# Approximations of Numerical Method for Neutral Stochastic Functional Differential Equations with Markovian Switching

Authors: Hua Yang; Feng Jiang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Stochastic systems with Markovian switching have been used in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. In this paper, we study the Euler-Maruyama (EM) method for neutral stochastic functional differential equations with Markovian switching. The main aim is to show that the numerical solutions will converge to the true solutions. Moreover, we obtain the convergence order of the approximate solutions.


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## Body

## 1. Introduction

Stochastic systems with Markovian switching have been successfully used in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance [1]. As well as deterministic neutral functional differential equations and stochastic functional differential equations (SFDEs), most neutral stochastic functional differential equations with Markovian switching (NSFDEsMS) cannot be solved explicitly, so numerical methods become one of the powerful techniques. A number of papers have studied the numerical analysis of the deterministic neutral functional differential equations, for example, [2, 3] and references therein. The numerical solutions of SFDEs and stochastic systems with Markovian switching have also been studied extensively by many authors. Here we mention some of them, for example, [4–16]. moreover, Many well-known theorems in SFDEs are successfully extended to NSFDEs, for example, [17–24] discussed the stability analysis of the true solutions.However, to the best of our knowledge, little is yet known about the numerical solutions for NSFDEsMS. In this paper we will extend the method developed by [5, 16] to NSFDEsMS and study strong convergence for the Euler-Maruyama approximations under the local Lipschitz condition, the linear growth condition, and contractive mapping. The three conditions are standard for the existence and uniqueness of the true solutions. Although the method of analysis borrows from [5], the existence of the neutral term and Markovian switching essentially complicates the problem. We develop several new techniques to cope with the difficulties which have risen from the two terms. Moreover, we also generalize the results in [19].In Section2, we describe some preliminaries and define the EM method for NSFDEsMS and state our main result that the approximate solutions strongly converge to the true solutions. The proof of the result is rather technical so we present several lemmas in Section 3 and then complete the proof in Section 4. In Section 5, under the global Lipschitz condition, we reveal the order of the convergence of the approximate solutions. Finally, a conclusion is made in Section 6.


## 2. Preliminaries and EM Scheme

Throughout this paper let(Ω,ℱ,ℙ) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions, that is, it is right continuous and increasing while ℱ0 contains all P-null sets. Let w(t)=(w1(t),…,wm(t))T be an m-dimensional Brownian motion defined on the probability space. Let |·| be the Euclidean norm in ℝn. Let ℝ+=[0,∞), and let τ>0. Denoted by C([-τ,0],ℝn) the family of continuous functions from [-τ,0] to ℝn with the norm ∥φ∥=sup-τ≤θ≤0|φ(θ)|. Let p>0 and Lℱ0p([-τ,0];ℝn) be the family of ℱ0-measurable C([-τ,0];ℝn)-valued random variables ξ such that 𝔼∥ξ∥p<∞. If x(t) is an ℝn-valued stochastic process on t∈[-τ,∞), we let xt={x(t+θ):-τ≤θ≤0} for t≥0.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
(2.1)P{r(t+Δ)=j∣r(t)=i}={γijΔ+o(Δ)ifi≠j,1+γijΔ+o(Δ)ifi=j,
where Δ>0. Here γij≥0 is the transition rate from i to j if i≠j while γii=-∑i≠j‍γij. We assumethat 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 simple jumps in any finite subinterval of ℜ+=[0,∞).In this paper, we consider then-dimensional NSFDEsMS
(2.2)d[x(t)-u(xt,r(t))]=f(xt,r(t))dt+g(xt,r(t))dw(t),t≥0,
with initial data x0=ξ∈Lℱ0p([-τ,0];ℝn) and r(0)=i0∈S, where f:C([-τ,0];ℝn)→ℝn, g:C([-τ,0];ℝn)→ℝn×m and u:C([-τ,0];ℝn)→ℝn. As a standing hypothesis we assume that both f and g are sufficiently smooth so that (2.2) has a unique solution. We refer the reader to Mao [10, 12] for the conditions on the existence and uniqueness of the solution x(t). The initial data ξ and i0 could be random, but the Markov property ensures that it is sufficient to consider only the case when both x0 and i0 are constants.To analyze the Euler-Maruyama (EM) method, we need the following lemma (see [6, 7, 10–12, 16]).Lemma 2.1.
GivenΔ>0, let rkΔ=r(kΔ) for k≥0. Then {rkΔ,k=0,1,2,…} is a discrete Markov chain with the one-step transition probability matrix
(2.3)P(Δ)=(Pij(Δ))N×N=eΔΓ.For the completeness, we give the simulation of the Markov chain as follows. Given a stepsizeΔ>0, we compute the one-step transition probability matrix
(2.4)P(Δ)=(Pij(Δ))N×N=eΔΓ.
Let r0Δ=i0 and generate a random number ξ1 which is uniformly distributed in [0,1]. Define
(2.5)r1Δ={i1,ifi1∈S-{N}suchthat∑j=1i1-1Pi0,j(Δ)≤ξ1<∑j=1i1Pi0,j(Δ),N,if∑j=1N-1Pi0,j(Δ)≤ξ1,
where we set ∑i=10‍Pi0,j(Δ)=0 as usual. Generate independently a new random number ξ2 which is again uniformly distributed in [0,1], and then define
(2.6)r2Δ={i2,ifi2∈S-{N}suchthat∑j=1i2-1Pr1Δ,j(Δ)≤ξ2<∑j=1i2Pr1Δ,j(Δ),N,if∑j=1N-1‍Pr1Δ,j(Δ)≤ξ2.Repeating this procedure a trajectory of{rkΔ,k=0,1,2,…} can be generated. This procedure can be carried out independently to obtain more trajectories.Now we can define the Euler-Maruyama (EM) approximate solution for (2.2) on the finite time interval [0,T]. Without loss of any generality, we may assume that T/τ is a rational number; otherwise we may replace T by a larger number. Let the step size Δ∈(0,1) be a fraction of τ and T, namely, Δ=τ/N=T/M for some integers N>τ and M>T. The explicit discrete EM approximate solution y¯(kΔ), k≥-N is defined as follows:
(2.7)y¯(kΔ)=ξ(kΔ),-N≤k≤0,y¯((k+1)Δ)=y¯(kΔ)+u(y¯kΔ,rkΔ)-u(y¯(k-1)Δ,rk-1Δ)+f(y¯kΔ,rkΔ)Δ+g(y¯kΔ,rkΔ)Δwk,0≤k≤M-1,
where Δwk=w((k+1)Δ)-w(kΔ) and y¯kΔ={y¯kΔ(θ):-τ≤θ≤0} is a C([-τ,0];ℝn)-valued random variable defined by
(2.8)y¯kΔ(θ)=y¯((k+i)Δ)+θ-iΔΔ[y¯((k+i+1)Δ)-y¯((k+i)Δ)]=Δ-(θ-iΔ)Δy¯((k+i)Δ)+θ-iΔΔy¯((k+i+1)Δ),
for iΔ≤θ≤(i+1)Δ, i=-N,-N+1,…,-1, where in order for y¯-Δ to be well defined, we set y¯(-(N+1)Δ)=ξ(-NΔ).That is,y¯kΔ(·) is the linear interpolation of y¯((k-N)Δ), y¯((k-N+1)Δ),…,y¯(kΔ). We hence have
(2.9)|y¯kΔ(θ)|=Δ-(θ-iΔ)Δ|y¯((k+i)Δ)|+θ-iΔΔ|y¯((k+i+1)Δ)|≤|y¯((k+i)Δ)|∨|y¯((k+i+1)Δ)|.
We therefore obtain
(2.10)|y¯kΔ(θ)|=max-N≤i≤0|y¯((k+i)Δ)|,foranyk=-1,0,1,…,M-1.
It is obvious that ∥y¯-Δ∥≤∥y¯0∥.In our analysis it will be more convenient to use continuous-time approximations. We hence introduce theC([-τ,0];ℝn)-valued step process
(2.11)y¯t=∑k=0M-2y¯kΔ1[kΔ,(k+1)Δ)(t)+y¯(M-1)Δ1[(M-1)Δ,MΔ](t),r¯(t)=∑k=0M-1rkΔ1[kΔ,(k+1)Δ)(t),
and we define the continuous EM approximate solution as follows: let y(t)=ξ(t) for -τ≤t≤0, while for t∈[kΔ,(k+1)Δ], k=0,1,…,M-1,
(2.12)y(t)=ξ(0)+u(y¯(k-1)Δ+t-kΔΔ(y¯kΔ-y¯(k-1)Δ),rkΔ)-u(y¯-Δ,r0Δ)+∫0tf(y¯s,r¯(s))ds+∫0tg(y¯s,r¯(s))dw(s).
Clearly, (2.12) can also be written as
(2.13)y(t)=y¯(kΔ)+u(y¯(k-1)Δ+t-kΔΔ(y¯kΔ-y¯(k-1)Δ),rkΔ)-u(y¯(k-1)Δ,rk-1Δ)+∫kΔtf(y¯s,r¯(s))ds+∫kΔtg(y¯s,r¯(s))dw(s).
In particular, this shows that y(kΔ)=y¯(kΔ), that is, the discrete and continuous EM approximate solutions coincide at the grid points. We know that y(t) is not computable because it requires knowledge of the entire Brownian path, not just its Δ-increments. However, y(kΔ)=y¯(kΔ), so the error bound for y(t) will automatically imply the error bound for y¯(kΔ). It is then obvious that
(2.14)∥y¯kΔ∥≤∥ykΔ∥,∀k=0,1,2,…,M-1.
Moreover, for any t∈[0,T],
(2.15)sup0≤t≤T∥y¯t∥=sup0≤k≤M-1∥y¯kΔ∥≤sup0≤k≤M-1∥ykΔ∥=sup0≤k≤M-1sup-τ≤θ≤0|y(kΔ+θ)|≤sup0≤t≤Tsup-τ≤θ≤0|y(t+θ)|≤sup-τ≤s≤T|y(s)|,
and letting [t/Δ] be the integer part of t/Δ, then
(2.16)∥y¯t∥=∥y¯[t/Δ]Δ∥≤∥y[t/Δ]Δ∥≤sup-τ≤s≤t|y(s)|.
These properties will be used frequently in what follows, without further explanation.For the existence and uniqueness of the solution of (2.2) and the boundedness of the solution’s moments, we impose the following hypotheses (e.g., see [11]).Assumption 2.2.
For each integerj≥1 and i∈S, there exists a positive constant Cj such that
(2.17)|f(φ,i)-f(ψ,i)2|∨|g(φ,i)-g(ψ,i)2|≤Cj∥φ-ψ∥2
for φ, ψ∈C([-τ,0];ℝn) with ∥φ∥∨∥ψ∥≤j.Assumption 2.3.
There is a constantK>0 such that
(2.18)|f(φ,i)|2∨|g(φ,i)|2≤K(1+∥φ∥2),
for φ∈C([-τ,0];ℝn) and i∈S.Assumption 2.4.
There exists a constantκ∈(0,1) such that for all φ, ψ∈C([-τ,0];ℝn) and i∈S,
(2.19)|u(φ,i)-u(ψ,i)|≤κ∥φ-ψ∥,
for φ, ψ∈C([-τ,0];ℝn) and u(0,i)=0.We also impose the following condition on the initial data.Assumption 2.5.
ξ
∈
L
ℱ
0
p
(
[
-
τ
,
0
]
;
ℝ
n
) for some p≥2, and there exists a nondecreasing function α(·) such that
(2.20)𝔼(sup-τ≤s≤t≤0|ξ(t)-ξ(s)|2)≤α(t-s),
with the property α(s)→0 as s→0.From Mao and Yuan [11], we may therefore state the following theorem.Theorem 2.6.
Letp≥2. If Assumptions 2.3–2.5 are satisfied, then
(2.21)𝔼(sup-τ≤t≤T|x(t)|p)≤Hκ,p,T,K,ξ,
for any T>0, where Hκ,p,T,K,ξ is a constant dependent on κ, p, T, K, ξ.The primary aim of this paper is to establish the following strong mean square convergence theorem for the EM approximations.Theorem 2.7.
If Assumptions2.2–2.5 hold,
(2.22)limΔ→0𝔼(sup0≤t≤T|x(t)-y(t)|2)=0.The proof of this theorem is very technical, so we present some lemmas in the next section, and then complete the proof in the sequent section.


