One kind of multiple dimensional Markovian BSDEs with stochastic linear growth generators

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R E S E A R C H

Open Access

One kind of multiple dimensional

Markovian BSDEs with stochastic linear

growth generators

Rui Mu and Zhen Wu

*

*_{Correspondence:}

[email protected]

School of Mathematics, Shandong University, Jinan, 250100, P.R. China

Abstract

In this article, we deal with a multiple dimensional coupled Markovian BSDEs system with stochastic linear growth generators with respect to volatility processes. An existence result is provided by virtue of approximation techniques.

MSC: 60H10; 39A05

Keywords: backward stochastic diﬀerential equations; stochastic diﬀerential equations; approximation techniques

1 Introduction

Backward stochastic differential equations (BSDEs) were proposed firstly by Bismut [] in linear case to solve the optimal control problems. Later this notion was generalized by Pardoux and Peng [] into the general nonlinear form, and the existence and uniqueness results were proved under the classical Lipschitz condition. A class of BSDEs was also introduced by Duffie and Epstein [] in point of view of recursive utility in economics. During the past twenty years, BSDEs theory has attracted many researchers’ interest and has been fully developed into various directions. Among the abundant literature, we refer readers to the florilegium book edited by El-Karoui and Mazliak [] for the early works before . Surveys on BSDEs theory also include [] which is written by El-Karoui, Hamadène and Matoussi collected in book [] (see Chapter ) and the book by Yong and Zhou [] (see Chapter ). Some applications on optimization problems can be found in []. About other applications, such as in the field of economics, we refer to El-Karoui, Peng and Quenez []. Recently, a complete review on BSDEs theory as well as some new results on nonlinear expectation were introduced in a survey paper by Peng [].

One possible extension to the pioneering work of [] is to relax as much as possible the uniform Lipschitz condition on the coeﬃcient. Mao [] provided an existence and uniqueness result under a weaker condition than the Lipschitz one. Hamadène introduced in [] a one-dimensional BSDE with local Lipschitz generator. Later Lepeltier and San Martin [] provided an existence result of minimum solution for one-dimensional BSDE where the generator functionf is continuous and of linear growth in terms of (y,z). When

f is uniformly continuous inzwith respect to (ω,t) and independent ofy, a uniqueness result was obtained by Jia []. BSDEs with polynomial growth generator were studied by

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Briand in []. The case of one-dimensional BSDEs with coeﬃcient which is monotonic inyand non-Lipschitz onzis shown in work []. About the BSDE with continuous and quadratic growth driver, a classical research should be the one by Kobylanski [] which investigated a one-dimensional BSDE with driver|f(t,y,z)≤C( +|y|+|z|_{) and bounded}

terminal value. This result was generated by Briand and Hu into the unbounded terminal value case in [].

There are plenty of works on one-dimensional BSDE. However, limited results have been obtained about the multi-dimensional case. We refer to Hamadène, Lepeltier and Peng [] for an existence result on BSDEs system of Markovian case where the driver is of linear growth on (y,z) and of polynomial growth on the state process. See Bahlali [, ] for high-dimensional BSDE with local Lipschitz coeﬃcient.

In the present article, we consider a high-dimensional BSDE under Markovian frame-work as follows:

Y_ti=gi(XT) +

T t

Hi

s,Xs,Ys, . . . ,Ysn,Zs, . . . ,Zns

ds– T

t

Z_sidBs

fori= , , . . . ,n, with processXas a solution of a stochastic diﬀerential equation (SDE for short). For eachi= , , . . . ,n, the coeﬃcientHiis continuous on (y, . . . ,yn,z, . . . ,zn) and

satisﬁes

Hi

t,x,y, . . . ,yn,z, . . . ,zn≤C +|x|zi+C +|x|γ₊_yi_, _γ _{> ,}

which means thatHiis of stochastic linear growth onZi, or in other words, it is of linear

growthωbyω. Similar situation was considered in [] in the background of nonzero-sum stochastic diﬀerential game problem. However, in [], the generatorHiis independent of

(y, . . . ,yn). According to our knowledge, this general form of high-dimensional coupled BSDEs system with stochastic linear growth generator has not been considered in litera-ture. This is the main motivation of the present work.

