sup0≤t≤τd∧T|X(t)-Y(t)|2]≤(6ɛ1+Cɛ2)E[sup0≤t≤τd∧T|X(t)-Y(t)|2]+(10+24ɛ1+2C1+ɛ2ɛ2)Kd[C1(d)+C2(d)]Th+(4+12ɛ1)C3(d)[h+∘(h)]+[1+(20+48ɛ1+4C1+ɛ2ɛ2)Kd]×∫0T‍Esup0≤u≤τ∧s|X(u)-Y(u)|2ds.
By choosing ɛ1>0, ɛ2>0 sufficiently small and letting 6ɛ1+Cɛ2=2/3, we have
(38)E[sup0≤t≤τ∧T|X(t)-Y(t)|2]≤3(10+24ɛ1+2C1+ɛ2ɛ2)Kd[C1(d)+C2(d)]Th+3(4+12ɛ1)C3(d)[h+∘(h)]+3[1+(20+48ɛ1+4C1+ɛ2ɛ2)Kd]×∫0T‍Esup0≤u≤τ∧s|X(u)-Y(u)|2ds.
Therefore, we apply Gronwall's inequality to get
(39)E[sup0≤t≤τ∧T|X(t)-Y(t)|2]≤{3(10+24ɛ1+2C1+ɛ2ɛ2)Kd[C1(d)+C2(d)]Th+3(4+12ɛ1)C3(d)[h+∘(h)]}e3[1+(20+(48/ɛ1)+4C((1+ɛ2)/ɛ2))Kd].
This completes the proof.Remark 8.
Under the local Lipschitz condition, Theorem4 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 1/2 by (39).Remark 9.
Whenr(t)≡0 or h≡0, (4) reduces to which was studied by Fan et al. [14], Xiao and Zhang [15], and Milošević and Jovanović [16]. 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 Theorem4, 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.
Forx∈Rn and i∈S, there exist a positive function V∈C2(Rn×S;R+), K>0, and two constants λ1>λ2≥0 such that
(40)lim|x|→∞V(x,i)=∞,(41)LV(x,y,i)≤K[1-λ1V(x,i)+λ2V(y,i)].Assumption 11.
There exists a positive constantLd such that, for all x,y∈Rn and i∈S with |x|∨|y|≤d,
(42)|V(x,i)-V(y,i)|∨|Vx(x,i)-Vx(y,i)|∨|Vxx(x,i)-Vxx(y,i)|≤Ld|x-y|.Now, let us state our convergence theorem.Theorem 12.
Let the assumptions of Theorem4 hold. Also assume that there exists a C2 function V:Rn×S→R+ satisfying (40)–(42). Then the Euler approximate solutions converges to the true solutions of (4) in the sense of the probability.
That is,(43)limh→0sup0≤t≤T|Y(t)-X(t)|2=0,inprobability.Proof.
The proof is rather technical, and we divide it into three steps.
Step1. We assume the existence of the nonnegative Lyapunov function V(x,i) satisfying (40). Applying the Itô’s formula, V(X(t),r(t)) yields
(44)dV(X(t),r(t))=LV(X(t),X(qt),r(t))dt+Vx(X(t),r(t))g(X(t),X(qt),r(t))dW(t)+∫Rn‍[V(X(t)+h(X(t),X(qt),r(t),u),r(t))+∫Rn‍-V(X(t),r(t))]N~(dt,du).
Integrating from 0 to t∧θd and taking expectations gives
(45)EV(X(t∧θd),r(t∧θd))=V(X0,r0)+E∫0t∧θd‍LV(X(s),X(qs),r(s))ds.
By (41), we have
(46)EV(X(t∧θd),r(t∧θd))≤V(X0,r0)+KE∫0t∧θd‍[1-λ1V(X(s),r(s))≤V(X0,r0)+KE∫0t∧θd‍+λ2V(X(qs),r(s))]ds≤V(X0,r0)+KT+K(λ1+λ2)∫0t∧θ‍sup0≤u≤sEV(X(u),r(u))ds.
