\(L^{p }\) (\(p>2\)) strong convergence of multiscale integration scheme for jump diffusion systems

(1)

R E S E A R C H

Open Access

L

p

(

p

> 2

)-strong convergence of

multiscale integration scheme for

jump-diffusion systems

Jiaping Wen

1*

*_{Correspondence:} [email protected] 1_{College of Mathematics and} Information Science, Henan Normal University, Xinxiang, P.R. China

Abstract

In this paper we shall prove theLp_(p_{> 2)-strong convergence of multiscale}

integration scheme for stochastic jump-diﬀusion systems with two-time-scale, which gives a numerical method for eﬀective dynamical systems.

MSC: 60H10; 34F05

Keywords: Fast-slow SDEs with jumps;Lp_(p_{> 2)-strong convergence; Multiscale} integration scheme

1 Introduction

This paper focuses on the following two-time-scale jump-diﬀusion SDEs:

⎧ ⎨ ⎩

dx

t =a(xt,yt)dt+b(xt)dBt+c(xt)dPt, x0=x0,

dy t =1f(x

t,yt)dt+√1g(x

t,yt)dWt+h(xt,yt)dNt, y0=y0,

(1)

wherex

t ∈Rnandyt ∈Rmare jump-diﬀusion processes. The functionsa(x,y)∈Rnand

f(x,y)∈Rm_{are the drift coefficients, the functions}_b₍_x₎_∈_Rn×d1 _and_g₍_x_,_y₎_∈_Rm×d2 _are the diffusion coefficients, and the functionsc(x)∈Rn_and_h₍_x_,_y₎_∈_Rm _{are the jump} co-efficients;BtandWtared1,d2-dimensional independent Wiener processes,Ptis a scalar simple Poisson process with intensityλ1, andNt is a scalar simple Poisson process with intensityλ2

.is a small parameter, which represents the ratio of time scale between the processesx

t andyt. With this time scale, the vectorxt is referred to as the “slow compo-nent” andy

t as the “fast component”. Under suitable assumptions the authors [1,2] proved that when→0, the slow componentx

t mean square converges to the solution of SDEs in the following form:

⎧ ⎨ ⎩

dx¯t=a¯(x¯t)dt+b(x¯t)dBt+c(x¯t)dPt,

¯

x0=x0,

(2)

with

¯

a(x) =

Rm

a(x,y)μx(dy), x∈Rn,

(2)

andμx_{is the invariant, ergodic measure generated by the following equation with frozen}_x_:

dyt=f(x,yt)dt+g(x,yt)dWt+h(x,yt)dNt, y0=y0.

Multiscale jump-diﬀusion stochastic diﬀerential equations arise in many applications and have already been studied widely. What is usually of interest for this kind of system (1) is the time evolution of the slow variablex

t. Thus a simplified equation, which is in-dependent of the fast variable and possesses the essential features of the system, is highly desirable. On the one hand, while averaging principle [1–6] plays an important role in the research of slow component by getting a reduced equation (2), the difficulty of obtaining the effective equation (2) lies in the fact that the coefficienta¯(·) is given via expectation with respect to measureμx₍_dy_{), which is usually difficult or impossible to obtain} analyt-ically, especially when the dimension mis large. On the other hand, even if we get the reduced equation, the equation cannot be solved explicitly. Therefore, the construction of the efficient computational methods is of great importance. Furthermore, the idea of multiscale integration schemes (cf. [7]) overcomes these difficulties exactly, which solves

¯

xt witha¯(·) being estimated on the ﬂy using an empirical average of the original slow co-eﬃcientsa(·) with respect to numerical solutions of the fast processes. This is one of our motivations.