The rest of this article is organized as follows. In Section , we give some notations and assumptions on the coefficient. The properties of the forward SDE are also provided. The main existence result of BSDEs is proved in Section  where a measure domination result plays an important role. This domination result holds true when we assume that the diffu-sion process of the SDE satisfies the uniform elliptic condition. For the proof of the main result, we adopt an approximation scheme following the well-known mollify technique. The irregular coefficients are approximated by a sequence of Lipschitz functions. Then, we obtain the uniform estimates of the sequence of solutions as well as the convergence result in some appropriate spaces. Finally, we verify that the limit of the solutions is exactly the solution to the original BSDE, which completes the proof.

2 Notations and assumptions

In this section, we give some basic notations, the preliminary assumptions throughout this paper, as well as some useful results. Let (,F, P) be a probability space on which we deﬁne anm-dimensional Brownian motionB= (Bt)≤t≤T with integerm≥. Let us

denote by F ={Ft, ≤t≤T}for ﬁxedT>  the natural ﬁltration generated by processB

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LetPbe theσ-algebra on [,T]×ofFt-progressively measurable sets. Letp∈[,∞)

be a real constant andt∈[,T] be ﬁxed. We then deﬁne the following spaces:Lp ₌_{_ξ _:

Ft-measurable and Rm-valued random variable s.t. E[|ξ|p] <∞}; Stp,T ={ϕ = (ϕs)t≤s≤T :

P-measurable and Rm_{-valued s.t. E[}_sup

s∈[t,T]|ϕs|p] < ∞} and Hpt,T = {ϕ = (ϕs)t≤s≤T :

P-measurable and Rm_{-valued s.t. E[(}T

t |ϕs|ds)

p

] <∞}. Hereafter, Sp

,T and H

p

,T are

simply denoted byS_TpandHp_T.

The following assumptions are in force throughout this paper. Letσ be the function deﬁned as

σ: [,T]×Rm−→Rm×m

which satisﬁes the following assumption.

Assumption .

(i) σ is uniformly Lipschitz w.r.t.x,i.e.,there exists a constantCsuch that,∀t∈[,T],

∀x,x ∈Rm_,_|_σ₍_t_,_x_{) –}_σ(_t_,_x₎_{| ≤}_C

|x–x|.

(ii) σ is invertible and bounded and its inverse is bounded,i.e.,there exists a constantCσ such that∀(t,x)∈[,T]×Rm_,_|_σ₍_t_,_x₎_|₊_|_σ–₍_t_,_x₎_{| ≤}_C

σ.

Remark .(Uniform elliptic condition) Under Assumption ., we can verify that there

exists a real constant>  such that for any (t,x)∈[,T]×Rm,

.I≤σ(t,x).σ(t,x)≤–.I, (.)

whereIis the identity matrix of dimensionm.

Suppose that we have a system whose dynamic is described by a stochastic diﬀerential equation as follows: for (t,x)∈[,T]×Rm_,

⎧ ⎨ ⎩

X_st,x=x+_tsσ(u,X_ut,x)dBu, s∈[t,T];

Xt,x

s =x, s∈[,t].

(.)

The solution Xt,x= (X_st,x)s≤T exists and is unique under Assumption . (cf.Karatzas

and Shreve [], p.). We recall a well-known result associated to integrability of the solution. For any ﬁxed (t,x)∈[,T]×Rm,p≥, it holds that, P-a.s.,

E

sup ≤s≤T

Xt_s,xp

≤C +|x|p, (.)

where the constantConly depends on the Lipschitz coefficient and the bound ofσ. For integern≥, we first present the following Borelian function as the terminal coef-ficient of then-dimensional BSDE that we are considering:

gi: Rm−→R, i= , , . . . ,n,

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Assumption . The function gi_,_i_{= , , . . . ,}_n_,_{is of polynomial growth with respect to x}_,

i.e.,there exist a constant Cgandγ≥such that

gi(x)≤Cg

 +|x|γ, ∀x∈Rm,for i= , , . . . ,n.

Now, we consider Borelian functionsHi,i= , , . . . ,n, from [,T]×Rm×Rn×Rnminto

Ras follows:

Hi

t,x,y, . . . ,yn,z, . . . ,zn, i= , , . . . ,n, which satisfy the following hypothesis.

Assumption .