Thus, for any t1∈[0,T], it follows that
(47)sup0≤t≤t1EV(X(t∧θd),r(t∧θd))≤V(X0,r0)+KT+(λ1+λ2)K∫0t1‍sup0≤t≤sEV(X(t∧θd),r(t∧θd))ds.
Using the Gronwall inequality, we obtain that
(48)sup0≤t≤TEV(x(t∧θd),r(t∧θd))≤[V(X0,r0)+KT]e(λ1+λ2)KT.
Let 𝒱d=inf{V(x,i):|x|≥d}. By (40), we have limd→∞𝒱d=∞. Noting that |X(θd)|=d, as θd<T, we derive from (48) that
(49)[V(X0,r0)+KT]e(λ1+λ2)KT≥sup0≤t≤TEV(x(t∧θd),r(t∧θd))≥E[V(X(θd),r(θd))I{θd<T}(w)]≥𝒱dP(θd<T).
That is,
(50)P(θd<T)≤[V(X0,r0)+KT]e(λ1+λ2)KT𝒱d.
Recall that 𝒱d→∞ as d→∞. For a given T, X0, and r(0), it follows that
(51)[V(X0,r0)+KT]e(λ1+λ2)KT𝒱d→0,
as d→∞. Let
(52)ɛ3=[V(X0,r0)+KT]e(λ1+λ2)KT𝒱d∈(0,1).
Thus we have
(53)P(θd<T)≤ɛ3.
Step2. We will give the estimate of P(ρd<T). By (14), applying the Itô’s formula to V(Y(t),r(t)) yields
(54)dV(Y(t),r(t))=Vx(Y(t),r(t))f(Z1(t),Z2(t),r¯(t))+12trace[g⊤(Z1(t),Z2(t),r¯(t))+12trace×Vxx(Y(t),r(t))g(Z1(t),Z2(t),r¯(t))]dt+Vx(Y(t),r(t))g(Z1(t),Z2(t),r¯(t))dW(t)+∫Rn‍[V(Y(t)+h(Z1(t),Z2(t),r¯(t),u),r(t))+∫Rn‍-V(Y(t),r(t))-Vx(Y(t),r(t))+∫Rn‍×h(Z1(t),Z2(t),r¯(t),u)]Π(du)dt+∫Rn‍[V(Y(t)+h(Z1(t),Z2(t),r¯(t),u),r(t))+∫Rn‍-V(Y(t),r(t)){Z2}]N~(dt,du)+∑j=1N‍γr(t)jV(Y(t),j)dt.
By (7), we have
(55)dV(Y(t),r(t))=LV(Z1(t),Z2(t),r¯(t))dt+[Vx(Y(t),r(t))-Vx(Z1(t),r¯(t))]×f(Z1(t),Z2(t),r¯(t))dt+12trace[g⊤(Z1(t),Z2(t),r¯(t))+12trace×[Vxx(Y(t),r(t))-Vxx(Z1(t),r¯(t))]+12trace×g(Z1(t),Z2(t),r¯(t))]dt+∫Rn‍{[V(Y(t)+h(Z1(t),Z2(t),r¯(t),u),r(t))+∫Rn‍-V(Z1(t)+h(Z1(t),Z2(t),r¯(t),u),r¯(t))]+∫Rn‍-[V(Y(t),r(t))-V(Z1(t),r¯(t))]+∫Rn‍-[Vx(Y(t),r(t))-Vx(Z1(t),r¯(t))]+∫Rn‍×h(Z1(t),Z2(t),r¯(t),u)[V(Y(t)+h(Z1(t),Z2(t),r¯(t),u),r(t))}Π(du)dt+Vx(Y(t),r(t))g(Z1(t),Z2(t),r¯(t))dW(t)+∫Rn‍[V(Y(t)+h(Z1(t),Z2(t),r¯(t),u),r(t))+∫Rn‍-V(Y(t),r(t)){(Z1(t),Z2(t),r¯(t),u)}]N~(dt,du)+∑j=1N‍γr(t)jV(Y(t),j)dt-∑j=1N‍γr¯(t)jV(Z1(t),j)dt.
