Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 675651, 32 pages
doi:10.1155/2012/675651
Research Article
Approximations of Numerical Method for Neutral
Stochastic Functional Differential Equations with
Markovian Switching
Hua Yang1 and Feng Jiang2
1
2
School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
Correspondence should be addressed to Feng Jiang, fjiang78@gmail.com
Received 9 April 2012; Accepted 17 September 2012
Academic Editor: Zhenyu Huang
Copyright q 2012 H. Yang and F. Jiang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
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.
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
2
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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 Section 2, 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 wt w1 t, . . . , wm tT 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
p
with the norm ϕ sup−τ≤θ≤0 |ϕθ|. Let p > 0 and LF0 −τ, 0; Rn be the family of F0 n
measurable C−τ, 0; R -valued random variables ξ such that Eξp < ∞. If xt is an Rn valued stochastic process on t ∈ −τ, ∞, we let xt {xt θ : −τ ≤ θ ≤ 0} for t ≥ 0.
Let rt, 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
⎧
⎨γij Δ oΔ
P rt Δ j | rt i ⎩1 γij Δ oΔ
if i /
j,
if i j,
2.1
where Δ > 0. Here γij ≥ 0 is the transition rate from i to j if i /
j while γii − i / j γij . 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 0, ∞.
In this paper, we consider the n-dimensional NSFDEsMS
dxt − uxt , rt fxt , rtdt gxt , rtdwt,
p
t ≥ 0,
2.2
with initial data x0 ξ ∈ LF0 −τ, 0; Rn and r0 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 xt.
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.
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3
Lemma 2.1. Given Δ > 0, let rkΔ rkΔ for k ≥ 0. Then {rkΔ , k 0, 1, 2, . . .} is a discrete Markov
chain with the one-step transition probability matrix
P Δ Pij Δ N×N eΔΓ .
2.3
For the completeness, we give the simulation of the Markov chain as follows. Given a
stepsize Δ > 0, we compute the one-step transition probability matrix
P Δ Pij Δ N×N eΔΓ .
2.4
Let r0Δ i0 and generate a random number ξ1 which is uniformly distributed in 0, 1. Define
r1Δ
⎧
⎪
⎪
⎪i1 ,
⎨
if i1 ∈ S − {N} such that
N−1
⎪
⎪
⎪
Pi0 ,j Δ ≤ ξ1 ,
⎩N, if
i
1 −1
i1
j1
j1
Pi0 ,j Δ ≤ ξ1 <
Pi0 ,j Δ,
2.5
j1
where we set 0i1 Pi0 ,j Δ 0 as usual. Generate independently a new random number ξ2
which is again uniformly distributed in 0, 1, and then define
r2Δ ⎧
⎪
⎪
⎪
⎨i2 ,
if i2 ∈ S − {N} such that
N−1
⎪
⎪
⎪
Pr1Δ ,j Δ ≤ ξ2 .
⎩N, if
i
2 −1
i2
j1
j1
Pr1Δ ,j Δ ≤ ξ2 <
Pr1Δ ,j Δ,
2.6
j1
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 ykΔ, k ≥ −N is defined as follows:
ykΔ ξkΔ, −N ≤ k ≤ 0,
Δ
f y kΔ , rkΔ Δ
yk 1Δ ykΔ u ykΔ , rkΔ − u y k−1Δ , rk−1
g ykΔ , rkΔ Δwk ,
0 ≤ k ≤ M − 1,
2.7
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where Δwk wk 1Δ − wkΔ and y kΔ {y kΔ θ : −τ ≤ θ ≤ 0} is a C−τ, 0; Rn -valued
random variable defined by
ykΔ θ yk iΔ θ − iΔ yk i 1Δ − yk iΔ
Δ
Δ − θ − iΔ
θ − iΔ
yk iΔ yk i 1Δ,
Δ
Δ
2.8
for iΔ ≤ θ ≤ i 1Δ, i −N, −N 1, . . . , −1, where in order for y −Δ to be well defined, we set
y−N 1Δ ξ−NΔ.
That is, ykΔ · is the linear interpolation of yk − NΔ, yk − N 1Δ, . . . , ykΔ.
We hence have
y θ Δ − θ − iΔ yk iΔ θ − iΔ yk i 1Δ
kΔ
Δ
Δ
≤ yk iΔ ∨ yk i 1Δ.
2.9
We therefore obtain
y
kΔ θ
max yk iΔ,
−N≤i≤0
for any k −1, 0, 1, . . . , M − 1.
2.10
It is obvious that y −Δ ≤ y 0 .
In our analysis it will be more convenient to use continuous-time approximations. We
hence introduce the C−τ, 0; Rn -valued step process
yt M−2
ykΔ 1kΔ,k
1Δ t y M−1Δ 1M−1Δ,MΔ t,
k0
rt M−1
2.11
rkΔ 1kΔ,k
1Δ t,
k0
and we define the continuous EM approximate solution as follows: let yt ξt for −τ ≤ t ≤
0, while for t ∈ kΔ, k 1Δ, k 0, 1, . . . , M − 1,
t − kΔ ykΔ − y k−1Δ , rkΔ − u y −Δ , r0Δ
yt ξ0 u y k−1Δ Δ
t
t
f y s , rs ds g y s , rs dws.
0
0
2.12
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5
Clearly, 2.12 can also be written as
t − kΔ Δ
yt ykΔ u y k−1Δ ykΔ − y k−1Δ , rkΔ − u y k−1Δ , rk−1
Δ
t
t
f ys , rs ds g ys , rs dws.
kΔ
2.13
kΔ
In particular, this shows that ykΔ ykΔ, that is, the discrete and continuous EM
approximate solutions coincide at the grid points. We know that yt is not computable
because it requires knowledge of the entire Brownian path, not just its Δ-increments.
However, ykΔ ykΔ, so the error bound for yt will automatically imply the error
bound for ykΔ. It is then obvious that
y ≤ ykΔ ,
kΔ
∀k 0, 1, 2, . . . , M − 1.