(i) For each(t,x,y_{, . . . ,}_yn_,_z_{, . . . ,}_zn₎_∈_[,_T_]_×_Rm_×_Rn_×_Rnm_,_{there exist constants}_C

, Chandγ> such that,for eachi= , , . . . ,n,

Hi

t,x,y, . . . ,yn,z, . . . ,zn≤C

 +|x|zi+Ch

 +|x|γ₊_yi_; _(.)

(ii) the mapping(y_{, . . . ,}_yn_,_z_{, . . . ,}_zn₎_∈_Rn_×_Rnm_−→_H

i(t,x,y, . . . ,yn,z, . . . ,zn)∈Ris

continuous for any ﬁxed(t,x)∈[,T]×Rm.

For a ﬁxed constanta∈Rm_{, and}_i_{= , , . . . ,}_n_{, let us consider the following BSDE:}

Y_ti=giX,_Ta+ T

t

Hi

s,X_s,a,Y_s, . . . ,Y_sn,Z_s, . . . ,Z_snds

– T

t

Zi_sdBs, t∈[,T]. (.)

From Assumptions . and ., we know that this is a multiple dimensional coupled BSDEs system under Markovian framework with unbounded terminal value.

3 Existence of solutions for the multiple dimensional coupled BSDEs system In this section, we provide an existence result of BSDEs (.) whenn=  as an example. Actually, the case forn>  can be dealt with in the same way without any diﬃculties.

3.1 Measure domination

Before we state our main theorem, let us ﬁrst recall a result related to measure domination.

Deﬁnition .(Lq_{-domination condition) Let}_q_∈_(,_∞_{) be ﬁxed. For given}_t

∈[,T],

a family of probability measures {ν(s,dx),s∈[t,T]} deﬁned on Rm is said to be Lq

-dominated by another family of probability measures{ν(s,dx),s∈[t,T]}if for anyδ∈

(,T–t], there exists an applicationφtδ: [t+δ,T]×R

m_→_R+_{such that:}

(i) ν(s,dx)ds=φtδ(s,x)ν(s,dx)dson[t+δ,T]×R

m_.

(ii) ∀k≥,φδ_t_(s,x)∈Lq_([_t

+δ,T]×[–k,k]m;ν(s,dx)ds).

Lemma . Let a∈Rm_{, (}_t_,_x₎_∈_[,_T_]_×_Rm_,_s_∈₍_t_,_T_]_and_μ(_t_,_x_;_s_,_dy₎_{the law of X}t,x s ,i.e.,

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Under Assumption.onσ,for any q∈(,∞),the family of laws{μ(t,x;s,dy),s∈[t,T]}is Lq-dominated by{μ(,a;s,dy),s∈[t,T]}for ﬁxed a∈Rm.

Proof See [], Lemma . and Corollary ., pp.-.

3.2 High-dimensional coupled BSDEs system

Our main result in this section is the following theorem.

Theorem . Let a∈Rm_{be ﬁxed}_._{Then under Assumptions}_{., .}_and_.,_{there exist two}

pairs ofP-measurable processes(Yi_,_Zi₎_{with values in}_R+m_,_i_{= , ,}_{and two deterministic}

functionsςi₍_t_,_x₎_{which are of polynomial growth}_,_i.e._,_|_ςi₍_t_,_x₎_{| ≤}_C_{( +}_|_x_|γ₎_with_γ _≥_, i= , ,such that

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

P-a.s., ∀t≤T,Yi

t =ςi(t,Xt,a)and Ziis dt-square integrableP-a.s.;

Y

t =g(X

,a t ) +

T

t H(s,Xs,a,Ys,Ys,Zs,Zs)ds–

T t ZsdBs;

Y

t =g(X

,a t ) +

T

t H(s,Xs,a,Ys,Ys,Zs,Zs)ds–

T t ZsdBs.

(.)

The result holds true as well for the case i> following the same way.

Proof The structure of this proof is as follows. We ﬁrst use the mollify technique on the generatorHito construct a sequence of BSDEs with generators which are Lipschitz

con-tinuous. Then, we provide uniform estimates of the solutions as well as the convergence property. Finally, we verify that the limits of the sequences are exactly the solutions for BSDE (.).

Step. Approximation.

Letξbe an element ofC∞(R+m_{, R) with compact support and satisfy}

R+m

ξy,y,z,zdydydzdz= .