Integrating from 0 to ρd∧t, taking expectations, and by (41), we have
(56)EV(Y(t∧ρd),r(t∧ρd))≤V(X0,r0)+KT+λ1KE∫0ρd∧t‍V(Y(s),r(s))ds+λ2KE∫0ρd∧t‍V(Y(qs),r(s))ds-λ1KE∫0ρd∧t‍[V(Z1(s),r¯(s))+V(Y(s),r(s))]ds+ℐ1+ℐ2+ℐ3+ℐ4+ℐ5,
where
(57)ℐ1=λ2KE∫0ρd∧t‍|V(Z2(s),r¯(s))-V(Y(qs),r(s))|ds,ℐ2=E∫0ρd∧t‍|Vx(Y(s),r(s))-Vx(Z1(s),r¯(s))|E∫0ρd∧t‍×f(Z1(s),Z2(s),r¯(s))ds,ℐ3=12E∫0ρd∧t‍|Vxx(Y(s),r(s))-Vxx(Z1(s),r¯(s))|12E∫0ρd∧t‍×|g(Z1(s),Z2(s),r¯(s))|2ds,ℐ4=E∫0ρd∧t‍∫Rn‍[|V(Y(s)+h(r¯(s)Z1(s),Z2(s),E(Y(s)+h(,∫0ρd∧t‍∫Rn‍r¯(s),u),r(s))E∫Rn‍∫0ρd∧t‍-V(Z1(s)E∫Rn‍∫0ρd∧t‍-V+h(Z1(s),Z2(s),r¯(s),u),r¯(s))|E∫Rn‍∫0ρd∧t‍+|V(Y(s),r(s))-V(Z1(s),r¯(s))|E∫Rn‍∫0ρd∧t‍+|Vx(Y(s),r(s))-Vx(Z1(s),r¯(s))|E∫Rn‍∫0ρd∧t‍×|h(Z1(s),Z2(s),r¯(s),u)|]π(du)ds,ℐ5=E∫0ρd∧t‍∑j=1N‍|γr(s)jV(Y(s),j)|ds-E∫0ρd∧t‍∑j=1N‍|γr¯(s)jV(Z1(s),j)|ds.
By (42) and Young’s inequality, we have
(58)ℐ1≤λ2KE∫0ρd∧t‍|V(Z2(s),r¯(s))-V(Y(qs),r¯(s))|ds+λ2KE∫0ρd∧t‍|V(Y(qs),r¯(s))-V(Y(qs),r(s))|ds≤λ2KLdE∫0ρd∧t‍|Z2(s)-Y(qs)|ds+λ2KE∫0T‍|V(Y(qs),r¯(s))-V(Y(qs),r(s))|ds≤λ2KLd∫0ρd∧t‍(E|Z2(s)-Y(qs)|2)1/2ds+λ2𝒥.
Let N=[T/h], the integer part of T/h, and let IG be the indicator function of the set G. Then
(59)𝒥≤K∑n=0N‍E∫nh(n+1)h‍|V(Y[qn],rnh)-V(Y[qn],r(s))|ds≤K∑n=0N‍E∫nh(n+1)h‍|V(Y[qn],rnh)|+|f(Y[qn],r(s))|I{r(s)≠rnh}ds.
By setting V1=sup|x|≤R,i∈SV(x,i) and using the Markov property, we have
(60)𝒥≤2KV1∑n=0N‍E∫nh(n+1)h‍I{r(s)≠rnh}ds=2KV1∑n=0N‍∫nh(n+1)h‍E[E[I{r(s)≠rnh}]∣rnh]ds=2KV1∑n=0N‍∫nh(n+1)h‍E[∑i∈S‍I{rnh=i}P(r(s)≠i∣rnh=i)]ds=2KV1×∑n=0N‍∫nh(n+1)h‍E[∑i∈S‍I{rnh=i}∑j≠i‍(γij(s-nh)+∘(s-nh))]=2KV1∑n=0N‍∫nh(n+1)h‍×ds≤2KV1∑n=0N‍∫nh(n+1)h‍E[(max1≤i≤N(-γii)h+∘(h))∑i∈S‍I{rnh=i}]ds=2KV1[max1≤i≤N(-γii)h+∘(h)]T.