2.14
Moreover, for any t ∈ 0, T ,
sup yt 0≤t≤T
sup y kΔ ≤
0≤k≤M−1
sup
sup ykΔ 0≤k≤M−1
sup ykΔ θ
0≤k≤M−1 −τ≤θ≤0
2.15
≤ sup sup yt θ
0≤t≤T −τ≤θ≤0
≤ sup ys,
−τ≤s≤T
and letting t/Δ be the integer part of t/Δ, then
y y
t
t/ΔΔ ≤ yt/Δ Δ ≤ sup ys .
−τ≤s≤t
2.16
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 integer j ≥ 1 and i ∈ S, there exists a positive constant Cj such that
2
2 2 f ϕ, i − f ψ, i ∨ g ϕ, i − g ψ, i ≤ Cj ϕ − ψ 2.17
for ϕ, ψ ∈ C−τ, 0; Rn with ϕ ∨ ψ ≤ j.
Assumption 2.3. There is a constant K > 0 such that
2 2
f ϕ, i ∨ g ϕ, i ≤ K 1 ϕ2 ,
for ϕ ∈ C−τ, 0; Rn and i ∈ S.
2.18
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Assumption 2.4. There exists a constant κ ∈ 0, 1 such that for all ϕ, ψ ∈ C−τ, 0; Rn and
i ∈ S,
u ϕ, i − u ψ, i ≤ κϕ − ψ ,
2.19
for ϕ, ψ ∈ C−τ, 0; Rn and u0, i 0.
We also impose the following condition on the initial data.
p
Assumption 2.5. ξ ∈ LF0 −τ, 0; Rn for some p ≥ 2, and there exists a nondecreasing function
α· such that
E
sup |ξt − ξs|
2
−τ≤s≤t≤0
≤ αt − s,
2.20
with the property αs → 0 as s → 0.
From Mao and Yuan 11, we may therefore state the following theorem.
Theorem 2.6. Let p ≥ 2. If Assumptions 2.3–2.5 are satisfied, then
E
sup |xt|
p
−τ≤t≤T
≤ Hκ,p,T,K,ξ ,
2.21
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 Assumptions 2.2–2.5 hold,
lim E
Δ→0
2
sup xt − yt
0.
2.22
0≤t≤T
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 Assumptions 2.3–2.5 hold, for any p ≥ 2, there exists a constant Hp such that
E
where Hp is independent of Δ.
p
sup yt
−τ≤t≤T
≤H p ,
3.1
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7
Proof. For t ∈ kΔ, k 1Δ, k 0, 1, 2, . . . , M − 1, set yt
: yt − uyk−1Δ t − kΔykΔ −
Δ
yk−1Δ /Δ, rk and
ht : E
p
,
sup ys
p
ht : E sup ys
.
−τ≤s≤t
3.2
0≤s≤t
Recall the inequality that for p ≥ 1 and any ε > 0, |x y|p ≤ 1 εp−1 |x|p ε1−p |y|p . Then we
have, from Assumption 2.4,
⎛
⎞p ⎞
−
y
y
−
kΔ
t
p
kΔ
k−1Δ
1−p ⎜
Δ ⎟ ⎟
ytp ≤ 1 εp−1 ⎜
, rk ⎠ ⎠
ε u⎝yk−1Δ ⎝ yt
Δ
⎛
≤ 1 εp−1
p
p
t − kΔ .
yt
ε1−p κp y
y
−
y
kΔ
k−1Δ k−1Δ
Δ
3.3
By y −Δ ≤ y 0 , noting k 0, 1, 2, . . . , M − 1,
t − kΔ p k 1Δ − t p
t − kΔ y
ykΔ − y k−1Δ ≤ ykΔ y k−1Δ k−1Δ Δ
Δ
Δ
p
t − kΔ
k 1Δ − t
sup ys sup ys
≤
Δ
Δ
−τ≤s≤t
−τ≤s≤t
p
≤ sup ys .
−τ≤s≤t
3.4
Consequently, choose ε κ/1 − κ, then
p
p
ytp ≤ 1 − κ1−p yt
.
κ sup ys
−τ≤s≤t
3.5
Hence,
p
ht ≤ Eξ E sup ys
p
0≤s≤t
3.6
≤ Eξp κht 1 − κ1−p ht,
which implies
ht ≤
ht
Eξp
.
1 − κ 1 − κp
3.7
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Journal of Applied Mathematics
Since
yt
y0
t
0
f y s , rs ds t
0
g y s , rs dws,
3.8
with y0
y0 − uy−Δ , by the Hölder inequality, we have
p t p
p p−1 t p
p−1
yt
f y , rs ds g y , rs dws .
y0
≤3
t
s
s
0
0
3.9
Hence, for any t1 ∈ 0, T ,
1 ≤ 3
ht
p−1
p
T p−1 E
Ey0
t1
0
p t p
f y , rs ds E sup g y , rs dws
.
s
s
0≤t≤t1 0
3.10
By Assumption 2.4 and the fact y −Δ ≤ y 0 , we compute that
p
p
Ey0 − u y −Δ , r0Δ Ey0
p
≤ E y0 κy −Δ p
≤ E y0 κy 0 3.11
≤ E|ξ0| κξp
≤ 1 κp Eξp .
Assumption 2.3 and the Hölder inequality give
E
t1
0
f y , rs p ds ≤ E
s
t1
0
2 p/2
K p/2 1 y s ds
≤ K p/2 2p−2/2 E
≤K
p/2 p−2/2
2
t1
0
p 1 ys ds
t1 p
T
E sup yt ds .
0
−τ≤t≤s
3.12
Journal of Applied Mathematics
9
Applying the Burkholder-Davis-Gundy inequality, the Hölder inequality and Assumption
2.3 yield
p t
p/2
t 1 2
g y s , rs ds
≤ Cp E
E sup g ys , rs dws
0≤t≤t1 0
0
≤ Cp T p−2/2 E
t1
0
≤ Cp T p−2/2 E
t1
0
g y , rs p ds
s
2 p/2
K p/2 1 ys ds
≤ Cp T p−2/2 K p/2 2p−2/2 E
t1
0
t1 p
ds ,
T
E sup yt
≤ Cp T
p−2/2
K
p 1 ys ds
p/2 p−2/2
2
−τ≤t≤s
0
3.13
where Cp is a constant dependent only on p. Substituting 3.11, 3.12, and 3.13 into 3.10
gives
1 ≤ 3p−1 1 κp Eξp K p/2 2p−2/2 T p Cp 2T p−2/2 K p/2 T
ht
3
p−1
K
p/2 p−2/2 p−1
2
T
p−2/2
Cp 2T K
p/2
! t1
E
0
: C1 C2
!
p
ds
sup yt
−τ≤t≤s
3.14
t1
hsds.