For (t,x,y_,_y_,_z_,_z₎_∈_[,_T_]_×_Rm_×_R+m_{, we set}

˜

Hn

t,x,y,y,z,z

=

R+mn

_H 

s,ϕn(x),p,p,q,q

×ξny–p,ny–p,nz–q,nz–qdpdpdqdq,

where the truncation functionϕn(x) = ((xj∨(–n))∧n)j=,,...,mforx= (xj)j=,,...,m∈Rm. We

next deﬁneψ∈C∞(R+m_{, R) by}

ψy,y,z,z= ⎧ ⎨ ⎩

, |y_|₊_|_y_|₊_|_z_|₊_|_z_|_≤_,

, |y_|₊_|_y_|₊_|_z_|₊_|_z_|_≥_.

Then we deﬁne the measurable function sequence (Hn)n≥as follows:∀(t,x,y,y,z,z)∈

[,T]×Rm×R+m,

Hn

t,x,y,y,z,z=ψ

y n,

y n,

z n,

z n

˜

Hn

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We have the following properties:

The construction of the approximating sequence (Hn

)n≥is carried out in the same way.

For eachn≥ and (t,x)∈[,T]×Rm_{, since}_H

nandHnare uniformly Lipschitz w.r.t.

(y_,_y_,_z_,_z_{), by the result of Pardoux-Peng (see []), we know that there exist two pairs of}

Meanwhile, properties (.)(a), (c) and the result of El-Karouiet al.(ref. []) yield that there exist two sequences of deterministic measurable applications ςn _(resp._ςn_{) : [,}_T_]_×

Rm_→_R_and_zn_(resp._zn_{) : [,}_T_]_×_Rm_→_Rm_{such that for any}_s_∈_[_t_,_T_],

Y_sn;(t,x)=ςns,X_st,x resp.Y_sn;(t,x)=ςns,X_st,x (.)

and

Z_sn;(t,x)=zns,X_st,x resp.Z_sn;(t,x)=zns,X_st,x.

Besides, we have the following deterministic expression: fori= ,  andn≥,

ςin(t,x) = E

For eachn≥, let us ﬁrst consider the following BSDE:

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For anyx∈Rm_and_n_≥_{, the mapping (}_y_,_z₎_∈_R_×_Rm_→_C

( +|ϕn(Xrt,x)|)|z|+Ch( +

|ϕn(Xtr,x)|γ +|y|) is Lipschitz continuous; therefore, the solution (Y¯n,Z¯n)∈St,T(R)×

H

t,T(Rm) exists and is unique. Moreover, it follows from the result of El-Karouiet al.

(see []) that Y¯n _{can be characterized through a deterministic measurable function}

¯

ςn: [,T]×Rm_→_{R, that is, for any}_s_∈_[_t_,_T_],

¯

Y_sn=ς¯ns,X_st,x. (.)

Next let us consider the process

Bn_s =Bs–

which is, thanks to Girsanov’s theorem, a Brownian motion under the probability Pn

on (,F) whose density with respect to P is ET:=ET(

increasing process. Therefore, the term_tTdL_s is positive. Considering (.), we have

eCht_ς¯n₍_t_,_x₎≤_eCht_ς¯n₍_t_,_x₎+

where En_{is the expectation under the probability P}n_{. Taking the expectation on both sides}

under the probability Pn_{and taking account of}_ς_¯n_{is deterministic and}_f+_{≤ |}_f_|_{for any}

integrable functionf, we obtain

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Sinceg_{is of polynomial growth, and}_e–Cht≤_{ for}_t∈_[,_T_{], we infer that,}∀₍_t_,_x₎∈_[,_T_]×

Rm_,

¯

ςn(t,x)≤CEn

sup

t≤s≤T

 +X_st,xγ≤CE

ET· sup t≤s≤T

 +X_st,xγ,

where the constant Cdepends only on T,Ch andCg. Next, by a result of Haussmann

([], p., see also [], Lemma .), there exist somep∈(, ) and a constantCwhich is

independent ofnsuch that E[|ET|p]≤Cuniformly. As a result of Young’s inequality and

estimate (.), we have

¯

ςn(t,x)≤Cp

E|ET|p

+ E

sup

t≤s≤T

 +X_st,x

pγ

p–_,

which yields

¯

ςn(t,x)≤C +|x|λ withλ=pγ/(p– ) > . (.)