Inserting (60) into (58), we have, by Lemma 5,
(61)ℐ1≤λ2KLd∫0ρd∧t‍(E|Z2(s)-Y(qs)|2)1/2ds+2λ2KV1max1≤i≤N(-γii)h+∘(h)T≤λ2KLdC2(d)Th1/2+2λ2KV1Tmax1≤i≤N(-γii)[h+∘(h)].
Similarly, by Assumptions 2, 3, and 11, Lemma 5, and Markov property, we have
(62)ℐ2≤E∫0ρd∧t‍|Vx(Y(s),r(s))-Vx(Z1(s),r(s))|E∫0ρd∧t‍×|f(Z1(s),Z2(s),r¯(s))|ds+E∫0ρd∧t‍|Vx(Z1(s),r(s))-Vx(Z1(s),r¯(s))|+E∫0ρd∧t‍×|f(Z1(s),Z2(s),r¯(s))|ds≤2KddE∫0ρd∧t‍|Vx(Y(s),r(s))-Vx(Z1(s),r(s))|ds+2KddE∫0ρd∧t‍|Vx(Z1(s),r(s))+2KddE∫0ρd∧t‍-Vx(Z1(s),r¯(s))|ds≤2KddLdE∫0ρd∧t‍|Y(s)-Z1(s)|ds+2dKdE∫0T‍|Vx(Z1(s),r(s))-Vx(Z1(s),r¯(s))|ds≤2dKdLdC1(d)Th1/2+4KddV2Tmax1≤i≤N(-γii)[h+∘(h)],
where V2=sup|x|≤R,i∈SVx(x,i). For the term ℐ3 in (56), by Assumptions 2, 3, and 11 and Lemma 5, we have
(63)ℐ3≤12E∫0ρd∧t‍|Vxx(Y(s),r(s))-Vxx(Z1(s),r(s))|12E∫0ρd∧t‍×|g(Z1(s),Z2(s),r¯(s))|2ds+12E∫0ρd∧t‍|Vxx(Z1(s),r(s))-Vxx(Z1(s),r¯(s))|+12E∫0ρd∧t‍×|g(Z1(s),Z2(s),r¯(s))|2ds≤2d2KdE∫0ρd∧t‍|Vxx(Y(s),r(s))-Vxx(Z1(s),r(s))|ds+2d2KdE∫0ρd∧t‍|Vxx(Z1(s),r(s))+2d2KdE∫0ρd∧t‍-Vxx(Z1(s),r¯(s))|ds≤2d2KdLdE∫0ρd∧t‍|Y(s)-Z1(s)|ds+2d2KdE∫0T‍|Vxx(Z1(s),r(s))+2d2KdE∫0T‍-Vxx(Z1(s),r¯(s))|ds≤2d2KdLdC1(d)Th1/2+4d2KdV3Tmax1≤i≤N(-γii)[h+∘(h)],
where V3=sup|x|≤R,i∈SVxx(x,i). For the term ℐ4 in (56), by Assumptions 2, 3, and 11 and Lemma 5, we have
(64)ℐ4≤E∫0ρd∧t‍∫Rn‍|V({Z1}Y(s)EV∫0ρd∧t‍∫Rn‍+h(Z1(s),Z2(s),r¯(s),u),r(s))E∫0ρd∧t‍-V∫Rn‍-V(Z1(s)+h(Z1(s),Z2(s),r¯(s),u),E∫0ρd∧t‍∫Rn‍r(s))|π(du)ds+E∫0ρd∧t‍∫Rn‍|V(Z1(s)+E∫0ρd∧t‍∫Rn‍+h(Z1(s),Z2(s),r¯(s),u),r(s))+E∫0ρd∧t‍∫Rn‍-V(Z1(s)+E∫0ρd∧t‍∫Rn‍-V+