0
Hence from 3.7, we have
t1
1
Eξp
C1 C2
hsds
ht1 ≤
1 − κ 1 − κp
0
C1
C2
Eξp
≤
1 − κ 1 − κp 1 − κp
t1
3.15
hsds.
0
By the Gronwall inequality we find that
"
hT ≤
#
p
C1
Eξp
eC2 T/1−κ .
1 − κ 1 − κp
3.16
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.
10
Journal of Applied Mathematics
Lemma 3.2. If Assumptions 2.3–2.5 hold, then for any integer l > 1,
E
2
sup ykΔ − y k−1Δ ≤ c1 c1 αΔ c1 lΔl−1/l : γΔ,
0≤k≤M−1
3.17
where c1 1/1 − 2κ, c1 8κ/1 − 2κH2, and c1 l is a constant dependent on l but
independent of Δ.
Proof. For θ ∈ iΔ, i 1Δ, where i −N, −N 1, . . . , −1, from 2.7,
i 1Δ − θ yk iΔ − yk − 1 iΔ
y kΔ − y k−1Δ ≤
Δ
θ − iΔ yk i 1Δ − yk iΔ
Δ
≤ yk iΔ − yk − 1 iΔ ∨ yk i 1Δ − yk iΔ,
3.18
so
y kΔ − y k−1Δ ≤ sup yk iΔ − yk − 1 iΔ.
3.19
−N≤i≤0
We therefore have
E
2
sup ykΔ − yk−1Δ ≤E
0≤k≤M−1
sup
−N≤i≤0
0≤k≤M−1
ykΔ − yk − 1Δ2 .
≤E
2
sup yk iΔ − yk − 1 iΔ
sup
−N≤k≤M−1
3.20
When −N ≤ k ≤ 0, by Assumption 2.5 and y−N 1Δ ξ−NΔ,
E
2
sup ykΔ − yk − 1Δ
−N≤k≤0
≤E
sup |ξkΔ − ξk − 1Δ|
−N≤k≤0
≤ αΔ.
2
3.21
Journal of Applied Mathematics
11
When 1 ≤ k ≤ M − 1, from 2.13, we have
Δ
Δ
Δ
− u yk−2Δ , rk−2
f y k−1Δ , rk−1
Δ
ykΔ − yk − 1Δ u yk−1Δ , rk−1
Δ
Δwk−1 .
g yk−1Δ , rk−1
3.22
Recall the elementary inequality, for any x, y > 0 and ε ∈ 0, 1, x y2 ≤ x2 /ε y2 /1 − ε.
Then
ykΔ − yk − 1Δ2
2
1 Δ
Δ
Δ
Δ
− u yk−2Δ , rk−1
u yk−2Δ , rk−1
− u yk−2Δ , rk−2
u y k−1Δ , rk−1
ε
2
1 Δ
Δ
Δ g yk−1Δ , rk−1
Δwk−1 f y k−1Δ , rk−1
1−ε
2 8κ2 2
2κ2 ≤
y k−1Δ − yk−2Δ yk−2Δ ε
ε
2
2
2 2 2
Δ
Δ
Δwk−1 .
f y k−1Δ , rk−1
g yk−1Δ , rk−1
Δ 1−ε
1−ε
≤
3.23
Consequently
E
2
sup ykΔ − yk − 1Δ
1≤k≤M−1
2
2
8κ2
2κ2
E
E
sup yk−1Δ − y k−2Δ sup y k−2Δ ≤
ε
ε
1≤k≤M−1
1≤k≤M−1
2
2
2
2Δ2
E
sup f yk−1Δ , rt E
sup g yk−1Δ , rt Δwk−1 .
1−ε
1−ε
1≤k≤M−1
1≤k≤M−1
3.24
We deal with these four terms, separately. By 3.21,
E
2
sup y k−1Δ − yk−2Δ 1≤k≤M−1
≤E
sup
1≤k≤M−1 −N≤i≤0
≤E
2
sup yk i − 1Δ − yk − 2 iΔ
sup
ykΔ − yk − 1Δ2
−N≤k≤M−1
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Journal of Applied Mathematics
2
2
E
sup ykΔ − yk − 1Δ
≤ E sup ykΔ − yk − 1Δ
−N≤k≤0
1≤k≤M−1
2
.
sup ykΔ − yk − 1Δ
≤ αΔ E
1≤k≤M−1
3.25
Noting that Esup−τ≤t≤T |yt|2 ≤ Esup−τ≤t≤T |yt|2 ≤ H2 where Hp has been defined
in Lemma 3.1, by Assumption 2.3 and 2.15,
E
2
Δ
sup f y k−1Δ , rk−1
1≤k≤M−1
≤E
2 sup K 1 y k−1Δ 1≤k≤M−1
≤ K KE
≤ K KE
≤ K KE
2
sup yk − 1 iΔ
sup
1≤k≤M−1 −N≤i≤0
ykΔ2
sup
−N≤k≤M−1
2
sup yt
3.26
−τ≤t≤T
≤ K1 H2.
By the Hölder inequality, for any integer l > 1,
E
2
Δ
Δwk−1 sup g yk−1Δ , rk−1
1≤k≤M−1
2
sup g y k−1Δ , rt sup |Δwk−1 |2
≤E
1≤k≤M−1
≤ E
1≤k≤M−1
2l/l−1
Δ
sup g yk−1Δ , rk−1
l−1/l E
1≤k≤M−1
≤ E
2 l/l−1
sup K 1 ykΔ 0≤k≤M−1
⎡
≤ ⎣K l/1−l E 1 1/l
sup |Δwk−1 |
2l
1≤k≤M−1
l−1/l E
M−1
|Δwk |
1/l
2l
k0
2
sup ykΔ 0≤k≤M−1
l/l−1 ⎤l−1/l 1/l
M−1
2l
⎦
E|Δwk |
k0
Journal of Applied Mathematics
≤ 2
1/l−1
K
l/1−l
13
1
E
2l/l−1
sup ykΔ l−1/l M−1
2l − 1!!Δ
0≤k≤M−1
"
≤ 2
1/l−1
K
l/1−l
2l
1
H
l−1
1/l
l
k0
#l−1/l
2l − 1!!T Δl−1
!1/l
≤ DlΔl−1/l ,
3.27
where Dl is a constant dependent on l.