For another signiﬁcant motivation, a substantial body of work has been done concerning multiscale integration scheme for fast-slow SDEs. Most of the existing research theories discuss the convergence inLp_{(0 <}_p_≤_{2), even in a weaker sense [4,}_5,_{8–10]. Nevertheless,} convergence in a stronger sense is what we want. In 2007, theL2_{averaging principle was} proposed for a system, in which slow and fast dynamics were driven by Brownian noises and Poisson noises in [1]. Subsequently, the authors gave a multiscale integration scheme for the result in [9]. In 2015, Xu and Miao extended the result of [1] to theLp₍_p_{> 2) case} under assumptions (H1)–(H5) in [2]. A natural question is as follows: Can we also establish theLp₍_p_{> 2) averaging principle by the multiscale integration scheme? It is well known} thatL1_{convergence and}_L2 _{convergence cannot conclude}_Lp₍_p_{> 2) convergence.} How-ever,L1_{convergence and}_L2_{convergence can be deduced by}_Lp₍_p_{> 2) convergence. Once} theLp ₍_p_{> 2) convergence has been established, then a much bigger degree of freedom} for parameterqin research ofLq_{(0 <}_q_<_p_{) convergence would be obtained.}

Based on the above discussion, the aim of this paper is to prove theLp ₍_p_{> 2)} conver-gence of the multiscale integration scheme under the following assumptions:

(H1) The measurable functionsa,b,c,f,g, andhsatisfy the global Lipschitz conditions, i.e., there is a positive constantLsuch that

a(u1,v1) –a(u2,v2) 2

+b(u1) –b(u2) 2

+c(u1) –c(u2) 2

+f(u1,v1) –f(u2,v2) 2

+g(u1,v1) –g(u2,v2) 2

+h(u1,v1) –h(u2,v2) 2

≤L|u1–u2|2+|v1–v2|2

for allui∈Rn,vi∈Rm,i= 1, 2. Here and below we use| · |to denote both the

(3)

Remark1.1 With the help of (H1), it immediately follows that there is a positive constant K such that

a(u,v)2+b(u)2+c(u)2+f(u,v)2+g(u,v)2+h(u,v)2

≤K1 +|u|2+|v|2

for (u,v)∈Rn_×_Rm_{. Thus}_a_,_b_,_c_,_f_,_g_{, and}_h_{satisfy the sublinear growth condition.}

(H2) a,g, andhare globally bounded.

Remark1.2 By (H1) and (H2), it is easy to derive thata¯in (2) is bounded and satisﬁes the Lipschitz condition [11].

(H3) There exist constantsβ1> 0and βj∈R,j= 2, 3, 4, which are all independent of (u1,v1,v2), such that

v1·f(u1,v1)≤–β1|v1|2+β2,

f(u1,v1) –f(u1,v2)

(v1–v2)≤β3|v1–v2|2,

(3)

and

h(u1,v1) –h(u1,v2)

(v1–v2)≤β4|v1–v2|2 (4)

for allu1∈Rnandv1,v2∈Rm.

(H4) η:= –(2β3+ 2λ2β4+Cg+λ2Ch) > 0, hereβ3andβ4are taken from (3) and (4),λ2is

fromN

t with intensityλ2/,Cg, andChare the Lipschitz coeﬃcients forgandh,

respectively, i.e.,

g(u1,v1) –g(u2,v2) 2

≤Cg

|u1–u2|2+|v1–v2|2

and

h(u1,v1) –h(u2,v2) 2

≤Ch

|u1–u2|2+|v1–v2|2

for allu1,u2∈Rn,v1,v2∈Rm.

(H5) There exists a constantγ > 0, which is independent of(u,v), such that

vTg(u,v)gT(u,v)v≥γ|v|2

for all(u,v)∈Rn_×_Rm_.

An example that satisﬁes (H1)–(H5) isa(u,v) = _1+(u+v)1 2,b(u) =e–u 2

,c(u) =sinu,f(u,v) = –1.5(λ2+ 1)ν,g(u,ν) = 3+sin√u+₂sinν, andh(u,ν) =sinu+√₂sinν.

It is worth pointing out that theLp₍_p_{> 2) averaging principle under assumptions (H1)–} (H5) had been established in [2].