## 3. Lemmas

Lemma 3.1.
If Assumptions2.3–2.5 hold, for any p≥2, there exists a constant H(p) such that
(3.1)𝔼(sup-τ≤t≤T|y(t)|p)≤H(p),
where H(p) is independent of Δ.Proof.
Fort∈[kΔ,(k+1)Δ], k=0,1,2,…,M-1, set y~(t):=y(t)-u(y¯(k-1)Δ+(t-kΔ)(y¯kΔ-y¯(k-1)Δ)/Δ,rkΔ) and
(3.2)h(t):=𝔼(sup-τ≤s≤t|y(s)|p),h~(t):=𝔼(sup0≤s≤t|y~(s)|p).
Recall the inequality that for p≥1 and any ɛ>0, |x+y|p≤(1+ɛ)p-1(|x|p+ɛ1-p|y|p). Then we have, from Assumption 2.4,
(3.3)|y(t)|p≤(1+ɛ)p-1(|y~(t)|p+ɛ1-p|u(y¯(k-1)Δ+(t-kΔ)(y¯kΔ-y¯(k-1)Δ)Δ,rkΔ)|p)≤(1+ɛ)p-1(|y~(t)|p+ɛ1-pκp∥y¯(k-1)Δ+t-kΔΔ(y¯kΔ-y¯(k-1)Δ)∥p).
By ∥y¯-Δ∥≤∥y¯0∥, noting k=0,1,2,…,M-1,
(3.4)∥y¯(k-1)Δ+t-kΔΔ(y¯kΔ-y¯(k-1)Δ)∥p≤|(k+1)Δ-tΔ∥y¯(k-1)Δ∥+t-kΔΔ∥y¯kΔ∥|p≤[(k+1)Δ-tΔ(sup-τ≤s≤t|y(s)|)+t-kΔΔ(sup-τ≤s≤t|y(s)|)]p≤sup-τ≤s≤t|y(s)|p.
Consequently, choose ɛ=κ/(1-κ), then
(3.5)|y(t)|p≤(1-κ)1-p|y~(t)|p+κ(sup-τ≤s≤t|y(s)|p).
Hence,
(3.6)h(t)≤𝔼∥ξ∥p+𝔼(sup0≤s≤t|y(s)|p)≤𝔼∥ξ∥p+κh(t)+(1-κ)1-ph~(t),
which implies
(3.7)h(t)≤𝔼∥ξ∥p1-κ+h~(t)(1-κ)p.
Since
(3.8)y~(t)=y~(0)+∫0tf(y¯s,r¯(s))ds+∫0tg(y¯s,r¯(s))dw(s),
with y~(0)=y¯(0)-u(y¯-Δ), by the Hölder inequality, we have
(3.9)|y~(t)|p≤3p-1[|y~(0)|p+tp-1∫0t|f(y¯s,r¯(s))|pds+|∫0tg(y¯s,r¯(s))dw(s)|p].
Hence, for any t1∈[0,T],
(3.10)h~(t1)≤3p-1[𝔼|y~(0)|p+Tp-1𝔼∫0t1‍|f(y¯s,r¯(s))|pds+𝔼(sup0≤t≤t1|∫0t‍g(y¯s,r¯(s))dw(s)|p)].
By Assumption 2.4 and the fact ∥y¯-Δ∥≤∥y¯0∥, we compute that
(3.11)𝔼|y~(0)|p=𝔼|y¯(0)-u(y¯-Δ,r0Δ)|p≤𝔼(|y¯(0)|+κ∥y¯-Δ∥)p≤𝔼(|y¯(0)|+κ∥y¯0∥)p≤𝔼(|ξ(0)|+κ∥ξ∥)p≤(1+κ)p𝔼∥ξ∥p.
Assumption 2.3 and the Hölder inequality give
(3.12)𝔼∫0t1|f(y¯s,r¯(s))|pds≤𝔼∫0t1Kp/2(1+∥y¯s∥2)p/2ds≤Kp/22(p-2)/2𝔼∫0t1(1+∥y¯s∥p)ds≤Kp/22(p-2)/2[T+∫0t1𝔼(sup-τ≤t≤s|y(t)|pds)].
Applying the Burkholder-Davis-Gundy inequality, the Hölder inequality and Assumption 2.3 yield
(3.13)𝔼(sup0≤t≤t1|∫0t‍g(y¯s,r¯(s))dw(s)|p)≤Cp𝔼(∫0t1|g(y¯s,r¯(s))|2ds)p/2≤CpT(p-2)/2𝔼∫0t1|g(y¯s,r¯(s))|pds≤CpT(p-2)/2𝔼∫0t1Kp/2(1+∥y¯s∥2)p/2ds≤CpT(p-2)/2Kp/22(p-2)/2𝔼∫0t1(1+∥y¯s∥p)ds≤CpT(p-2)/2Kp/22(p-2)/2[T+∫0t1𝔼(sup-τ≤t≤s|y(t)|p)ds],
where Cp is a constant dependent only on p. Substituting (3.11), (3.12), and (3.13) into (3.10) gives
(3.14)h~(t1)≤3p-1[(1+κ)p𝔼∥ξ∥p+Kp/22(p-2)/2Tp+Cp(2T)(p-2)/2Kp/2T]+3p-1[Kp/22(p-2)/2Tp-1+Cp(2T)(p-2)/2Kp/2]∫0t1𝔼(sup-τ≤t≤s|y(t)|p)ds=:C1+C2∫0t1h(s)ds.
Hence from (3.7), we have
(3.15)h(t1)≤𝔼∥ξ∥p1-κ+1(1-κ)p[C1+C2∫0t1h(s)ds]≤𝔼∥ξ∥p1-κ+C1(1-κ)p+C2(1-κ)p∫0t1h(s)ds.
By the Gronwall inequality we find that
(3.16)h(T)≤[𝔼∥ξ∥p1-κ+C1(1-κ)p]eC2T/(1-κ)p.
From the expressions of C1 and C2, we know that they are positive constants dependent only on ξ, κ, K, p, and T, but independent of Δ. The proof is complete.Lemma 3.2.
If Assumptions2.3–2.5 hold, then for any integer l>1,
(3.17)𝔼(sup0≤k≤M-1∥y¯kΔ-y¯(k-1)Δ∥2)≤c1′+c1α(Δ)+c¯1(l)Δ(l-1)/l=:γ(Δ),
where c1=1/(1-2κ), c1′=(8κ/(1-2κ))H(2), and  c¯1(l) is a constant dependent on l but independent of Δ.Proof.
Forθ∈[iΔ,(i+1)Δ], where i=-N,-(N+1),…,-1, from (2.7),
(3.18)|y¯kΔ-y¯(k-1)Δ|≤(i+1)Δ-θΔ|y¯((k+i)Δ)-y¯((k-1+i)Δ)|+θ-iΔΔ|y¯((k+i+1)Δ)-y¯((k+i)Δ)|≤|y¯((k+i)Δ)-y¯((k-1+i)Δ)|∨|y¯((k+i+1)Δ)-y¯((k+i)Δ)|,
so
(3.19)∥y¯kΔ-y¯(k-1)Δ∥≤sup-N≤i≤0|y¯((k+i)Δ)-y¯((k-1+i)Δ)|.
We therefore have
(3.20)𝔼(sup0≤k≤M-1∥y¯kΔ-y¯(k-1)Δ∥2)≤𝔼[sup0≤k≤M-1(sup-N≤i≤0|y¯((k+i)Δ)-y¯((k-1+i)Δ)|2)]≤𝔼(sup-N≤k≤M-1|y¯(kΔ)-y¯((k-1)Δ)|2).
When -N≤k≤0, by Assumption 2.5 and y¯(-(N+1)Δ)=ξ(-NΔ),
(3.21)𝔼(sup-N≤k≤0|y¯(kΔ)-y¯((k-1)Δ)|2)≤𝔼(sup-N≤k≤0|ξ(kΔ)-ξ((k-1)Δ)|2)≤α(Δ).
When 1≤k≤M-1, from (2.13), we have
(3.22)y¯(kΔ)-y¯((k-1)Δ)=u(y¯(k-1)Δ,rk-1Δ)-u(y¯(k-2)Δ,rk-2Δ)+f(y¯(k-1)Δ,rk-1Δ)Δ+g(y¯(k-1)Δ,rk-1Δ)Δwk-1.
Recall the elementary inequality, for any x, y>0 and ɛ∈(0,1), (x+y)2≤x2/ɛ+y2/(1-ɛ). Then