h(Z1(s),Z2(s),r¯(s),u),r¯(s))|+E∫0ρd∧t‍∫Rn‍×π(du)ds+E∫0ρd∧t‍∫Rn‍|V(Y(s),r(s))-V(Z1(s),r(s))|+E∫0ρd∧t‍∫Rn‍×π(du)ds+E∫0ρd∧t‍∫Rn‍|V(Z1(s),r(s))-V(Z1(s),r¯(s))|+E∫0ρd∧t‍∫Rn‍×π(du)ds+E∫0ρd∧t‍∫Rn‍|Vx(Y(s),r(s))-Vx(Z1(s),r(s))|+E∫0ρd∧t‍∫Rn‍×|h(Z1(s),Z2(s),r¯(s),u)|π(du)ds+E∫0ρd∧t‍∫Rn‍|Vx(Z1(s),r(s))-Vx(Z1(s),r¯(s))|+E∫0ρd∧t‍∫Rn‍×|h(Z1(s),Z2(s),r¯(s),u)|π(du)ds≤2LdE∫0ρd∧t‍∫Rn‍|Y(s)-Z1(s)|π(du)ds+2dKd∫Rn‍π(du)×E∫0ρd∧t‍|Vx(Y(s),r(s))-Vx(Z1(s),r(s))|ds+∫Rn‍π(du)E×∫0ρd∧t‍|V(Z1(s),r(s))-V(Z1(s),r¯(s))|ds+2dKd∫Rn‍π(du)×E∫0ρd∧t‍|Vx(Z1(s),r(s))-Vx(Z1(s),r¯(s))|ds+E∫0ρd∧t‍∫Rn‍|V(Z1(s)+E∫0ρd∧t‍∫Rn‍+h(Z1(s),Z2(s),r¯(s),u),r(s))+E∫0ρd∧t‍∫Rn‍-V(Z1(s)+E∫0ρd∧t‍∫Rn‍+h(Z1(s),Z2(s),r¯(s),u),r¯(s))|+E∫0ρd∧t‍∫Rn‍×π(du)ds≤2[∫Rn‍π(du)+2dKd∫Rn‍π(du)]×C1(d)LdTh1/2+2[∫Rn‍π(du)V1+2dKd∫Rn‍π(du)V2]×Tmax1≤i≤N(-γii)[h+∘(h)]+ℒ,
where
(65)ℒ≤E∫0ρd∧t‍∫Rn‍|V(Z1(s)+h(Z1(s),Z2(s),r¯(s),u),r(s))E∫0ρd∧t‍∫Rn‍-V(Z1(s),r¯(s))|π(du)ds+E∫0ρd∧t‍∫Rn‍|V(Z1(s),r(s))+E∫0ρd∧t‍∫Rn‍-V(Z1(s),r¯(s))|π(du)ds+E∫0ρd∧t‍∫Rn‍|V(Z1(s),r¯(s))+E∫0ρd∧t‍∫Rn‍-V(Z1(s)+h(Z1(s),Z2(s),r¯(s),u),+E∫0ρd∧t‍∫Rn‍r¯(s)Z1(s))|π(du)ds≤4dLdKd∫Rn‍π(du)T+2V1T∫Rn‍π(du)max1≤i≤N(-γii)[h+∘(h)].
Inserting (65) into (64), we have
(66)ℐ4≤2[∫Rn‍π(du)+2dKd∫Rn‍π(du)]C1(d)LdTh1/2+4dLdKd∫Rn‍π(du)T+4[∫Rn‍π(du)V1+dKd∫Rn‍π(du)V2]×Tmax1≤i≤N(-γii)[h+∘(h)].
For the term ℐ5, by Assumption 11, equation (56), and Lemma 5, we have
(67)ℐ5≤E∫0ρd∧t‍∑j=1N‍|γr(s)j||V(Y(s),j)-V(Z1(s),j)|ds+E∫0ρd∧t‍∑j=1N‍|γr(s)j-γr¯(s)j||V(Z1(s),j)|ds≤Nmax1≤i≤N(-γii)LdC1(d)Th1/2+2V1NTmax1≤i≤N(-γii).