(4)

1.Macro solver. Lettbe a ﬁxed step, and letXnbe a numerical approximation to the coarse variablex¯at timetn=nt. The simplest choice is the Euler–Maruyama scheme

Xn+1=Xn+A(Xn)t+b(Xn)Bn+c(Xn)Pn, X0=x0, (5)

whereA(Xn) is estimated by an empirical average

A(Xn) = 1

M

m=1

aXn,Ymn

. (6)

2.Micro solver. To getA(Xn) used in the macro solver, we adopt the Euler–Maruyama scheme to generateYn

m:

Y_m+1n =Y_mn+1 f

Xn,Ymn

δt+√1 g

Xn,Ymn

W_mn+hXn,Ymn

N_mn, (7)

with ﬁxedXn, and we denote the solution byYmn,m= 0, 1, . . . ,M, whereWmnare Brownian increments over a time intervalδt, andNn

mare Poisson increments with intensity λ2

. Due to ergodicity [2] of the fast dynamics, we can select, among other selections,Yn

0 =y0. Note that the eﬀective dynamics do not rely on. Meanwhile, since the discrete solution

Yn

m obtained by the micro-solver is for Xn ﬁxed, it only depends on the ratio δt. Thus, without loss of generality, we may take= 1. Then we have

Y_m+1n =Y_mn+fXn,Ymn

δt+gXn,Ymn

W_mn+hXn,Ymn

N_mn, (8)

whereWn

m=W(m+1)n δt–Wmnδt are the Brownian increments, andNmn=N(m+1)n δt–Nmnδt are the Poisson increments with intensityλ2.

Simultaneously,Y_mn are numerically generated discrete solutions of the family of SDEs as well:

dzn_t =fXn,znt

dt+gXn,ztn

dW_tn+hXn,ztn

dN_tn, (9)

with initial conditionszn₀=Y₀n=y0and a time stepδt(the choice of a ﬁxedY0nfor all n simpliﬁes our estimates; in practice, we could takeYn

0 =YMn–1for alln> 0).

We also present a discrete auxiliary processX¯n, the Euler solution to the eﬀective dy-namics (2):

¯

Xn+1=X¯n+a¯(X¯n)t+b(X¯n)Bn+c(X¯n)Pn. (10)

(5)

¯

Xn(see Lemma2.4below); (II) the diﬀerence between the processXnand the auxiliary processX¯n(see Lemma3.8below).

We now describe the structure of the present paper. In Sect.2, we introduce some a priori estimates to testify the error between the processx¯tnand the auxiliary processX¯n. In

Sect.3, we devote ourselves to proving the error between the processXnand the auxiliary processX¯n. In Sect.4, based on the above two estimates, we can derive our main result (see Theorem4.1).

Throughout this paper, we will denote byCorKa generic positive constant which may change its value from line to line. In chains of inequalities, we will adoptC,C,C, . . . or

C1,C2,K1,K2, . . . to avoid confusion.

2 Some a priori estimates

In this section, we shall give some a priori estimates in the ﬁrst three lemmas. Then we can apply the obtained results to estimate the diﬀerence between the processx¯tnand the

auxiliary processX¯n.

For convenience, we will extend the discrete numerical solutionX¯nof (10) to continuous time. We ﬁrst deﬁne the ‘step functions’

Z(t) = k

¯

Xk1[kt,(k+1)t)(t), (11)

where 1Gis the indicator function for the setG. Then we deﬁne

¯

X(t) =x0+

t 0

¯

aZ(s)ds+

t 0

bZ(s)dBs+

t 0

cZ(s)dPs. (12)

(Note that by construction Z(t–) =Z(t) fort=kt.) It is not diﬃcult to verifyZ(tk) =

¯

X(tk) =X¯k. The aim of this section is to prove a convergence result forX¯(t) because the discrete numerical solution is interpolated toX¯(t). Then, we can obtain the convergence result forX¯kstraightly.

Firstly, we show that the discrete numerical solutionX¯kand the continuous approxima-tionX¯(t) have 2pbounded moments in the ﬁrst two lemmas.

Lemma 2.1 For any p> 1and T> 0,there exist positive constantst∗and C1such that,

for all0 <t≤t∗,

E| ¯Xk|2p≤C1

1 +|x0|2p

(13)

for kt≤T,where C1is independent of(k,t).