(3.23)|y¯(kΔ)-y¯((k-1)Δ)|2≤1ɛ|u(y¯(k-1)Δ,rk-1Δ)-u(y¯(k-2)Δ,rk-1Δ)+u(y¯(k-2)Δ,rk-1Δ)-u(y¯(k-2)Δ,rk-2Δ)|2+11-ɛ|f(y¯(k-1)Δ,rk-1Δ)Δ+g(y¯(k-1)Δ,rk-1Δ)Δwk-1|2≤2κ2ɛ∥y¯(k-1)Δ-y¯(k-2)Δ∥2+8κ2ɛ∥y¯(k-2)Δ∥2+21-ɛ|f(y¯(k-1)Δ,rk-1Δ)|2Δ2+21-ɛ|g(y¯(k-1)Δ,rk-1Δ)Δwk-1|2.
Consequently
(3.24)𝔼(sup1≤k≤M-1|y¯(kΔ)-y¯((k-1)Δ)|2)≤2κ2ɛ𝔼(sup1≤k≤M-1∥y¯(k-1)Δ-y¯(k-2)Δ∥2)+8κ2ɛ𝔼(sup1≤k≤M-1∥y¯(k-2)Δ∥2)+2Δ21-ɛ𝔼(sup1≤k≤M-1|f(y¯(k-1)Δ,r¯(t))|2)+21-ɛ𝔼(sup1≤k≤M-1|g(y¯(k-1)Δ,r¯(t))Δwk-1|2).
We deal with these four terms, separately. By (3.21),
(3.25)𝔼(sup1≤k≤M-1∥y¯(k-1)Δ-y¯(k-2)Δ∥2)≤𝔼(sup1≤k≤M-1sup-N≤i≤0|y¯((k+i-1)Δ)-y¯((k-2+i)Δ)|2)≤𝔼(sup-N≤k≤M-1|y¯(kΔ)-y¯((k-1)Δ)|2)≤𝔼(sup-N≤k≤0|y¯(kΔ)-y¯((k-1)Δ)|2)+𝔼(sup1≤k≤M-1|y¯(kΔ)-y¯((k-1)Δ)|2)≤α(Δ)+𝔼(sup1≤k≤M-1|y¯(kΔ)-y¯((k-1)Δ)|2).
Noting that 𝔼[sup-τ≤t≤T|y¯(t)|2]≤𝔼[sup-τ≤t≤T|y(t)|2]≤H(2) (where H(p) has been defined in Lemma 3.1), by Assumption 2.3 and (2.15),
(3.26)𝔼(sup1≤k≤M-1|f(y¯(k-1)Δ,rk-1Δ)|2)≤𝔼(sup1≤k≤M-1K(1+∥y¯(k-1)Δ∥2))≤K+K𝔼(sup1≤k≤M-1sup-N≤i≤0|y¯((k-1+i)Δ)|2)≤K+K𝔼(sup-N≤k≤M-1|y¯(kΔ)|2)≤K+K𝔼(sup-τ≤t≤T|y¯(t)|2)≤K(1+H(2)).
By the Hölder inequality, for any integer l>1,
(3.27)𝔼(sup1≤k≤M-1|g(y¯(k-1)Δ,rk-1Δ)Δwk-1|2)≤𝔼(sup1≤k≤M-1|g(y¯(k-1)Δ,r¯(t))|2sup1≤k≤M-1|Δwk-1|2)≤[𝔼(sup1≤k≤M-1|g(y¯(k-1)Δ,rk-1Δ)|2l/(l-1))](l-1)/l[𝔼(sup1≤k≤M-1|Δwk-1|2l)]1/l≤[𝔼(sup0≤k≤M-1(K(1+∥y¯kΔ∥2))l/(l-1))](l-1)/l[𝔼(∑k=0M-1|Δwk|2l)]1/l≤[Kl/(1-l)𝔼(1+(sup0≤k≤M-1∥y¯kΔ∥2))l/(l-1)](l-1)/l[(∑k=0M-1𝔼|Δwk|2l)]1/l≤[21/(l-1)Kl/(1-l)(1+𝔼(sup0≤k≤M-1∥y¯kΔ∥2l/(l-1)))](l-1)/l[(∑k=0M-1(2l-1)!!Δl)]1/l≤[21/(l-1)Kl/(1-l)(1+H(2ll-1))](l-1)/l[(2l-1)!!TΔl-1]1/l≤D(l)Δ(l-1)/l,
where D(l) is a constant dependent on l.
Substituting (3.25), (3.26), and (3.27) into (3.24), choosing ɛ=κ, and noting Δ∈(0,1), we have
(3.28)𝔼(sup1≤k≤M-1|y¯(kΔ)-y¯((k-1)Δ)|2)≤2κ1-2κα(Δ)+8κ1-2κH(2)+2K(1+H(2))+2D(l)(1-2κ)2Δ(l-1)/l.
Combining (3.21) with (3.28), from (3.20), we have
(3.29)𝔼(sup0≤k≤M-1∥y¯kΔ-y¯(k-1)Δ∥2)≤𝔼(sup-N≤k≤M-1|y¯(kΔ)-y¯((k-1)Δ)|2)≤𝔼(sup-N≤k≤0|y¯(kΔ)-y¯((k-1)Δ)|2)+𝔼(sup1≤k≤M-1|y¯(kΔ)-y¯((k-1)Δ)|2)≤11-2κα(Δ)+8κ1-2κH(2)+2K(1+H(2))+2D(l)(1-2κ)2Δ(l-1)/l,
as required.Lemma 3.3.
If Assumptions2.3–2.5 hold,
(3.30)𝔼(sup0≤s≤T∥ys-y¯s∥2)≤c2′+c2α(2Δ)+c¯2(l)Δ(l-1)/l=:β(Δ),
where c2, c2′ are a constant independent of  l  and Δ, c¯2(l) is a constant dependent on  l  but independent of Δ.Proof.
Fix anys∈[0,T] and θ∈[-τ,0]. Let ks∈{0,1,2,…,M-1}, kθ∈{-N,-N+1,…,-1}, and ksθ∈{-N,-N+1,…,M-1} be the integers for which s∈[ksΔ,(ks+1)Δ], θ∈[kθΔ,(kθ+1)Δ], and s+θ∈[ksθΔ,(ksθ+1)Δ], respectively. Clearly,
(3.31)0≤s+θ-(ks+kθ)≤2Δ,ksθ-(ks+kθ)∈{0,1,2}.
From (2.7),
(3.32)y¯s=y¯ksΔ(θ)=y¯((ks+kθ)Δ)+θ-kθΔΔ(y¯((ks+kθ+1)Δ)-y¯(ks+kθ)Δ),
which yields
(3.33)|ys-y¯s|=|y(s+θ)-y¯ksΔ(θ)|≤|y(s+θ)-y¯((ks+kθ)Δ)|+θ-kθΔΔ|y¯((ks+kθ+1)Δ)-y¯((ks+kθ)Δ)|≤|y(s+θ)-y¯((ks+kθ)Δ)|+|y¯((ks+kθ+1)Δ)-y¯((ks+kθ)Δ)|,
so by (3.20) and Lemma 3.2, noting y¯(MΔ)=y¯((M-1)Δ) from (2.15),
(3.34)𝔼(sup0≤s≤T∥ys-y¯s∥2)≤2𝔼[sup0≤s≤T(sup-τ≤θ≤0|y(s+θ)-y¯((ks+kθ)Δ)|2)]+2𝔼[sup0≤ks≤M-1(sup-N≤kθ≤0|y¯((ks+kθ+1)Δ)-y¯((ks+kθ)Δ)|2)]≤2𝔼(sup-τ≤θ≤0,0≤s≤T|y(s+θ)-y¯((ks+kθ)Δ)|2)+2γ(Δ).
Therefore, it is a key to compute 𝔼(sup-τ≤θ≤0,0≤s≤T|y(s+θ)-y¯((ks+kθ)Δ)|2). We discuss the following four possible cases.
Case1 (ks+kθ≥0). We again divide this case into three possible subcases according to ksθ-(ks+kθ)∈{0,1,2}.
Subcase  1 (ksθ-(ks+kθ)=0). From (2.13),
(3.35)y(s+θ)-y¯((ks+kθ)Δ)=u(y¯(ksθ-1)Δ+s+θ-ksθΔΔ(y¯ksθΔ-y¯(ksθ-1)Δ),rksθΔ)-u(y¯(ksθ-1)Δ,rksθ-1Δ)+∫ksθΔs+θf(y¯v,r¯(v))dv+∫ksθΔs+θg(y¯v,r¯(v))dw(v),
which yields