Combing these inequalities and (53), we derive that, for t1∈[0,T],
(68)sup0≤t≤t1EV(Y(t∧ρd),r(t∧ρd))≤V(X0,r0)+[K+4dLdKd∫Rn‍π(du)+2V1Nmax1≤i≤N(-γii)]T+(λ1+λ2)K×∫0t1‍sup0≤t≤sEV(Y(t∧ρd))ds+M1h1/2+M2[h+∘(h)],
where
(69)M1=LdC1(d)T[{Kd∫Rn‍π(du)}2dKd+2d2Kd+2∫Rn‍π(du)LdC1(d)T+4dKd∫Rn‍π(du)+Nmax1≤i≤N(-γii)]+λ2KLdC2(d)T,M2=[{(4dKd+4dKd∫Rn‍π(du))}(2λ2K+4∫Rn‍π(du))V1+(4dKd+4dKd∫Rn‍π(du))V2+4d2KdV3{(4dKd+4dKd∫Rn‍π(du))}]Tmax1≤i≤N(-γii).
For arbitrary 0≤t1≤T, by the Gronwall inequality, we get
(70)sup0≤t≤TEV(Y(t∧ρd),r(t∧ρd))≤{{[K+4dLdKd∫Rn‍π(du)+2V1Nmax1≤i≤N(-γii)]}V(X0,r0)+[K+4dLdKd∫Rn‍π(du)+2V1Nmax1≤i≤N(-γii)]T+M1h1/2+M2[h+∘(h)]{[K+4dLdKd∫Rn‍π(du)+2V1Nmax1≤i≤N(-γii)]}}e(λ1+λ2)KT.
Noting that |Y(ρd)|=d, as ρd<T, we derive from (70) that
(71){{[K+4dLdKd∫Rn‍π(du)+2V1Nmax1≤i≤N(-γii)]}V(X0,r0)+[K+4dLdKd∫Rn‍π(du)+2V1Nmax1≤i≤N(-γii)]T+M1h1/2+M2[h+∘(h)]{[K+4dLdKd∫Rn‍π(du)+2V1Nmax1≤i≤N(-γii)]}}e(λ1+λ2)KT≥EV(Y(t∧ρd),r(t∧ρd))≥E[V(Y(ρd),r(ρd))I{ρd<T}(w)]≥𝒱dP(ρd<T).
So we have
(72)P(ρd<T)≤{V(X0,r0)+[K+4dLdKd∫Rn‍π(du)]T}e(λ1+λ2)KT𝒱d+{[2V1Nmax1≤i≤N(-γii)]T}e(λ1+λ2)KT𝒱d+{M1h1/2+M2[h+∘(h)]}e(λ1+λ2)KT𝒱d.Step  3. Let ϵ,δ∈(0,1) be arbitrarily small. By setting
(73)Ω¯={w:sup0≤t≤T|Y(t)-X(t)|2≥δ}
and using Theorem 4, we have
(74)Cd[h+∘(h)]≥E[sup0≤t≤T|Y(t)-X(t)|2Iτd≥T(w)]≥E[sup0≤t≤T|Y(t)-X(t)|2Iτd≥T(w)IΩ¯(w)]≥δE[Iτd≥T(w)IΩ¯(w)]=δP({τd≥T}∩Ω¯)≥δ[P(Ω¯)-P(τd<T)].
Combing (53) and (72) together, one gets
(75)P(τd<T)≤P(ρd<T)+P(θd<T)≤ɛ3+{V(X0,r0)+[K+4dLdKd∫Rn‍π(du)]T}e(λ1+λ2)KT𝒱d+{[2V1Nmax1≤i≤N(-γii)]T}e(λ1+λ2)KT𝒱d+{M1h1/2+M2[h+∘(h)]}e(λ1+λ2)KT𝒱d.