Proof By construction (12), we have

¯

Xk+1=x0+

(k+1)t 0

¯

aZ(s)ds+

(k+1)t 0

bZ(s)dBs+

(k+1)t 0

(6)

Then we obtain

E| ¯Xk+1|2p ≤C|x0|2p+CE

₀(k+1)ta¯Z(s)ds

2p +CE

(k+1)t 0

bZ(s)dBs

2p

+CE

(k+1)t 0

cZ(s)dPs

2p

:=C|x0|2p+I1+I2+I3

(14)

for (k+ 1)t≤T. ForI2andI3, usingP˜t:=Pt–λ1t, Burkhölder’s inequality [12], Hölder’s inequality, Remark1.1, and (11), we have

I2≤CE

(k+1)t 0

bZ(s)2ds

p

≤C

(k+1)t 0

EbZ(s)2pds

≤C

(k+1)t 0

E1 +Z(s)2pds

≤C+Ct

k

i=0

E| ¯Xi|2p (15)

and

I3 =E

(k+1)t

0

cZ(s)dP˜s+λ1

(k+1)t 0

cZ(s)ds

2p

≤CE

(k+1)t 0

cZ(s)dP˜s

2p+CEλ1

(k+1)t 0

cZ(s)ds

2p

≤CE (k+1)t

0

cZ(s)2ds

p

+CEλ1

(k+1)t 0

cZ(s)ds

2p

≤C

(k+1)t 0

EcZ(s)2pds+C

(k+1)t 0

EcZ(s)2pds

≤C

(k+1)t 0

E1 +Z(s)2pds

≤C+Ct

k

i=0

E| ¯Xi|2p. (16)

Similarly, we may deal withI1and have

I1≤C+Ct k

i=0

E| ¯Xi|2p. (17)

Choosingtsuﬃciently small and by (14)–(17), we have

E| ¯Xk+1|2p≤C

1 +|x0|2p

+Ct

k

i=0

E| ¯Xi|2p,

(7)

E sup t∈[0,T]

X¯(t)2p≤C2

1 +|x0|2p

, (18)

where C2is independent oft.

Proof From (12), we have

X¯(t)2p≤C|x0|2p+C

₀ta¯Z(s)ds

2p +C

t 0

bZ(s)dBs

2p+C

t 0

cZ(s)dPs

2p.

Thus, by the deﬁnition ofP˜t, we have

E sup t∈[0,T]

X¯(t)2p≤C|x0|2p+CE sup t∈[0,T]

t

0

¯

aZ(s)ds

2p

+CE sup t∈[0,T]

t

0

bZ(s)dBs

2p

+CE sup t∈[0,T]

₀tcZ(s)dP˜s

2p+CE sup t∈[0,T]

λ1

t 0

cZ(s)ds

2p . (19)

By the same method as in the previous lemma, we obtain

E sup t∈[0,T]

X¯(t)2p≤C1 +|x0|2p

+C

T 0

EZ(s)2pds.

Applying (11) and Lemma2.1over the interval [0,T], we obtain result (18).

Secondly, we show that the continuous-time approximation remains close to the step functions Z(s) in a strong sense.

E sup t∈[0,T]

X¯(t) –Z(t)2p≤C3tp

1 +|x0|2p

, (20)

Proof Considert∈[kt, (k+ 1)t]⊆[0,T], we have

¯

X(t) –Z(t) =X¯(t) –X¯k=

t kt

¯

aZ(s)ds+

t kt

bZ(s)dBs+

t kt

cZ(s)dPs.

Thus

X¯(t) –Z(t)2p≤C

t kt

¯

aZ(s)ds

2p +C

t kt

bZ(s)dBs

2p+C

t kt

cZ(s)dPs

(8)

for eacht∈[kt, (k+ 1)t]. Then

sup t∈[0,T]

X¯(t) –Z(t)2p≤ max

k=0,1,...T/t–1_τ_∈_[ksup_t,(k+1)_t]

C

τ kt

¯

aZ(s)ds

2p

+C

τ kt

bZ(s)dBs

2p+C

τ kt

cZ(s)dP˜s

2p

+Cλ1

τ kt

cZ(s)ds

2p

. (21)

Now, taking expectations on both sides of (21), then using Burkhölder’s inequality on the martingale integrals and by Hölder’s inequality, we have

E sup t∈[0,T]

X¯(t) –Z(t)2p≤ max k=0,1,...T/t–1

Ct2p–1

(k+1)t kt

Ea¯Z(s)2pds

+Ctp–1

(k+1)t kt

EbZ(s)2pds

+Ctp–1

(k+1)t kt

EcZ(s)2pds

+Ct2p–1

(k+1)t kt

EcZ(s)2pds

.