(3.36)𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y(s+θ)-y¯((ks+kθ)Δ)|2)≤3𝔼((s+θ-ksθΔΔ)sup-τ≤θ≤0,0≤s≤T,ksθ≥0|u(y¯(ksθ-1)Δ+s+θ-ksθΔΔ(y¯ksθΔ-y¯(ksθ-1)Δ,rksθΔ))-u(y¯(ksθ-1)Δ,rksθ-1Δ)-u(y¯(ksθ-1)Δ,rksθ-1Δ)(+s+θ-ksθΔΔ)|2)+3𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0|∫ksθΔs+θ‍f(y¯r,r¯(r))dr|2)+3𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0|∫ksθΔs+θ‍g(y¯r,r¯(r))dw(r)|2).
From Assumption 2.4, (3.24), (3.26), and Lemma 3.2, we have
(3.37)𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0|u(y¯(ksθ-1)Δ+s+θ-ksθΔΔ(y¯ksθΔ-y¯(ksθ-1)Δ,rksθΔ))-u(y¯(ksθ-1)Δ,rksθ-1Δ)|2)≤2κ2𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0∥s+θ-ksθΔΔ(y¯ksθΔ-y¯(ksθ-1)Δ)∥2)+8κ2H(2)≤2κ2𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0∥(y¯ksθΔ-y¯(ksθ-1)Δ)∥2)+8κ2H(2)≤2κ2𝔼(sup0≤ksθ≤M-1∥(y¯ksθΔ-y¯(ksθ-1)Δ)∥2)+8κ2H(2)≤2κ2γ(Δ)+8κ2H(2).
By the Hölder inequality, Assumption2.3, and Lemma 3.1,
(3.38)𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0|∫ksθΔs+θf(y¯r,r¯(r))dr|2)≤Δ𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0∫ksθΔs+θ|f(y¯r,r¯(r))|2dr)≤KΔ𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0∫ksθΔs+θ(1+∥y¯r∥2)dr)≤KΔ𝔼[∫0T(1+sup0≤r≤T∥y¯r∥2)dr]≤KΔ∫0T[1+𝔼(sup-τ≤t≤Ty(t)|2)]dr≤KΔ∫0T[1+H(2)]dr≤KT[1+H(2)]Δ.
Setting v=s+θ and kv=ksθ and applying the Hölder inequality yield
(3.39)𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0|∫ksθΔs+θg(y¯r,r¯(r))dw(r)|2)=𝔼(sup0≤v≤T,0≤kv≤M-1|g(y¯kvΔ,r¯(v))(w(v)-w(kvΔ))|2)≤[𝔼(sup0≤v≤T,0≤kv≤M-1|g(y¯kvΔ,r¯(v))|2l/(l-1))](l-1)/l×[𝔼(sup0≤v≤T,0≤kv≤M-1|w(v)-w(kvΔ)|2l)]1/l.
The Doob martingale inequality gives
(3.40)𝔼(sup0≤v≤T,0≤kv≤M-1|w(v)-w(kvΔ)|2l)=𝔼(sup0≤kv≤M-1(supkvΔ≤v≤(kv+1)Δ|w(v)-w(kvΔ)|2l))≤𝔼(∑kv=0M-1(supkvΔ≤v≤(kv+1)Δ|w(v)-w(kvΔ)|2l))=∑kv=0M-1𝔼(supkvΔ≤v≤(kv+1)Δ|w(v)-w(kvΔ)|2l)≤(2l2l-1)2l∑kv=0M-1𝔼|w((kv+1)Δ)-w(kvΔ)|2l.
By (3.27), we therefore have
(3.41)𝔼(sup-τ≤θ≤0,0≤s≤T,ksθ≥0|∫ksθΔs+θ‍g(y¯r,r¯(r))dw(r)|2)≤(2l2l-1)2[𝔼(sup0≤kv≤M-1|g(y¯kvΔ,r¯(v))|2l/(l-1))](l-1)/l×[∑kv=0M-1‍𝔼|w((kv+1)Δ)-w(kvΔ)|2l]1/l≤(2l2l-1)2D(l)Δ(l-1)/l.
Substituting (3.37), (3.38), and (3.41) into (3.36) and noting Δ∈(0,1) give
(3.42)𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y(s+θ)-y¯((ks+kθ)Δ)|2)≤6κ2γ(Δ)+24κ2H(2)+3(KT(1+H(2))+(2l2l-1)2D(l))Δ(l-1)/l=:6κ2γ(Δ)+24κ2H(2)+c1(l)Δ(l-1)/l.Subcase  2 (ksθ-(ks+kθ)=1). From (2.13),
(3.43)y(s+θ)-y¯((ks+kθ)Δ)=y(ksθΔ)-y¯((ks+kθ)Δ)+y(s+θ)-y¯(ksθΔ)≤u(y¯(ks+kθ)Δ,rks+kθΔ)-u(y¯(ks+kθ-1)Δ,rks+kθ-1Δ)+f(y¯(ks+kθ)Δ,rks+kθΔ)Δ+g(y¯(ks+kθ)Δ,rks+kθΔ)Δwks+kθ+y(s+θ)-y¯(ksθΔ),
so we have
(3.44)𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y(s+θ)-y¯((ks+kθ)Δ)|2)≤4{𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|u(y¯(ks+kθ)Δ,rks+kθΔ)-u(y¯(ks+kθ-1)Δ,rks+kθ-1Δ)|2)+𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|f(y¯(ks+kθ)Δ,rks+kθΔ)Δ|2)+𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|g(y¯(ks+kθ)Δ,rks+kθΔ)Δwks+kθ|2)+𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y(s+θ)-y¯(ksθΔ)|2)}.
Since
(3.45)𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|u(y¯(ks+kθ)Δ,rks+kθΔ)-u(y¯(ks+kθ-1)Δ,rks+kθ-1Δ)|2)≤2κ2γ(Δ)+8κ2H(2),
from (3.26), (3.27), and the Subcase 1, noting Δ∈(0,1), we have
(3.46)𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y(s+θ)-y¯((ks+kθ)Δ)|2)≤4{2κ2γ(Δ)+8κ2H(2)+K[1+H(2)]Δ2+D(l)Δ(l-1)/l+3(2κ2γ(Δ)+8κ2H(2))(Δ)+c1(l)Δ(l-1)/l}=:32κ2γ(Δ)+128κ2H(2)+c2(l)Δ(l-1)/l.Subcase  3 (ksθ-(ks+kθ)=2). From (2.13), we have
(3.47)y(s+θ)-y¯((ks+kθ)Δ)=y(s+θ)-y((ksθ-1)Δ)+y((ksθ-1)Δ)-y¯((ks+kθ)Δ),
so from the Subcase 2, we have
(3.48)𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y(s+θ)-y¯((ks+kθ)Δ)|2)≤2𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y(s+θ)-y¯((ksθ-1)Δ)|2)+2𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y((ksθ-1)Δ)-y¯((ks+kθ)Δ)|2)≤2[32κ2γ(Δ)+128κ2H(2)+c2(l)Δ(l-1)/l]+2[2κ2γ(Δ)+8κ2H(2)+K[1+H(2)]Δ2+D(l)Δ(l-1)/l]=:68κ2γ(Δ)+272κ2H(2)+c3(l)Δ(l-1)/l.