Next, by the comparison theorem of BSDEs and property (.)(b), we actually have, for anys∈[t,T],

Y_sn;(t,x)=ςns,X_st,x≤ ¯Y_sn=ς¯ns,X_st,x,

and by choosings=t, we get thatςn₍_t_,_x₎_≤_C_{( +}_|_x_|λ_{), (}_t_,_x₎_∈_[,_T_]_×_Rm_{. In a similar way,}

we can also show that for any (t,x)∈[,T]×Rm_,_ςn₍_t_,_x₎_≥_–_C_{( +}_|_x_|λ_{). As a conclusion,}

ςn_{is of polynomial growth on (}_t_,_x_{) uniformly in}_n_,_i.e._{, there exist a constant}_C_{, which is}

independent ofn, andλ>  such that

ςn₍_t_,_x₎_≤_C_{ +}_|_x_|λ

. (.)

Combining (.) and (.), we deduce that, for anyα> ,i= , ,

E

sup

t≤s≤T

Y_sin;(t,x)α

≤C. (.)

On the other hand, by applying Itô’s formula to (Yin;(t,x)₎ _{and considering the uniform}

estimate (.), we can infer in a regular way that, for anyt∈[,T],i= , ,

E

T

t

_Zin;(t,x)

s

 ds

≤C. (.)

Step. For ﬁxeda∈Rm, there exists a subsequence of ((Ysn;(,a),Zsn;(,a))≤s≤T)n≥which

converges in spaceS

T(R)×HT(Rm) respectively to (Ys,Zs)≤s≤T, a solution of BSDE (.).

Let us recall expression (.) for casei=  and apply property (.)(b) combined with the uniform estimates (.), (.), (.) and Young’s inequality to show that, for  <q< ,

E

T



Fn

s,X_s,aqds

≤CE

T



 +ϕn

X_s,aqZ_sn;(,a)q+ +ϕn

X_s,aγq+Y_sn;(,a)qds

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≤C

Therefore, there exists a subsequence{nk}(for notation simpliﬁcation, we still denote it

by{n}) and aB([,T])⊗B(Rm_{)-measurable deterministic function}_F

where on the right-hand side, noticing (.), we obtain

E

At the same time, Lemma . associated with theL

q

q–_{-domination property implies}

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Finally,

t ), P-a.s. Taking account of (.) and Lebesgue’s dominated convergence

the-orem, we obtain that the sequence ((Ytn;(,a))≤t≤T)n≥converges toY= (ς(t,Xt,a))≤t≤T script (,a) for convenience) and considering (.)(b), we get

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Taking nowt=  in (.), expectation on both sides and the limit w.r.t.nandm, we deduce

due to (.), (.) and the convergence of (.). Asε is arbitrary, then the sequence (Zn₎

n≥is convergent inHTto a processZ.

Now, returning to inequality (.), taking the supremum over [,T] and using BDG’s inequality, we obtain that

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The sequenceI_n,n≥, converges to . On the one hand, forn≥, property (.)(b) in

On the other hand, considering property (.)(d), we obtain that _H__n

asn→ ∞. Thanks to Lebesgue’s dominated convergence theorem, the sequenceIn

 of

The last inequality is a straightforward result of estimates (.), (.) and (.). The third sequenceIn

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Ac-In addition, considering that

sup

k≥

H

t,X,_ta,Ynk

t ,Y

nk

t ,Z

nk

t ,Z

nk

t ∈H q

T(R) for  <q< ,

which follows from (.). Finally, by the dominated convergence theorem, one can get that

H

t,X_t,a,Ynk

t ,Y

nk

t ,Z

nk

t ,Z

nk

t

k→∞ −−−→H

t,X_t,a,Y_t,Y_t,Z_t,Z_t inHq_T, which yields the convergence ofIn

 in (.) to .

It follows that the sequence (Hn(t,Xt,a,Ytn,Ytn,Ztn,Ztn)≤t≤T)n≥converges to (H(t, Xt,a,Yt,Yt,Zt,Zt))≤t≤TinL([,T]×,dt⊗dP) and thenF(t,Xt,a) =H(t,X,ta,Yt,Yt,

Z

t,Zt), dt ⊗dP-a.e. In the same way, we have F(t,X,tx) =H(t,Xt,a,Yt,Yt,Zt,Zt),

dt⊗dP-a.e. Thus, the processes (Yi_,_Zi_),_i_{= , , are the solutions of the backward}

equa-tion (.).