Applying Remarks1.1and1.2, we have

E sup t∈[0,T]

X¯(t) –Z(t)2p≤ max k=0,1,...T/t–1

Ctp–1+t2p–1

(k+1)t

kt

1 +EZ(s)2pds

.

ButZ(s)≡ ¯Xkon [kt, (k+ 1)t), hence, it follows from Lemma2.1that

E sup t∈[0,T]

X¯(t) –Z(t)2p≤Ctp+t2p1 +C1

1 +|x0|2p

,

which yields result (20).

Lastly, we prove a strong convergence result forX¯(t).

E sup t∈[0,T]

X¯(t) –x¯(t)2p≤C4tp

1 +|x0|2p

, (22)

Proof By construction (12), we get

¯

X(t) –x¯(t) =

t 0

¯

aZ(s)–a¯x¯(s)ds+

t 0

bZ(s)–bx¯(s)dBs

+

t 0

(9)

Hence, we have

E sup t∈[0,t1]

X¯(t) –x¯(t)2p ≤CE sup t∈[0,t1]

₀ta¯Z(s)–a¯x¯(s)ds

2p

+CE sup t∈[0,t1]

₀tbZ(s)–bx¯(s)dBs

2p

+CE sup t∈[0,t1]

₀tcZ(s)–cx¯(s)dPs

2p

:=C(I1+I2+I3) (23)

for any 0≤t1≤T. By the deﬁnition ofP˜t, Burkhölder’s inequality, Hölder’s inequality, and (H1), we obtain

I2≤CE

t1 0

bZ(s)–bx¯(s)2ds

p

≤C

t1 0

EbZ(s)–bx¯(s)2pds

≤C

t1 0

EZ(s) –x¯(s)2pds, (24)

I3≤CE sup t∈[0,t1]

₀tcZ(s)–cx¯(s)dP˜s

2p

+CE sup t∈[0,t1]

λ1

t 0

cZ(s)–cx¯(s)ds

2p

≤CE

t1 0

cZ(s)–cx¯(s)2ds

p

+CE sup t∈[0,t1]

λ1

t 0

cZ(s)–cx¯(s)ds

2p

≤C

t1 0

EcZ(s)–cx¯(s)2pds

≤C

t1 0

EZ(s) –x¯(s)2pds. (25)

Dealing withI1similarly and combining (23)–(25), it follows that

E sup t∈[0,T]

X¯(t) –x¯(t)2p≤C

t1 0

EZ(s) –x¯(s)2pds

≤C

t1

0 E

X¯(s) –x¯(s)2pds+C

t1

0 E

X¯(s) –Z(s)2pds.

Applying Lemma2.3, we obtain

E sup t∈[0,T]

X¯(t) –x¯(t)2p≤C5tp

1 +|x0|2p

+C6

t1 0

Esup t∈[0,s]

X¯(t) –x¯(t)2pds.

(10)

3 Strong convergence of the scheme

In this section, some a priori estimates would be provided in the ﬁrst seven lemmas. Then we can use the established estimates to get the error between the processXnand the aux-iliary processX¯n.

Now, we ﬁrstly show the 2pth moment estimates for the processeszn_t,Xn, andYmn.

Lemma 3.1 For any p> 1and T> 0,there exists a positive constant K1such that

sup 0≤t≤T

Ezn_t2p≤K1. (26)

Proof For|zn

t|2p, direct computation with Itô’s formula gives that

zn_t2p=|y0|2p+ 2p

t 0

zn_s2p–2fXn,zns

,zn_sds+ 2p

t 0

zn_s2p–2gXn,zsn

,z_sndW_sn

+ 2p

t 0

zn_s2p–2hXn,zns

,zn_sdN_sn+ 2p(p– 1)

t 0

z_sn2(p–2)gXn,zns

,zn_s2ds

+ 2p(p– 1)λ2

t 0

zn_s2(p–2)hXn,zns

,zn_s2ds+p

t 0

zn_s2p–2gXn,zns 2

ds

+pλ2

t 0

zn_s2p–2hXn,zsn 2

ds. (27)