From these three subcases, we have
(3.49)𝔼(sup-τ≤θ≤0,0≤s≤T,ks+kθ≥0|y(s+θ)-y¯((ks+kθ)Δ)|2)≤106κ2γ(Δ)+424κ2H(2)+[c1(l)+c2(l)+c3(l)]Δ(l-1)/l.
Case  2  (ks+kθ=-1 and 0≤s+θ≤Δ). In this case, applying Assumption 2.5 and case 1, we have
(3.50)𝔼(sup-τ≤θ≤0,0≤s≤T|y(s+θ)-y¯((ks+kθ)Δ)|2)≤2𝔼(sup-τ≤θ≤0,0≤s≤T|y(s+θ)-y¯(0)|2)+2E|y(0)-y(-Δ)|2≤212κ2γ(Δ)+848κ2H(2)+2[c1(l)+c2(l)+c3(l)]Δ(l-1)/l+2α(Δ).
Case3 (ks+kθ=-1 and -Δ≤s+θ≤0). In this case, using Assumption 2.5,
(3.51)𝔼(sup-τ≤θ≤0,0≤s≤T|y(s+θ)-y¯((ks+kθ)Δ)|2)≤α(Δ).
Case  4 (ks+kθ≤-2). In this case, s+θ≤0, so using Assumption 2.5,
(3.52)𝔼(sup-τ≤θ≤0,0≤s≤T|y(s+θ)-y¯((ks+kθ)Δ)|2)≤α(2Δ).
Substituting these four cases into (3.34) and noting the expression of γ(Δ), there exist c2, c2′, and c¯2(l) such that
(3.53)𝔼(sup0≤s≤T∥ys-y¯s∥2)≤c2α(2Δ)+c2′+c¯2(l)Δ(l-1)/l,
This proof is complete.Remark 3.4.
It should be pointed out that much simpler proofs of Lemmas3.2 and 3.3 can be obtained by choosing l=2 if we only want to prove Theorem 2.7. However, here we choose l>1 to control the stochastic terms β(Δ) and γ(Δ) by Δ(l-1)/l in Section 3, which will be used to show the order of the strong convergence.Lemma 3.5.
If Assumption2.4 holds,
(3.54)𝔼(sup0≤t≤T|u(y¯t,r¯(t))-u(y¯t,r(t))|2)≤8κ2H(2)LΔ∶=ζΔ
where L is a positive constant independent of Δ.Proof.
Letn=[T/Δ], the integer part of T/Δ, and 1G be the indication function of the set G. Then, by Assumption 2.4 and Lemma 3.1, we obtain
(3.55)𝔼(sup0≤t≤T|u(y¯t,r¯(t))-u(y¯t,r(t))|2)≤max0≤k≤n𝔼(sups∈[tk,tk+1)|u(y¯s,r¯(s))-u(y¯s,r(s))|2)≤2max0≤k≤n𝔼(sups∈[tk,tk+1)|u(y¯s,r¯(s))-u(y¯s,r(s))|21{r(s)≠r(tk)})≤4max0≤k≤n𝔼(sups∈[tk,tk+1)(|u(y¯s,r¯(s))|2+|u(y¯s,r(s))|2)1{r(s)≠r(tk)})≤8κ2max0≤k≤n(𝔼(sup0≤t≤T∥y¯t∥2))𝔼(1{r(s)≠r(tk)})≤8κ2H(2)𝔼(1{r(s)≠r(tk)})=8κ2H(2)𝔼[𝔼(1{r(s)≠r(tk)}∣(tk))],
where in the last step we use the fact that y¯tk and 1{r(s)≠r(tk)} are conditionally independent with respect to the σ- algebra generated by r(tk). But, by the Markov property,
(3.56)𝔼(1{r(s)≠r(tk)}∣r(tk))=∑i∈S1{r(tk)=i}P(r(s)≠i∣r(tk)=i)=∑i∈S1{r(tk)=i}∑j≠i‍(γij(s-tk)+o(s-tk))≤∑i∈S1{r(tk)=i}(max1≤i≤N(-γij)Δ+o(Δ))≤LΔ,
where L is a positive constant independent of Δ. Then
(3.57)𝔼(sup0≤t≤T|u(y¯t,r¯(t))-u(y¯t,r(t))|2)≤8κ2H(2)LΔ.
This proof is complete.Lemma 3.6.
If Assumption2.3 holds, there is a constant C, which is independent of Δ such that
(3.58)𝔼∫0T|f(y¯s,r(s))-f(y¯s,r¯(s))|2ds≤CΔ,(3.59)𝔼∫0T‍|g(y¯s,r(s))-g(y¯s,r¯(s))|2ds≤CΔ.Proof.
Letn=[T/Δ], the integer part of T/Δ. Then
(3.60)𝔼∫0T|f(y¯s,r(s))-f(y¯s,r¯(s))|2ds=∑k=0n𝔼∫tktk+1|f(y¯tk,r(s))-f(y¯tk,r(tk))|2ds,
with tn+1 being T. Let 1G be the indication function of the set G. Moreover, in what follows, C is a generic positive constant independent of Δ, whose values may vary from line to line. With these notations we derive, using Assumption 2.3, that
(3.61)𝔼∫tktk+1|f(y¯tk,r(s))-f(y¯tk,r(tk))|2ds≤2𝔼∫tktk+1[|f(y¯tk,r(s))|2+|f(y¯tk,r(tk))|2]1{r(s)≠r(tk)}ds≤C𝔼∫tktk+1(1+∥y¯tk∥2)1{r(s)≠r(tk)}ds≤C∫tktk+1𝔼[𝔼[(1+∥y¯tk∥2)1{r(s)≠r(tk)}∣r(tk)]]ds≤C∫tktk+1𝔼[𝔼[(1+∥y¯tk∥2)∣r(tk)]𝔼[1{r(s)≠r(tk)}∣r(tk)]]ds,
where in the last step we use the fact that y¯tk and 1{r(s)≠r(tk)} are conditionally independent with respect to the σ- algebra generated by r(tk). But, by the Markov property,
(3.62)𝔼[1{r(s)≠r(tk)}∣r(tk)]=∑i∈S1{r(tk)=i}P(r(s)≠i∣r(tk)=i)=∑i∈S1{r(tk)=i}∑j≠i(γij(s-tk)+o(s-tk))≤∑i∈S1{r(tk)=i}(max1≤i≤N(-γij)Δ+o(Δ))≤CΔ.
So, by Lemma 3.1,
(3.63)𝔼∫tktk+1|f(y¯tk,r(s))-f(y¯tk,r(tk))|2ds≤CΔ∫tktk+1(1+𝔼∥y¯tk∥2)ds≤CΔ2.
Therefore
(3.64)𝔼∫0T|f(y¯s,r(s))-f(y¯s,r¯(s))|2ds≤CΔ.
Similarly, we can show (3.59).