4 Generalizations

As we can see from Assumption .(i), this model deals with BSDE (.) with stochastic linear growth generators. However, whenxtakes value ofX,a_{, the coeﬃcient of the}

com-ponent yi _{in (.) is deterministic. Therefore, one possible generalization is to improve}

this model by considering the following assumption instead of Assumption ..

Assumption .

(i) For each(t,x,y, . . . ,yn,z, . . . ,zn)∈[,T]×Rm×Rn×Rnm,there exist a constantC and <γ < such that,for eachi= , , . . . ,n,

Hi

t,x,y, . . . ,yn,z, . . . ,zn≤C +|x|zi+C +|x|γyi; (.) (ii) the mapping(y_{, . . . ,}_yn_,_z_{, . . . ,}_zn₎_∈_Rn_×_Rnm_−→_H

i(t,x,y, . . . ,yn,z, . . . ,zn)∈Ris

continuous for any ﬁxed(t,x)∈[,T]×Rm_.

Then we will obtain a similar conclusion below. However, notice that we can only con-veniently deal with the case whenγ is strictly smaller than .

Theorem . Let a∈Rmbe ﬁxed.Then,under Assumptions., .and.,there exist two pairs ofP-measurable processes(Yi_,_Zi₎_{with values in}_R+m_,_i_{= , ,}_{and two deterministic}

functionsςi(t,x)with the growth property as follows,|ςi(t,x)| ≤eC(+|x|γ₎

with <γ < ,

i= , ,such that equations(.)hold true.

Sketch of the proof One can follow the same method of the proof of Theorem .. We only point out here some diﬀerence in Step  related to the growth property of the deterministic functionςn_.

In Step , we consider the following BSDE sequence:

¯

Yn s =g

X_Tt,x+ T

t

C +ϕn

Xt,x r Z¯rn

+C +ϕn

X_rt,xγY¯_rndr– T

s

¯

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After changing probability following the same line as (.), we obtain

¯

Y_sn=gX_Tt,x+ T

t

C +ϕn

X_rt,xγY¯_rndr– T

s

¯

Z_rndBn_r, s∈[t,T]. (.)

ConsideringY¯n

s =ς¯n(s,Xst,x) for the deterministic functionς¯n, we obviously have

¯

ςn(t,x) = EngX_Tt,xe T

t C(+|ϕn(X t,x

r )|γ)sign(Y¯rn)dr

≤EneCsup≤s≤T(+|Xst,x|γ)

= EeCsup≤s≤T(+|Xst,x|γ)E

T

≤eC(+|x|γ).

This inequality is obtained from the integrability ofETinLpfor somep∈(, ) and the

fact that ( +|x|) <e+|x|for anyx. Therefore, by comparison theorem, we knowςn(t,x)≤

eC(+|x|γ)_{. In a similar way, we can also show that}_ςn₍_t_,_x₎_≥_e–C(+|x|γ)_.

Combining this growth property and the fact thatYsn:(t,x)=ςn(s,Xst,x), we conclude that,

for anyα>  and  <γ< ,

E

sup

t≤s≤T

Y_sin;(t,x)α

≤C,

since

E

sup

t≤s≤T

eC(+|Xst,x|γ)

≤C (.)

is true only for positiveγ strictly smaller than . Whenγ touches  or bigger than , then estimate (.) is not necessarily true for arbitraryT > . In this case, (.) may explode, and this is the reason why we only considerγ ∈(, ) in Assumption .. However, for γ ≥, we can still expect that (.) holds true forTsmall enough.

The rest of the proof will have no practical diﬃculties, therefore, we omit it.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

RM carried out the model formulation, the proof and the edit work. ZW reviewed and edited the manuscript. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors thank the anonymous reviewers for their useful comments and suggestions. The ﬁrst author was supported in part by the Natural Science Foundation for Young Scientists of Jiangsu Province, P.R. China (No. BK20140299). The second author was supported in part by National Natural Science Foundation of China (11221061 and 61174092), 111 project (B12023), the National Natural Science Fund for Distinguished Young Scholars of China (11125102) and the Chang Jiang Scholar Program of Chinese Education Ministry.

Received: 8 January 2015 Accepted: 12 August 2015 References

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doi:10.1080/17442508.2014.915973