We have by (H3)

fXn,zns

,zn_s≤–β1zns 2

+β2. (28)

By Young’s inequality and (H2), we have

2λ2

hXn,zns

,zn_s≤λ

2 2

β1

hXn,zns 2

+β1zns 2_≤

C+β1zns 2

, (29)

gXn,zns

,zn_s2≤Cz_sn2, (30)

and

hXn,zsn

,z_sn2≤Czn_s2. (31)

Taking expectations on both sides of (27) and combining (28)–(31), we have

Ezn t

2p

≤ |y0|2p–pβ1E

t 0

zn s

2p

ds+Cp,λ2,β1,β2E

t 0

zn s

2p–2

ds.

Moreover, taking k> 0 small enough for Young’s inequality in the form ab≤k|b|m+

Ck,m|a|m/m–1, we have

Ezn_s2p≤ |y0|2p–Cp,λ2,β1,β2E

t 0

zn_s2pds+C_p,λ2,β1,β2t,

(11)

The proof of the following lemma is similar to Sect.2. We omit the details.

Lemma 3.2 For any p> 1and small enought,there exists a positive constant K2such

that

sup 0≤n≤T/tE|

Xn|2p≤K2, (32)

where K2is independent oft.

Lemma 3.3 For small enoughδt and p> 1,there exists a positive constant K3such that

sup 0≤n≤T

t

0<m<M

EY_mn2p≤K3, (33)

where K3is independent of(M,δt).

Proof Now we give a deﬁnition ofYn t by

Y_tn:=y0+

t 0

fXn,Yˆsn

ds+

t 0

gXn,Yˆsn

dW_sn+

t 0

hXn,Yˆsn

dN_sn,

whereYˆn

t :=Yknfort∈[kδt, (k+ 1)δt),k= 1, 2, . . .M, andYˆtnk=Y

n tk=Y

n

k (tk=kδt). Thus we have

Y_k+1n =y0+

(k+1)δt 0

fXn,Yˆsn

ds+

(k+1)δt 0

gXn,Yˆsn

dW_sn

+

(k+1)δt 0

hXn,Yˆsn

dN_sn. (34)

Taking the 2pth moment and expectations on both sides of (34), we get

EY_k+1n 2p =Ey0+

(k+1)δt 0

fXn,Yˆsn

ds+

(k+1)δt 0

gXn,Yˆsn

dW_sn

+

(k+1)δt 0

hXn,Yˆsn

dN_sn

2p

≤C|y0|2p+CE

₀(k+1)δtfXn,Yˆsn

ds

2p +CE

(k+1)δt 0

gXn,Yˆsn

dW_sn

2p

+CE

(k+1)δt 0

hXn,Yˆsn

dN_sn

2p

. (35)

Using Hölder’s inequality, Remark1.1, and the deﬁnition ofYˆ_tn, we have

E

(k+1)δt 0

fXn,Yˆsn

ds

2p

≤C

(k+1)δt 0

EfXn,Yˆsn 2p

(12)

≤C

(k+1)δt

0 E

1 +|Xn|2p+Yˆsn 2p

ds

≤C+CE|Xn|2p+Cδt k

i=0

EY_in2p. (36)

By the deﬁnition ofN˜t, Burkhölder’s inequality, Hölder’s inequality, and (H2), we obtain

E

(k+1)δt 0

gXn,Yˆsn

dW_sn

2p

≤CE

(k+1)δt 0

gXn,Yˆsn 2

ds

p

≤C

(k+1)δt

0 E

gXn,Yˆsn 2p

ds

≤Cp,T (37)

and

E

(k+1)δt 0

hXn,Yˆsn

dN_sn

2p

=E

(k+1)δt 0

hXn,Yˆsn

dN˜_sn+λ2

(k+1)δt 0

hXn,Yˆsn

ds

2p

≤CE (k+1)δt

0

hXn,Yˆsn 2

ds

p

+CEλ2

(k+1)δt 0

hXn,Yˆsn

ds

2p

≤C

(k+1)δt 0

EhXn,Yˆsn 2p

ds

≤Cp,T,λ2. (38)

Substituting (36)–(38) into (35) gives that

EY_k+1n 2p≤C1 +|y0|2p

+CE|Xn|2p+Cδt k

i=0

EY_in2p.