## 4. Convergence of the EM Methods

In this section we will use the lemmas above to prove Theorem2.7. From Lemma 3.1 and Theorem 2.6, there exists a positive constant H~ such that
(4.1)𝔼(sup-τ≤t≤T|x(t)|p)∨𝔼(sup-τ≤t≤T|y(t)|p)≤H~.
Let j be a sufficient large integer. Define the stopping times
(4.2)uj∶=inf{t≥0:∥xt∥≥j},vj∶=inf{t≥0:∥yt∥≥j},ρj∶=uj∧vj,
where we set inf∅=∞ as usual. Let
(4.3)e(t)∶=x(t)-y(t).
Obviously,
(4.4)𝔼(sup0≤t≤T|e(t)|2)=𝔼(sup0≤t≤T|e(t)|21{uj>T,vj>T})+𝔼(sup0≤t≤T|e(t)|21{uj≤Torvj≤T}).
Recall the following elementary inequality:
(4.5)aγb1-γ≤γa+(1-γ)b,∀a,b>0,γ∈[0,1].
We thus have, for any δ>0,
(4.6)𝔼(sup0≤t≤T|e(t)|21{uj≤Torvj≤T})≤𝔼[(δsup{0≤t≤T}|e(t)|p)2/p(δ-2/(p-2)1{uj≤Torvj≤T})(p-2)/p]≤2δp𝔼(sup0≤t≤T|e(t)|p)+p-2pδ2/(p-2)ℙ(uj≤Torvj≤T).
Hence
(4.7)𝔼(sup0≤t≤T|e(t)|2)≤𝔼(sup0≤t≤T|e(t)|21{ρj>T})+2δp𝔼(sup0≤t≤T|e(t)|p)+p-2pδ2/(p-2)ℙ(uj≤Torvj≤T).
Now,
(4.8)ℙ(uj≤T)≤𝔼(1{uj≤T}∥xt∥pjp)≤1jp𝔼(sup-τ≤t≤T|x(t)|p)≤H~jp.
Similarly,
(4.9)ℙ(vj≤T)≤H~jp.
Thus
(4.10)ℙ(vj≤Toruj≤T)≤ℙ(vj≤T)+ℙ(uj≤T)≤2H~jp.
We also have
(4.11)𝔼(sup0≤t≤T|e(t)|p)≤2p-1𝔼(sup0≤t≤T(|xt|p+|yt|p))≤2pH~.
Moreover,
(4.12)𝔼(sup0≤t≤T|e(t)|21{ρj>T})=𝔼(sup0≤t≤T|e(t∧ρj)|21{ρj>T})≤𝔼(sup0≤t≤T|e(t∧ρj)|2).
Using these bounds in (4.7) yields
(4.13)𝔼(sup0≤t≤T|e(t)|2)≤𝔼(sup0≤t≤T|e(t∧ρj)|2)+2p+1δH~p+2(p-2)H~pδ2/(p-2)jp.
Setting v:=t∧ρj and for any ɛ∈(0,1), by the Hölder inequality, when v∈[kΔ,(k+1)Δ], for k=0,1,2,…,M-1,
(4.14)|e(v)|2=|x(v)-y(v)|2≤|u(xv,r(v))-u(y¯(k-1)Δ+(v-kΔ)Δ(y¯kΔ-y¯(k-1)Δ),rkΔ)+∫0v[f(xs,r(s))-f(y¯s,r¯(s))]ds+∫0v[g(xs,r(s))-g(y¯s,r¯(s))]dw(s)|2≤1ɛ|u(xv,r(v))-u(y¯(k-1)Δ+(v-kΔ)Δ(y¯kΔ-y¯(k-1)Δ),rkΔ)|2+21-ɛ[T∫0v[f(xs,r(s))-f(y¯s,r¯(s))]2ds+|∫0v[g(xs,r(s))-g(y¯s,r¯(s))]dw(s)|2].
Assumption 2.4 yields
(4.15)|u(xv,r(v))-u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),rkΔ)|2≤2κ2∥xv-y¯(k-1)Δ-v-kΔΔ(y¯kΔ-y¯(k-1)Δ)∥2+2|u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),r(v))-u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),rkΔ)|2≤2κ2∥|xv-yv|+|yv-y¯v|+|y¯kΔ-y¯(k-1)Δ|-v-kΔΔ|y¯kΔ-y¯(k-1)Δ|∥2+2|u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),r(v))-u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),rkΔ)|2≤2κ2ɛ∥xv-yv∥2+4κ21-ɛ(∥yv-y¯v∥2+∥y¯kΔ-y¯(k-1)Δ∥2)+2|u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),r(v))-u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),rkΔ)|2.
Then, we have
(4.16)|e(v)|2≤2κ2ɛ2∥xv-yv∥2+4κ2ɛ(1-ɛ)(∥yv-y¯v∥2+∥y¯kΔ-y¯(k-1)Δ∥2)+2|u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),r(v))-u(y¯(k-1)Δ+v-kΔΔ(y¯kΔ-y¯(k-1)Δ),rkΔ)|2+21-ɛ[T∫0v‍[f(xs,r(s))-f(y¯s,r¯(s))]2ds+|∫0v‍[g(xs,r(s))-g(y¯s,r¯(s))]dw(s)|2].
Hence, for any t1∈[0,T], by Lemmas 3.2–3.5,