Substituting 3.25, 3.26, and 3.27 into 3.24, choosing ε κ, and noting Δ ∈ 0, 1,
we have
2
sup ykΔ − yk − 1Δ
E
1≤k≤M−1
3.28
2κ
8κ
2K1 H2 2Dl l−1/l
≤
αΔ H2 Δ
.
1 − 2κ
1 − 2κ
1 − 2κ2
Combining 3.21 with 3.28, from 3.20, we have
E
2
sup ykΔ − yk−1Δ 0≤k≤M−1
≤E
sup
−N≤k≤M−1
≤E
≤
ykΔ − yk − 1Δ2
2
sup ykΔ − yk − 1Δ
−N≤k≤0
E
2
sup ykΔ − yk − 1Δ
3.29
1≤k≤M−1
1
8κ
2K1 H2 2Dl l−1/l
αΔ H2 Δ
,
1 − 2κ
1 − 2κ
1 − 2κ2
as required.
Lemma 3.3. If Assumptions 2.3–2.5 hold,
E
2
sup ys − y s 0≤s≤T
≤ c2 c2 α2Δ c2 lΔl−1/l : βΔ,
3.30
where c2 , c2 are a constant independent of l and Δ, c2 l is a constant dependent on l but independent
of Δ.
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Journal of Applied Mathematics
Proof. Fix any s ∈ 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 ∈ ks Δ, ks 1Δ, θ ∈ kθ Δ, kθ 1Δ, and s θ ∈ ksθ Δ, ksθ 1Δ, respectively. Clearly,
0 ≤ s θ − ks kθ ≤ 2Δ,
3.31
ksθ − ks kθ ∈ {0, 1, 2}.
From 2.7,
y s yks Δ θ
yks kθ Δ 3.32
θ − kθ Δ yks kθ 1Δ − yks kθ Δ ,
Δ
which yields
ys − y ys θ − y θ
s
ks Δ
θ − kθ Δ ≤ ys θ − yks kθ Δ yks kθ 1Δ − yks kθ Δ
Δ
≤ ys θ − yks kθ Δ yks kθ 1Δ − yks kθ Δ,
3.33
so by 3.20 and Lemma 3.2, noting yMΔ yM − 1Δ from 2.15,
E
2
sup ys − y s ≤ 2E sup
0≤s≤T
0≤s≤T
2E
≤ 2E
2
sup ys θ − yks kθ Δ
−τ≤θ≤0
sup
0≤ks ≤M−1
sup
−τ≤θ≤0,0≤s≤T
2
sup yks kθ 1Δ − yks kθ Δ
−N≤kθ ≤0
ys θ − yks kθ Δ2
2γΔ.
3.34
Therefore, it is a key to compute Esup−τ≤θ≤0,0≤s≤T |ys θ − yks kθ Δ|2 . We discuss the
following four possible cases.
Case 1 ks kθ ≥ 0. We again divide this case into three possible subcases according to
ksθ − ks kθ ∈ {0, 1, 2}.
Journal of Applied Mathematics
15
Subcase 1 ksθ − ks kθ 0. From 2.13,
s θ − ksθ Δ y ksθ Δ − yksθ −1Δ , rkΔsθ
ys θ − yks kθ Δ u yksθ −1 Δ Δ
s
θ Δ
− u yksθ −1Δ , rksθ−1 f yv , rv dv
ksθ Δ
s
θ
ksθ Δ
3.35
g y v , rv dwv,
which yields
E
ys θ − yks kθ Δ2
sup
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
≤ 3E
sup
s θ − ksθ Δ Δ
u y
y
−
y
,
r
ksθ −1Δ
ksθ Δ
ksθ −1Δ ksθ
Δ
−τ≤θ≤0,0≤s≤T,ksθ ≥0
2
−u y ksθ −1Δ , rkΔsθ−1 ⎛
2 ⎞
s
θ sup
3E⎝
f yr , rr dr ⎠
−τ≤θ≤0,0≤s≤T,ksθ ≥0 ksθ Δ
⎛
2 ⎞
s
θ sup
3E⎝
g y r , rr dwr ⎠.
−τ≤θ≤0,0≤s≤T,ksθ ≥0 ksθ Δ
3.36
From Assumption 2.4, 3.24, 3.26, and Lemma 3.2, we have
E
2
Δ
s
θ−k
sθ
Δ
Δ
u y
−u yksθ −1Δ , rksθ−1 yksθ Δ −yksθ −1Δ , rksθ
ksθ −1Δ Δ
sup
−τ≤θ≤0,0≤s≤T,ksθ ≥0
2
s θ − ksθ Δ 8κ2 H2
sup
yksθ Δ − y ksθ −1Δ ≤ 2κ E
Δ
−τ≤θ≤0,0≤s≤T,ksθ ≥0
2
2
8κ2 H2
≤ 2κ E
sup
y ksθ Δ − y ksθ −1Δ 2
−τ≤θ≤0,0≤s≤T,ksθ ≥0
≤ 2κ E
2
sup
2
yksθ Δ − y ksθ −1Δ 0≤ksθ ≤M−1
8κ2 H2
≤ 2κ2 γΔ 8κ2 H2.
3.37
16
Journal of Applied Mathematics
By the Hölder inequality, Assumption 2.3, and Lemma 3.1,
2 ⎞
s
θ f y r , rr dr ⎠
E⎝
sup
−τ≤θ≤0,0≤s≤T,ksθ ≥0 ksθ Δ
⎛
≤ ΔE
s
θ
sup
−τ≤θ≤0,0≤s≤T,ksθ ≥0
ksθ Δ
≤ KΔE
f y , rr 2 dr
r
sup
s
θ −τ≤θ≤0,0≤s≤T,ksθ ≥0
ksθ Δ
2 1 yr dr
T 2
1 sup yr
dr
≤ KΔE
0
≤ KΔ
0≤r≤T
1
E
T
T
sup yt|
2
−τ≤t≤T
0
≤ KΔ
3.38
dr
1 H2dr
0
≤ KT 1 H2Δ.