Using Lemma3.2and discrete Gronwall’s inequality, we get the result.

Next, we give the 2pth moment deviation between two successive iterations of the micro-solver.

sup 0≤n≤T

t

0<m<M

EY_m+1n –Y_mn2p≤K4(δt)p, (39)

(13)

Proof It is clear that

Y_m+1n –Y_mn=fXn,Ymn

δt+gXn,Ymn

W_mn+hXn,Ymn

N˜_mn+λ2h

Xn,Ymn

δt. (40)

Taking the 2pth moment and expectations on both sides of (40), we get

EY_m+1n –Y_mn2p

=EfXn,Ymn

δt+gXn,Ymn

W_mn+hXn,Ymn

N˜_mn+λ2h

Xn,Ymn

δt2p

≤Cδt2pEfXn,Ymn 2P

+C(δt)pEgXn,Ymn 2p

+Cδt2pEhXn,Ymn 2p

+CδtpEhXn,Ymn 2p

.

By Remark1.1and (H2), we have

EY_m+1n –Y_mn2p≤Cδt2p1 +E|Xn|2p+EYmn 2p

+Cδtp.

Using Lemmas3.2and3.3, for small enoughδt, we get

EY_m+1n –Y_mn2p≤K4δtp.

Lemma 2.1 in [9] shows thatzn

t is statistically equivalent to a shifted and rescaled version ofy

t, withxbeing a parameter, that is,zkt ∼yt–tk/.

It is proved in [2] that dynamic (9) is ergodic with a unique invariant measureμXn

(As-sumptions H3–H5), which possesses exponentially mixing property in the following sense. LetPXn₍_t_,_z_,_E_{) denote the transition probability of (9). Then there exist positive constants}

η,α< 1 such that

PXn₍_t_,_z_,_E_{) –}_μXn₍_E₎_<_ηαt

for everyE∈B(Rm_).

Then we establish the mixing properties of the auxiliary processeszn

t. Note thata¯(Xn) is the average ofa(Xn,y) with respect toμXn, which is the invariant measure induced byznt. We denotezn

m=znmδt.

E1 M

M

m=1

aXn,znm

–a¯(Xn)

2p

≤K5

–log_αMδt+ 1

Mδt +

1

M

, (41)

where K5is independent of(M,δt).

Proof By (H2) and Remark (1.2), we have

E1 M

M

m=1

aXn,znm

–a¯(Xn)

2p

=E

1

M

m=1

aXn,znm

–a¯(Xn)

2p–2

×1

M

m=1

aXn,zmn

–a¯(Xn)

(14)

≤E

It remains to estimate the mean-square term, and the proof for the term is similar to the method in [9] (Lemma 2.6). We omit the details. Thus we obtain the desired result (41). Afterwards, we establish the 2pth moments deviation between (9) and its numerical approximation (8).

Lemma 3.6 Let z_tnbe the family of processes deﬁned by(9).For small enoughδt and p> 1,

there exists a positive constant K8such that

max

Hence, we have

E sup inequality and Hölder’s inequality on the two martingale terms to get

(15)

and

By Hölder’s inequality, we have

E sup

Combining (44)–(49) and applying the Lipschitz condition in (H1), we have

E sup

which, with the help of continuous Gronwall’s inequality, yields the result.

Lemma 3.7 There exists a positive constant K6such that,for all p> 1and0≤n≤ Tt,

Proof By deﬁnition, we have

(16)

where

t is the family of processes deﬁned by (9).I1nis the diﬀerence between the ensem-ble average ofa(Xn,·) with respect to the (exact) invariant measure ofznt and its empirical average over M equi-distanced sample points.In

2 is the diﬀerence between empirical av-erages ofa(Xn,·) over M equi-distanced sample points, once for the processznt and once for its Euler approximationY_mn.