(4.17)𝔼[sup0≤t≤t1|e(t∧ρj)|2]≤2κ2ɛ2𝔼(sup0≤t≤t1∥xt∧ρj-yt∧ρj∥2)+4κ2ɛ(1-ɛ)[𝔼(sup0≤t≤t1∥yt∧ρj-y¯t∧ρj∥2)+𝔼(sup0≤k≤M-1∥y¯kΔ∧ρj-y¯(k-1)Δ∧ρj∥2)]+2𝔼[sup0≤k≤M-1|u(y¯(k-1)Δ+kΔ∧ρj-kΔΔ(y¯kΔ-y¯(k-1)Δ),r(kΔ∧ρj))-u(y¯(k-1)Δ+kΔ∧ρj-kΔΔ(y¯kΔ-y¯(k-1)Δ),r¯(kΔ∧ρj))|2]+2T1-ɛ𝔼∫0t1∧ρj[f(xs,r(s))-f(y¯s),r¯(s)]2ds+21-ɛ𝔼[sup0≤t≤t1|∫0t∧ρj[g(xs,r(s))-g(y¯s,r¯(s))]dw(s)|2]≤2κ2ɛ2𝔼(sup0≤t≤t1∥xt∧ρj-yt∧ρj∥2)+4κ2ɛ(1-ɛ)(γ(Δ)+β(Δ))+2ζΔ+2T1-ɛ𝔼∫0t1∧ρj[f(xs,r(s))-f(y¯s,r¯(s))]2ds+21-ɛ𝔼[sup0≤t≤t1|∫0t∧ρj[g(xs,r(s))-g(y¯s,r¯(s))]dw(s)|2].
Since x(t)=y(t)=ξ(t) when t∈[-τ,0], we have
(4.18)𝔼(sup0≤t≤t1∥xt∧ρj-yt∧ρj∥2)≤𝔼(sup-τ≤θ≤0sup0≤t≤t1|x(t∧ρj+θ)-y(t∧ρj+θ)|2)≤𝔼(sup-τ≤t≤t1|x(t∧ρj)-y(t∧ρj)|2)=𝔼(sup0≤t≤t1|x(t∧ρj)-y(t∧ρj)|2).
By Assumption 2.2, Lemma 3.3, and Lemma 3.6, we may compute
(4.19)𝔼∫0t1∧ρj|f(xs,r(s))-f(y¯s,r¯(s))|2ds≤𝔼∫0t1∧ρj|f(xs,r(s))-f(y¯s,r(s))+f(y¯s,r(s))-f(y¯s,r¯(s))|2ds≤2Cj𝔼∫0t1∧ρj∥xs-ys+ys-y¯s∥2ds+2CΔ≤4Cj𝔼∫0t1∧ρj∥xs-ys∥2ds+4Cj𝔼∫0t1∧ρj∥ys-y¯s∥2ds+2CΔ≤4Cj𝔼∫0t1∧ρjsup-τ≤θ≤0|x(s+θ)-y(s+θ)|2ds+4CjTβ(Δ)+2CΔ≤4Cj𝔼∫0t1sup-τ≤θ≤0|x(s∧ρj+θ)-y(s∧ρj+θ)|2ds+4CjTβ(Δ)+2CΔ≤4Cj𝔼∫0t1sup-τ≤r≤s|x(r∧ρj)-y(r∧ρj)|2ds+4CjTβ(Δ)+2CΔ=4Cj∫0t1𝔼sup0≤r≤s|x(r∧ρj)-y(r∧ρj)|2ds+4CjTβ(Δ)+2CΔ.
By the Doob martingale inequality, Lemma 3.3, Lemma 3.6, and Assumption 2.2, we compute
(4.20)𝔼[sup0≤t≤t1|∫0t∧ρj[g(xs,r(s))-g(y¯s,r¯(s))]dw(s)|2]=𝔼[sup0≤t≤t1|∫0t∧ρj[g(xs,r(s))-g(y¯s,r(s))+g(y¯s,r(s))-g(y¯s,r¯(s))]dw(s)|2]≤8Cj𝔼∫0t1∧ρj∥xs-y¯s∥2ds+8CΔ≤16Cj∫0t1𝔼(sup0≤r≤s|x(r∧ρj)-y(r∧ρj)|2)ds+16CjTβ(Δ)+8CΔ.
Therefore, (4.17) can be written as
(4.21)(1-2κ2ɛ2)𝔼[sup0≤t≤t1|e(t∧ρj)|2]≤4κ2ɛ(1-ɛ)[β(Δ)+γ(Δ)]+8CjT(T+4)1-ɛβ(Δ)+4CΔ(T+8)1-ɛ+2ζΔ+8Cj(T+4)1-ɛ∫0t1‍𝔼(sup0≤v≤s|e(v∧ρj)|2)ds.
Choosing ɛ=(1+2κ)/2 and noting κ∈(0,1), we have
(4.22)𝔼(sup0≤t≤t1|e(t∧ρj)|2)≤16κ2(1+2κ)(1-2κ)2(1+32κ)[β(Δ)+γ(Δ)]+16CjT(1+2κ)2(T+4)(1-2κ)2(1+32κ)β(Δ)+8CΔ(T+8)(1+2κ)2(1-2κ)2(1+32κ)+2(1+2κ)2(1-2κ)2(1+32κ)ζΔ+16Cj(1+2κ)2(T+4)(1-2κ)2(1+32κ)∫0t1‍𝔼(sup0≤s≤v|e(s∧ρj)|2)dv≤16(1-2κ)2[β(Δ)+γ(Δ)]+16CjT(T+4)(1-2κ)2β(Δ)+8CΔ(T+8)(1-2κ)2+2(1-2κ)ζΔ+16Cj(T+4)(1-2κ)2∫0t1𝔼(sup0≤s≤v|e(s∧ρj)|2)dv.
By the Gronwall inequality, we have
(4.23)𝔼[sup0≤t≤t1|e(t∧ρj)|2]≤[16(1-2κ)2(β(Δ)+γ(Δ))+16CjT(T+4)(1-2κ)2β(Δ)+8CΔ(T+8)(1-2κ)2+2ζΔ(1-2κ)]×e(16/(1-2κ)2)CjT(T+4).
By (4.13),
(4.24)𝔼(sup0≤t≤T|e(t)|2)≤[16(1-2κ)2(β(Δ)+γ(Δ))+16CjT(T+4)(1-2κ)2β(Δ)+8CΔ(T+8)(1-2κ)2+2ζΔ(1-2κ)]×e(16/(1-2κ)2)CjT(T+4)+2p+1δH~p+2(p-2)H~pδ2/(p-2)jp.
Given any ϵ>0, we can now choose δ sufficient small such that 2p+1δH~/p≤ϵ/3, then choose j sufficient large such that
(4.25)2(p-2)H~pδ2/(p-2)jp<ϵ3,
and finally choose Δ so small such that
(4.26)[16(1-2κ)2(β(Δ)+γ(Δ))+16CjT(T+4)(1-2κ)2β(Δ)+8CΔ(T+8)(1-2κ)2+2ζΔ(1-2κ)]×e(16/(1-2κ)2)CjT(T+4)<ϵ3
and thus 𝔼(sup0≤t≤T|e(t)|2)≤ϵ as required.Remark 4.1.
Obviously, according to Theorem2.7, for neutral stochastic delay differential equations with Markovian switching [19], we can easily obtain that the numerical solutions converge to the true solutions in mean square under Assumptions 2.2–2.4.