Setting v s θ and kv ksθ and applying the Hölder inequality yield
2 ⎞
s
θ g y r , rr dwr ⎠
E⎝
sup
−τ≤θ≤0,0≤s≤T,ksθ ≥0 ksθ Δ
⎛
E
2
g ykv Δ , rv wv − wkv Δ
sup
0≤v≤T,0≤kv ≤M−1
≤ E
2l/l−1
g y kv Δ , rv sup
l−1/l
0≤v≤T,0≤kv ≤M−1
1/l
× E
|wv − wkv Δ|
sup
2l
.
0≤v≤T,0≤kv ≤M−1
The Doob martingale inequality gives
E
sup
|wv − wkv Δ|
0≤v≤T,0≤kv ≤M−1
E
2l
sup
sup
0≤kv ≤M−1
kv Δ≤v≤kv 1Δ
|wv − wkv Δ|
2l
3.39
Journal of Applied Mathematics
M−1
≤E
2l
sup
kv Δ≤v≤kv 1Δ
kv 0
17
|wv − wkv Δ|
M−1
E
sup
2l
|wv − wkv Δ|
kv 0
kv Δ≤v≤kv 1Δ
2l M−1
E|wkv 1Δ − wkv Δ|2l .
≤
2l
2l − 1
kv 0
3.40
By 3.27, we therefore have
⎛
2 ⎞
s
θ sup
E⎝
g yr , rr dwr ⎠
k
Δ
−τ≤θ≤0,0≤s≤T,ksθ ≥0
sθ
≤
2l
2l − 1
l−1/l
2 2l/l−1
E
sup g ykv Δ , rv 0≤kv ≤M−1
1/l
×
M−1
3.41
E|wkv 1Δ − wkv Δ|2l
kv 0
≤
2l
2l − 1
2
DlΔl−1/l .
Substituting 3.37, 3.38, and 3.41 into 3.36 and noting Δ ∈ 0, 1 give
E
sup
ys θ − yks kθ Δ2
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
≤ 6κ γΔ 24κ H2 3 KT 1 H2 2
2
2l
2l − 1
2
Dl Δl−1/l
3.42
: 6κ2 γΔ 24κ2 H2 c1 lΔl−1/l .
Subcase2 ksθ − ks kθ 1. From 2.13,
ys θ − yks kθ Δ yksθ Δ − yks kθ Δ ys θ − yksθ Δ
≤ u y ks kθ Δ , rkΔs kθ − u y ks kθ −1Δ , rkΔs kθ −1 f y ks kθ Δ , rkΔs kθ Δ
g y ks kθ Δ , rkΔs kθ Δwks kθ ys θ − yksθ Δ,
3.43
18
Journal of Applied Mathematics
so we have
E
ys θ − yks kθ Δ2
sup
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
( ≤4 E
2
u y ks kθ Δ , rkΔs kθ − u y ks kθ −1Δ , rkΔs kθ −1 sup
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
E
2
f y ks kθ Δ , rkΔs kθ Δ
sup
E
3.44
2
g yks kθ Δ , rkΔs kθ Δwks kθ sup
E
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
ys θ − yksθ Δ2
sup
)
.
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
Since
E
2
u yks kθ Δ , rkΔs kθ − u yks kθ −1Δ , rkΔs kθ −1 sup
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
≤ 2κ2 γΔ 8κ2 H2,
3.45
from 3.26, 3.27, and the Subcase 1, noting Δ ∈ 0, 1, we have
E
sup
ys θ − yks kθ Δ2
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
*
≤ 4 2κ2 γΔ 8κ2 H2 K1 H2Δ2 DlΔl−1/l
+
3 2κ2 γΔ 8κ2 H2 Δ c1 lΔl−1/l
3.46
: 32κ2 γΔ 128κ2 H2 c2 lΔl−1/l .
Subcase 3 ksθ − ks kθ 2. From 2.13, we have
ys θ − yks kθ Δ ys θ − yksθ − 1Δ yksθ − 1Δ − yks kθ Δ,
3.47
Journal of Applied Mathematics
19
so from the Subcase 2, we have
E
ys θ − yks kθ Δ2
sup
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
≤ 2E
ys θ − yksθ − 1Δ2
sup
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
2E
sup
yksθ − 1Δ − yks kθ Δ2
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
≤ 2 32κ2 γΔ 128κ2 H2 c2 lΔl−1/l
3.48
!
2 2κ2 γΔ 8κ2 H2 K1 H2Δ2 DlΔl−1/l
!
: 68κ2 γΔ 272κ2 H2 c3 lΔl−1/l .
From these three subcases, we have
E
ys θ − yks kθ Δ2
sup
−τ≤θ≤0,0≤s≤T,ks kθ ≥0
3.49
≤ 106κ2 γΔ 424κ2 H2 c1 l c2 l c3 lΔl−1/l .
Case 2 ks kθ −1 and 0 ≤ s θ ≤ Δ. In this case, applying Assumption 2.5 and case 1, we
have
E
sup
−τ≤θ≤0,0≤s≤T
ys θ − yks kθ Δ2
≤ 2E
sup
−τ≤θ≤0,0≤s≤T
ys θ − y02
2
2Ey0 − y−Δ
3.50
≤ 212κ2 γΔ 848κ2 H2 2c1 l c2 l c3 lΔl−1/l 2αΔ.
Case 3 ks kθ −1 and −Δ ≤ s θ ≤ 0. In this case, using Assumption 2.5,
E
sup
−τ≤θ≤0,0≤s≤T
ys θ − yks kθ Δ2
≤ αΔ.
3.51
20
Journal of Applied Mathematics
Case 4 ks kθ ≤ −2. In this case, s θ ≤ 0, so using Assumption 2.5,
E
sup
−τ≤θ≤0,0≤s≤T
ys θ − yks kθ Δ2
≤ α2Δ.
3.52
Substituting these four cases into 3.34 and noting the expression of γΔ, there exist c2 , c2 ,
and c2 l such that
E
2
sup ys − y s 0≤s≤T
≤ c2 α2Δ c2 c2 lΔl−1/l ,
3.53
This proof is complete.