The estimation ofIn

1 is given in Lemma3.5,

Then we estimateIn 2 by (H2)

Using Lemma3.6, we obtain

I₂n≤CK8δtp. (53)

Combining (51)–(53), we get

Ea¯(Xn) –A(Xn)

Finally, we estimate the diﬀerence between the processXnand the auxiliary processX¯n.

(17)

Proof SetEn=Esupl≤n| ¯Xl–Xl|2p, then

We split the ﬁrst term on the right-hand side:

En≤CEsup

(18)

Now using Burkhölder’s inequality and (H2) on the two martingale terms, we get

The second term on the right-hand side can be bounded as follows:

Esup

Combining (55)–(60) with Lemma3.7, we obtain a discrete linear integral inequality

En≤C

with the initial conditionE0= 0. It follows that, for suﬃciently smallt,

En≤CK6

4 Main result

Now we can state and prove our main theorem readily.

(19)

Proof We begin our proof with subtracting and adding the termX¯n:

E sup 0≤n≤T/t

_X_n_–_x_¯₍_t_n₎2p

≤CE sup

0≤n≤T/t|Xn–

¯

Xn|2p+CE sup 0≤n≤T/t

X¯n–x¯(tn) 2p

≤Ktp+K

–log_αMδt+ 1

Mδt +

1

M+δt

p

,

where we have used the result of Lemmas2.4and3.8.

5 Conclusions

In this paper, theLp₍_p_{> 2)-strong convergence of the multiscale integration scheme has} been studied for the two-time-scale jump-diﬀusion systems. By Lemmas2.4and3.8, we obtained our desired main result. The results in [2,9] are extended in this paper. First, we provide a numerical method for theLp (p> 2) averaging principle in [2]; second, in [9], the authors only studiedL2_{convergence of the multiscale integration scheme, and we} extended the result into theLp₍_p_{> 2) case.}

Acknowledgements

The ﬁrst author is very grateful to Associate Professor Jie Xu for his encouragement and useful discussions.

Funding

The author acknowledges the support provided by NSFs of China No. U1504620 and Youth Science Foundation of Henan Normal University Grant No. 2014QK02.

Availability of data and materials

Not applicable.

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Consent for publication

Not applicable.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 14 November 2018 Accepted: 13 January 2019

References

1. Givon, D.: Strong convergence rate for two-time-scale jump-diﬀusion stochastic diﬀerential systems. Multiscale Model. Simul.6(2), 577–594 (2007)

2. Xu, J., Miao, Y.:Lp₍_p_{> 2)-strong convergence of an averaging principle for two-time-scales jump-diﬀusion stochastic} diﬀerential equations. Nonlinear Anal. Hybrid Syst.18, 33–47 (2015)

3. Khasminskii, R.Z.: On the principle of averaging the Itô’s stochastic diﬀerential equations. Kybernetika4, 260–279 (1968) (in Russian)

4. E, W., Liu, D., Vanden-Eijnden, E.: Analysis of multiscale methods for stochastic diﬀerential equations. Commun. Pure Appl. Math.58(11), 1544–1585 (2005)

5. Liu, D.: Strong convergence of principle of averaging for multiscale stochastic dynamical systems. Commun. Math. Sci.8(4), 999–1020 (2010)

6. Li, Z., Yan, L.: Stochastic averaging for two-time-scale stochastic partial diﬀerential equations with fractional Brownian motion. Nonlinear Anal. Hybrid Syst.31, 317–333 (2019)

7. Vanden-Eijnden, E.: Numerical techniques for multi-scale dynamical systems with stochastic eﬀects. Commun. Math. Sci.1(2), 385–391 (2003)

(20)

9. Givon, D., Kevrekidis, I.G.: Multiscale integration schemes for jump-diﬀusion systems. Multiscale Model. Simul.7(2), 495–516 (2008)

10. Liu, D.: Analysis of multiscale methods for stochastic dynamical systems with multiple time scales. Multiscale Model. Simul.8(3), 944–964 (2010)

11. Cerrai, S., Freidlin, M.I.: Averaging principle for a class of stochastic reaction-diﬀusion equations. Probab. Theory Relat. Fields144(1–2), 147–177 (2009)