## 5. Convergence Order of the EM Method

To reveal the convergence order of the EM method, we need the following assumptions.Assumption 5.1.
There exists a constantQ such that for all φ, ψ∈C([-τ,0];ℝn),  i∈S, and t∈[0,T],
(5.1)|f(φ,i)-f(ψ,i)|2∨|g(φ,i)-g(ψ,i)|2≤Q∥φ-ψ∥2.It is easy to see from the global Lipschitz condition that, for anyφ∈C([-τ,0];ℝn),
(5.2)|f(φ,i)|2∨|g(ψ,i)|2≤2(|f(0,i)|2∨|g(0,i)|2)+Q∥φ∥2.
In other words, the global Lipschitz condition implies linear growth condition with the growth coefficient
(5.3)K=2[|f(0,i)|2∨|g(0,i)|2∨Q].Assumption 5.2.
ξ
∈
L
ℱ
0
p
(
[
-
τ
,
0
]
;
ℝ
n
) for some p≥2, and there exists a positive constant λ such that
(5.4)𝔼(sup-τ≤s≤t≤0|ξ(t)-ξ(s)|2)≤λ(t-s).We can state another theorem, which reveals the order of the convergence.Theorem 5.3.
If Assumptions5.1, 5.2, and 2.4 hold, for any positive constant l>1,
(5.5)𝔼(sup0≤t≤T|x(t)-y(t)|2)≤O(Δ1-1/l).Proof.
Sinceα(Δ) may be replaced by λΔ, from Lemmas 3.2 and 3.3, there exist constants c~1(l) and c~2(l) such that β(Δ)≤c~1(l)Δ(l-1)/l and γ(Δ)≤c~2(l)Δ(l-1)/l. Here we do not need to define the stopping times uj and vj, and we may repeat the proof in Section 4 and directly compute
(5.6)𝔼(sup0≤t≤T|e(t)|2)≤[16(1-2κ)2(c~1(l)+c~2(l))+16QT(T+4)(1-2κ)2c~1(l)+8C(T+8)(1-2κ)2+2ζ(1-2κ)]×e(16/(1-2κ)2)QT(T+4)Δ(l-1)/l≤O(Δ(l-1)/l).
The proof is complete.


## 6. Conclusion

The EM method for neutral stochastic functional differential equations with Markovian switching is studied. The results show that the numerical solution converges to the true solution under the local Lipschitz condition. In addition, the results also show that the order of convergence of the numerical method is close to1, although the order of the strong convergence in mean square for the EM scheme applied to both SDEs and SFDEs is one [6, 7, 11] under the global Lipschitz condition. Hence, we can control the numerical solution’s error; this method may value some path-dependent options more quickly and simply [25].



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*Source: 675651-2012-10-30.xml*