Remark 3.4. It should be pointed out that much simpler proofs of Lemmas 3.2 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 βΔ and γΔ by Δl−1/l in Section 3, which will be used
to show the order of the strong convergence.
Lemma 3.5. If Assumption 2.4 holds,
E
2
sup u yt , rt − u yt , rt ≤ 8κ2 H2LΔ : ζΔ
3.54
0≤t≤T
where L is a positive constant independent of Δ.
Proof. Let n 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
E
2
sup u y t , rt − u y t , rt 0≤t≤T
≤ max E
0≤k≤n
2
sup u y s , rs − u ys , rs s∈tk ,tk
1 ≤ 2max E
0≤k≤n
2
sup u ys , rs − u y s , rs 1{rs / rtk }
s∈tk ,tk
1 ≤ 4max E
sup
0≤k≤n
s∈tk ,tk
1 ≤ 8κ max E
2
0≤k≤n
2 |uys , rs| u ys , rs 1{rs / rtk }
2
sup y t 0≤t≤T
2
E 1{rs / rtk }
Journal of Applied Mathematics
21
≤ 8κ2 H2E 1{rs / rtk }
8κ2 H2E E 1{rs / rtk } | tk ,
3.55
where in the last step we use the fact that ytk and 1{rs / rtk } are conditionally independent
with respect to the σ− algebra generated by rtk . But, by the Markov property,
1{rtk i} P rs /
i | rtk i
E 1{rs / rtk } | rtk i∈S
1{rtk i}
≤
γij s − tk os − tk j/
i
i∈S
3.56
1{rtk i} max −γij Δ oΔ
1≤i≤N
i∈S
≤ LΔ,
where L is a positive constant independent of Δ. Then
E
2
sup u y t , rt − u y t , rt ≤ 8κ2 H2LΔ.
3.57
0≤t≤T
This proof is complete.
Lemma 3.6. If Assumption 2.3 holds, there is a constant C, which is independent of Δ such that
E
T
0
E
T
0
f y , rs − f y , rs 2 ds ≤ CΔ,
s
s
3.58
g y , rs − g y , rs 2 ds ≤ CΔ.
s
s
3.59
Proof. Let n T/Δ, the integer part of T/Δ. Then
E
T
0
n
f y , rs − f y , rs 2 ds E
s
s
k0
tk
1 2
f ytk , rs − f ytk , rtk ds,
tk
3.60
22
Journal of Applied Mathematics
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
tk
1 2
E
f ytk , rs − f ytk , rtk ds
tk
≤ 2E
tk
1 "
tk
≤ CE
2 #
|fytk , rs|2 f ytk , rtk 1{rs / rtk } ds
tk
1 tk
2 1 y tk 1{rs / rtk } ds
3.61
tk
1 " "
##
2 ≤C
E E 1 ytk 1{rs / rtk } | rtk ds
tk
tk
1 " "
#
#
2 ≤C
E E 1 ytk | rtk E 1{rs / rtk } | rtk ds,
tk
where in the last step we use the fact that ytk and 1{rs / rtk } are conditionally independent
with respect to the σ− algebra generated by rtk . But, by the Markov property,
E 1{rs / rtk } | rtk 1{rtk i} P rs / i | rtk i
i∈S
1{rtk i}
γij s − tk os − tk i∈S
j/
i
≤
1{rtk i} max −γij Δ oΔ
i∈S
3.62
1≤i≤N
≤ CΔ.
So, by Lemma 3.1,
E
tk
1 tk
1 2 2
1 Ey tk ds
f y tk , rs − f y tk , rtk ds ≤ CΔ
tk
tk
3.63
≤ CΔ .
2
Therefore
E
T
0
Similarly, we can show 3.59.
f y , rs − f y , rs 2 ds ≤ CΔ.
s
s
3.64
Journal of Applied Mathematics
23
4. Convergence of the EM Methods
In this section we will use the lemmas above to prove Theorem 2.7. From Lemma 3.1 and
, such that
Theorem 2.6, there exists a positive constant H
E
sup |xt|
p
−τ≤t≤T
∨E
p
sup yt
−τ≤t≤T
,
≤ H.
4.1
Let j be a sufficient large integer. Define the stopping times
vj : inf t ≥ 0 : yt ≥ j ,
uj : inf t ≥ 0 : xt ≥ j ,
ρj : uj ∧ vj ,
4.2
where we set inf ∅ ∞ as usual. Let
et : xt − yt.
4.3
Obviously,
E
sup |et|
2
2
E
E
sup |et| 1{uj >T,vj >T }
0≤t≤T
0≤t≤T
2
sup |et| 1{uj ≤T
or vj ≤T }
4.4
.
0≤t≤T
Recall the following elementary inequality:
aγ b1−γ ≤ γa 1 − γ b,
4.5
∀a, b > 0, γ ∈ 0, 1.
We thus have, for any δ > 0,
E
2
sup |et| 1{uj ≤T or vj ≤T }
⎡
2/p
p
δ−2/p−2 1{uj ≤T
≤ E⎣ δ sup |et|
{0≤t≤T }
0≤t≤T
≤
2δ
E
p
sup |et|p
0≤t≤T
p−2/p
or vj ≤T }
⎤
⎦
4.6
p−2
2/p−2 P uj ≤ T or vj ≤ T .
pδ
Hence
E
sup |et|
2
≤E
0≤t≤T
2
sup |et| 1{ρj >T }
0≤t≤T
P uj ≤ T or vj ≤ T .
2/p−2
p−2
pδ
2δ
p
E sup |et|
p
0≤t≤T
4.7
24
Journal of Applied Mathematics
Now,
xt p
P uj ≤ T ≤ E 1{uj ≤T } p
j
1
p
≤ p E sup |xt|
j
−τ≤t≤T
4.8
,
H
.
jp
≤
Similarly,
,
H
P vj ≤ T ≤ p .
j
4.9
Thus
P vj ≤ T or uj ≤ T ≤ P vj ≤ T P uj ≤ T
4.10
,
2H
.
jp
≤
We also have
E
sup |et|
p
≤2
p−1
E
0≤t≤T
p sup |xt | yt p
4.11
0≤t≤T
p,
≤ 2 H.