Hence on using (74), we conclude that
(76)P(Ω¯)=P(sup0≤t≤T|Y(t)-X(t)|2≥δ)≤{V(X0,r0)+[K+4dLdKd∫Rn‍π(du)]T}e(λ1+λ2)KT𝒱d+{[2V1Nmax1≤i≤N(-γii)]T}e(λ1+λ2)KT𝒱d+{M1h1/2+M2[h+∘(h)]}e(λ1+λ2)KT𝒱d+Cdδ[h+∘(h)]+ɛ3.
Recalling that 𝒱d→∞ as d→∞, we can choose d sufficiently large for
(77){V(X0,r0)+[K+4dLdKd∫Rn‍π(du)]T}e(λ1+λ2)KT𝒱d+{[2V1Nmax1≤i≤N(-γii)]T}e(λ1+λ2)KT𝒱d<ɛ3
and then choose h sufficiently small for
(78){M1h1/2+M2[h+∘(h)]}e(λ1+λ2)KT𝒱d+Cdδ[h+∘(h)]<ɛ3
to obtain
(79)P(sup0≤t≤T|Y(t)-X(t)|2≥δ)<ɛ.
The proof of Theorem 12 is now complete.


## 5. An Example

In this section, we construct one example to demonstrate the effectiveness of this theory. Letr(t) be a right-continuous Markov chain taking values in S={1,2} with the generator
(80)Γ=(γij)2×2=[-222-2]
Let N~(dt,du) be a compensated Poisson random measures and is given by π(du)dt=λf(u)dudt, where λ=2 and
(81)f(u)=12πue-(lnu)2/2,0≤u<∞
is the density function of a lognormal random variable. Of course N~(dt,du) and r(t) are assumed to be independent. Consider a linear stochastic pantograph delay equations with Markovian switching and pure jumps
(82)dX(t)=a(r(t))X(t)dt+∫|u|≤1‍ub(r(t))X(0.5t)N~(dt,du).
Here a(1)=-3, a(2)=-1, b(1)=1/10, and b(2)=1/6. Then (82) can be regarded as the result of the two equations
(83)dX(t)=-3X(t)dt+110∫|u|≤1‍uX(0.5t)N~(dt,du),dX(t)=-X(t)dt+16∫|u|≤1‍uX(0.5t)N~(dt,du),
switching among each other according to the movement of the Markov chain r(t). Obviously, (82) satisfies Assumptions 2 and 3. Given the stepsize h, we can have the Euler method
(84)Yn+1=Yn+a(rnh)Ynh+∫tntn+1‍∫|u|≤1‍ub(rnh)Y[0.5n]N~(dt,du),
with Y0=X(0). Let Z¯1(t)=Yn, Z¯2(t)=Y[0.5n], and r¯(t)=rnh, for t∈[tn,tn+1). Then we define the continuous Euler approximate solution
(85)Y(t)=Y(0)+∫0t‍a(r¯(s))Z¯1(s)ds+∫0t‍∫|u|≤1‍ub(r¯(s))Z¯2(s)N~(ds,du).
Since the conditions of Theorem 4 are satisfied, then the approximate solution (85) will converge to the true solution of (82) in the sense of the L2-norm. To examine the convergence in the sense of the probability, we construct a function
(86)V(x,i)={β1x2,ifi=1,β2x2,ifi=2.
Then
(87)LV(x,y,i)={(2β2-8β1)x2+110∫|u|≤1‍u2π(du)β1y2,ifi=1,(2β1-4β2)x2+16∫|u|≤1‍u2π(du)β2y2,ifi=2.
From the properties of the lognormal distribution, we have
(88)LV(x,y,1)≤-(8-2β2β1)β1x2+15e2β1y2,(89)LV(x,y,2)≤-(4-2β1β2)β2x2+13e2β2y2.
If 6/(12-e2)<(β2/β1)<4-(1/10)e2, then it follows that Assumptions 10 and 11 are satisfied. Consequently, the approximate solution (85) will converge to the true solution of (82) for any t∈[0,T] in the sense of Theorem 12.



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*Source: 718627-2013-06-12.xml*