Moreover,
E
2
sup |et| 1{ρj >T }
2
sup e t ∧ ρj 1{ρ >T }
E
0≤t≤T
j
0≤t≤T
≤E
2
.
sup e t ∧ ρj
4.12
0≤t≤T
Using these bounds in 4.7 yields
E
sup |et|
0≤t≤T
2
≤E
2
sup e t ∧ ρj 0≤t≤T
,
, 2 p−2 H
2p
1 δH
2/p−2 p .
p
pδ
j
4.13
Journal of Applied Mathematics
25
Setting v : t ∧ ρj and for any ε ∈ 0, 1, by the Hölder inequality, when v ∈ kΔ, k 1Δ,
for k 0, 1, 2, . . . , M − 1,
2
|ev|2 xv − yv
v − kΔ ≤ uxv , rv − u y k−1Δ y kΔ − yk−1Δ , rkΔ
Δ
v
v
2
fxs , rs − f ys , rs ds gxs , rs − g y s , rs dws
0
0
2
1 v − kΔ ≤ uxv , rv − u y k−1Δ y kΔ − y k−1Δ , rkΔ ε
Δ
v
v
2 2
2
T
fxs , rs−f y s , rs ds
gxs , rs−g y s , rs dws .
1−ε
0
0
4.14
Assumption 2.4 yields
2
v − kΔ Δ uxv , rv − u y
y
−
y
k−1Δ
kΔ
k−1Δ , rk Δ
2
v − kΔ 2
≤ 2κ xv − y k−1Δ −
y kΔ − y k−1Δ Δ
v − kΔ 2u y k−1Δ ykΔ − y k−1Δ , rv
Δ
2
v − kΔ −u y k−1Δ y kΔ − y k−1Δ , rkΔ Δ
2
v − kΔ xv − yv yv − y y − y
−
≤ 2κ2 −
y
y kΔ
v
kΔ
k−1Δ k−1Δ Δ
v − kΔ 2u y k−1Δ ykΔ − y k−1Δ , rv
Δ
2
v − kΔ −u yk−1Δ y kΔ − yk−1Δ , rkΔ Δ
2 2 2κ2 yv − y 2 xv − yv 2 4κ
≤
−
y
y
v
kΔ
k−1Δ ε
1−ε
v − kΔ 2u y k−1Δ ykΔ − y k−1Δ , rv
Δ
2
v − kΔ
Δ −uyk−1Δ ykΔ − yk−1Δ , rk .
Δ
4.15
26
Journal of Applied Mathematics
Then, we have
2 2
2 2κ2 4κ2
yv − y v y kΔ − y k−1Δ |ev| ≤ 2 xv − yv ε1
−
ε
ε
v − kΔ 2u yk−1Δ y kΔ − yk−1Δ , rv
Δ
2
v − kΔ −u y k−1Δ y kΔ − y k−1Δ , rkΔ Δ
v
v
2 2
2
T
fxs , rs−f y s , rs ds
gxs , rs−g ys , rs dws .
1−ε
0
0
4.16
2
Hence, for any t1 ∈ 0, T , by Lemmas 3.2–3.5,
2
2
2κ2
≤ 2 E sup xt∧ρj − yt∧ρj E sup e t ∧ ρj
ε
0≤t≤t1
0≤t≤t1
2
4κ2
E sup yt∧ρj − y t∧ρj ε1 − ε
0≤t≤t1
2
E
sup ykΔ∧ρj − yk−1Δ∧ρj 0≤k≤M−1
kΔ ∧ ρj − kΔ y kΔ −y k−1Δ , r kΔ ∧ ρj
2E sup u y k−1Δ Δ
0≤k≤M−1
2 ⎤
kΔ ∧ ρj − kΔ y kΔ − y k−1Δ , r kΔ ∧ ρj ⎦
−u yk−1Δ Δ
t1 ∧ρj
2
2T
E
fxs , rs − f y s , rs ds
1−ε
0
⎡
2 ⎤
t∧ρj 2
E⎣ sup gxs , rs − g y s , rs dws ⎦
1−ε
0≤t≤t1 0
2
2κ2
4κ2 ≤ 2 E sup xt∧ρj − yt∧ρj γΔ βΔ 2ζΔ
ε1
−
ε
ε
0≤t≤t1
t1 ∧ρj
2
2T
E
fxs , rs − f y s , rs ds
1−ε
0
⎡
2 ⎤
t∧ρj 2
E⎣ sup gxs , rs − g y s , rs dws ⎦.
1−ε
0≤t≤t1 0
4.17
Journal of Applied Mathematics
27
Since xt yt ξt when t ∈ −τ, 0, we have
E
2
sup xt∧ρj − yt∧ρj ≤E
0≤t≤t1
2
sup sup x t ∧ ρj θ − y t ∧ ρj θ −τ≤θ≤0 0≤t≤t1
≤E
−τ≤t≤t1
E
2
sup x t ∧ ρj − y t ∧ ρj 4.18
2
sup x t ∧ ρj − y t ∧ ρj .
0≤t≤t1
By Assumption 2.2, Lemma 3.3, and Lemma 3.6, we may compute
E
t1 ∧ρj
0
fxs , rs − f y , rs 2 ds
s
≤E
t1 ∧ρj
0
≤ 2Cj E
fxs , rs − f y , rs f y , rs − f y , rs 2 ds
s
s
s
t1 ∧ρj
0
≤ 4Cj E
t1 ∧ρj
xs − ys ys − y 2 ds 2CΔ
s
xs − ys 2 ds 4Cj E
0
≤ 4Cj E
t1 ∧ρj
0
≤ 4Cj E
t1
t1 ∧ρj
0
ys − y 2 ds 2CΔ
s
2
sup xs θ − ys θ ds 4Cj T βΔ 2CΔ
−τ≤θ≤0
2
sup x s ∧ ρj θ − y s ∧ ρj θ ds
0 −τ≤θ≤0
4Cj T βΔ 2CΔ
t1
2
sup x r ∧ ρj − y r ∧ ρj ds 4Cj T βΔ 2CΔ
≤ 4Cj E
0 −τ≤r≤s
4Cj
t1
0
2
E sup x r ∧ ρj − y r ∧ ρj ds 4Cj T βΔ 2CΔ.
0≤r≤s
4.19
By the Doob martingale inequality, Lemma 3.3, Lemma 3.6, and Assumption 2.2, we compute
⎡
2 ⎤
t∧ρj E⎣ sup gxs , rs − g y s , rs dws ⎦
0≤t≤t1 0
⎡
2 ⎤
t∧ρj E⎣ sup gxs , rs − g y s , rs g ys , rs − g y s , rs dws ⎦
0≤t≤t1 0
28
Journal of Applied Mathematics
≤ 8Cj E
t1 ∧ρj
0
xs − y 2 ds 8CΔ
s
t1 2
≤ 16Cj
E sup x r ∧ ρj − y r ∧ ρj ds 16Cj T βΔ 8CΔ.
0≤r≤s
0
4.20
Therefore, 4.17 can be written as
2κ2
1− 2
ε
2
E sup e t ∧ ρj 0≤t≤t1
≤
8Cj T T 4
4κ2 4CΔT 8
βΔ γΔ βΔ ε1 − ε
1−ε
1−ε
8Cj T 4
2ζΔ 1−ε
t1 2
ds.
E sup e v ∧ ρj
0
0≤v≤s
4.21
Choosing ε 1 E
√
2κ/2 and noting κ ∈ 0, 1, we have
2
sup e t ∧ ρj 0≤t≤t1
√ √ 2
T
1
2κ T 4
16κ2 1 2κ
16C
j
≤
βΔ
γΔ
2
2
√
√
√
√ βΔ
1 − 2κ
1 3 2κ
1 − 2κ
1 3 2κ
√ 2
√ 2
8CΔT 8 1 2κ
2 1 2κ
√ 2 √ √ 2 √ ζΔ
1 − 2κ
1 3 2κ
1 − 2κ
1 3 2κ
2
16Cj 1 2κ T 4 t1
2
E sup e s ∧ ρj dv
√ 2 √ 0≤s≤v
1 − 2κ
1 3 2κ 0
≤
√
16Cj T T 4
16
8CΔT 8
√ 2 βΔ γΔ √ 2 βΔ √ 2
1 − 2κ
1 − 2κ
1 − 2κ
1−
2
√
16Cj T 4
ζΔ √ 2
2κ
1 − 2κ
t1 2
E sup e s ∧ ρj dv.
0
0≤s≤v
4.22
Journal of Applied Mathematics
29
By the Gronwall inequality, we have
⎡
2
⎢
E sup e t ∧ ρj ≤ ⎣
0≤t≤t1
16Cj T T 4
16
√ 2 βΔ γΔ √ 2 βΔ
1 − 2κ
1 − 2κ
4.23
⎤
√
8CΔT 8
2ζΔ
⎥
16/1− 2κ2 Cj T T 4
.
√ ⎦ × e
√ 2 1 − 2κ
1 − 2κ
By 4.13,
sup |et|2
E
0≤t≤T
⎡
16Cj T T 4
16
8CΔT 8
√ 2 βΔ γΔ √ 2 βΔ √ 2
1 − 2κ
1 − 2κ
1 − 2κ
⎢
≤ ⎣
⎤
,
2 p−2 H
4.24
√
,
2ζΔ
2p
1 δH
⎥
16/1− 2κ2 Cj T T 4
2/p−2 p .
√ ⎦ × e
p
pδ
j
1 − 2κ
, ≤ /3, then choose
Given any > 0, we can now choose δ sufficient small such that 2p
1 δH/p
j sufficient large such that
,
2 p−2 H
pδ2/p−2 j p
<
,
3
4.25
and finally choose Δ so small such that
⎡
⎤
T
4
16C
T
16
8CΔT
8
2ζΔ
j
⎢
⎥
⎣
√ ⎦
√ 2 βΔ γΔ √ 2 βΔ √ 2 1 − 2κ
1 − 2κ
1 − 2κ
1 − 2κ
×e
√ 2
16/1− 2κ Cj T T 4
<
4.26
3
and thus Esup0≤t≤T |et|2 ≤ as required.
Remark 4.1. Obviously, according to Theorem 2.7, 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.
30
Journal of Applied Mathematics
Assumption 5.1. There exists a constant Q such that for all ϕ, ψ ∈ C−τ, 0; Rn , i ∈ S, and
t ∈ 0, T ,
f ϕ, i − f ψ, i 2 ∨ g ϕ, i − g ψ, i 2 ≤ Qϕ − ψ 2 .
5.1
It is easy to see from the global Lipschitz condition that, for any ϕ ∈ C−τ, 0; Rn ,
2 2
f ϕ, i ∨ g ψ, i ≤ 2 f0, i2 ∨ g0, i2 Qϕ2 .
5.2
In other words, the global Lipschitz condition implies linear growth condition with the
growth coefficient
!
2
2 K 2 f0, i ∨ g0, i ∨ Q .
5.3
p
Assumption 5.2. ξ ∈ LF0 −τ, 0; Rn for some p ≥ 2, and there exists a positive constant λ such
that
E
sup |ξt − ξs|
2
−τ≤s≤t≤0
≤ λt − s.
5.4
We can state another theorem, which reveals the order of the convergence.
Theorem 5.3. If Assumptions 5.1, 5.2, and 2.4 hold, for any positive constant l > 1,
E
2
sup xt − yt
≤ O Δ1−1/l .
5.5
0≤t≤T
Proof. Since αΔ may be replaced by λΔ, from Lemmas 3.2 and 3.3, there exist constants c1 l
and c2 l such that βΔ ≤ c1 lΔl−1/l and γΔ ≤ c2 lΔl−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
E
sup |et|2
0≤t≤T
⎡
⎤
16
2ζ
16QT T 4
8CT 8
⎢
⎥
≤ ⎣
c1 l
c2 l
√ ⎦
√ 2 √ 2 c1 l
√ 2 1 − 2κ
1 − 2κ
1 − 2κ
1 − 2κ
√ 2
2κ QT T 4
× e16/1−
≤ O Δl−1/l .
Δl−1/l
5.6
The proof is complete.
Journal of Applied Mathematics
31
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 to 1, 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.
Acknowledgments
the work was supported by the Fundamental Research Funds for the Central Universities
under Grant 2012089, China Postdoctoral Science Foundation funded project under Grant
2012M511615, the Research Fund for Wuhan Polytechnic University, and the State Key
Program of National Natural Science of China Grant no. 61134012.
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