R E S E A R C H
Open Access
Optimal consumption of the stochastic
Ramsey problem for non-Lipschitz diffusion
Chuandi Liu
**Correspondence: [email protected] School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China
Abstract
The stochastic Ramsey problem is considered in a growth model with the production function of a Cobb-Douglas form. The existence of a unique classical solution is proved for the Hamilton-Jacobi-Bellman equation associated with the optimization problem. A synthesis of the optimal consumption policy in terms of its solution is proposed.
MSC: 49L20; 49L25; 91B62
Keywords: Hamilton-Jacobi-Bellman equation; viscosity solutions; Ramsey problem; Cobb-Douglas production function
1 Introduction
We are concerned with the stochastic Ramsey problem in a growth model discussed by Merton []. For recent contribution in this direction, we refer to []. A firm produces goods according to the Cobb-Douglas production functionxγ for capitalx, where <γ < (cf.
Barro and Sala-i-Martin []). The stock of capitalXtat timetis modeled by the stochastic differential equation
dXt=Xtγdt+σXtdBt, t> ,X=x> ,σ= ,
on a complete probability space (,F,P) carrying a standard Brownian motion{Bt} en-dowed with the natural filtrationFtgenerated byσ(Bs,s≤t).
The capital stock can be consumed and the flow of consumption at timetis assumed to be written asctXt. The rate of consumptionc={ct}per capital stock is called an admissible policy if{ct}is an{Ft}-progressively measurable process such that
≤ct≤ for allt≥ a.s. (.)
We denote byA the set of admissible policies. Given a policyc∈A, the capital stock process{Xt}obeys the equation
dXt=Xγt –ctXt
dt+σXtdBt, X=x> . (.)
The objective is to find an optimal policyc∗={c∗t}so as to maximize the expected dis-counted utility of consumption
Jx(c) =E
∞
e–αtU(ctXt)dt
(.)
overc∈A, whereα> is a discount rate and U(x) is a utility function inC(,∞)∩
C[,∞), which is assumed to have the following properties:
U(∞) =U() = , U(+) =U(∞) =∞, U < . (.)
The Hamilton-Jacobi-Bellman (HJB for short) equation associated with this problem is given by
αu(x) = σ
xu (x) +xγ
u(x) +U˜x,u(x), x> , (.)
where
˜
U(x,y) =max≤c≤ U(cx) –cxy
forx,y> . (.)
This kind of economic growth problem has been studied by Kamien and Schwartz [] and Sethi and Thompson [, Chapter ]. However, the optimization problem is unsolved. It is not guaranteed that (.) admits a unique positive solution{Xt}and the value function is a viscosity solution of the HJB equation. The main difficulty stems from the fact that (.) is a degenerate nonlinear equation of elliptic type with the non-Lipschitz coefficientxγ in
(,∞). It is also analytically studied by [], nevertheless in the finite time horizon. The resulting HJB equation is a parabolic partial differential equation (PDE, for short), which is very different from the elliptic PDE dealt with in the present paper.
In this paper, we provide the existence results on a unique positive solution{Xt}to (.) and a classical solutionuof (.) by the theory of viscosity solutions. For the detail of the theory of viscosity solutions, we mention the works [, ] and []. An optimal policy is shown to exist in terms ofu.
This paper is organized as follows. In Section , we show that (.) admits a unique positive solution. In Section , we show the existence of a viscosity solutionuof the HJB equation (.). Section is devoted to theC-regularity of its solution. In Section , we
present a synthesis of the optimal consumption policy.
2 Preliminaries
In this section, we show the existence of a unique solution{Xt}to (.).
Proposition . There exists a unique positive solution{Xt}={Xx
t}to(.),for each c∈A, such that
E[Xt]≤ ( –γ)t+x–γ /(–γ)
, (.)
EXt≤eσt ( –λ)t+x(–λ)/(–λ), λ:= ( +γ)/, (.)
∀ε> ,∃Cε> s.t. EXtx–Xty≤Cε|x–y|+ε
+t/(–γ)+x+y, x,y> . (.) Proof We setxt=Xt–γ. Then, by Ito’s formula and (.),
dxt= ( –γ)X–
γ
t dXt+
σ
( –γ)(–γ)X
–γ
t dt
= ( –γ)
–
ct+
σ
γ
xt
By linearity, (.) has a unique solution{xt}. Since
dxtˆ = ( –γ)
–
ct+σ
γ
ˆ
xt
dt+ ( –γ)σxtˆ dBt, xˆ=x–γ (.)
has a positive solution{ˆxt}, we see by the comparison theorem [, Chapter , Theo-rem .] thatxt≥ ˆxt> holds almost surely (a.s.). Therefore, (.) admits a unique positive solution{Xt}of the formXt=xt/(–γ), which satisfiessup≤t≤TE[X
t] <∞for eachT≥. Letθtbe the right-hand side of (.) andφt=E[Xt]. Obviously, we see thatθtis a unique solution of
dθt=θtγdt, θ=x> .
By (.) and Jensen’s inequality,
dφt=dE[Xt] =E
Xtγ –ctXt
dt≤φtγdt.
Sinceθ=φ=x, we deduceφt≤θt, which implies (.).
Similarly, lettbe the right-hand side of (.) andt=E[Xt]. By substitution, it is easy to see that¯t:=e–σ
t
tis a unique solution of
d¯t= ¯tλdt, ¯=x> .
Hence
dt=eσ
t
¯λt +σ¯t
dt≥λt +σt
dt.
Furthermore, by (.), Ito’s formula and Jensen’s inequality,
dt =dE
Xt =EXλ
t – ctXt+σXt
dt
≤λt +σt
dt.
Thus, we deducet≤tand=, which implies (.).
Next, let{Yt}denote the solution{Xty}of (.) withY=yandyt=Yt–γ. Then, by (.)
d(xt–yt) = –( –γ)
ct+σ
γ
(xt–yt)dt+ ( –γ)σ(xt–yt)dBt,
which implies
xt–yt= (x–y)exp
–( –γ)
t
csds+σ
γt
+ ( –γ)σBt–σ
( –γ)
t
.
Settingβ= /( –γ) > , we have
E|xt–yt|β≤ |x–y|βE
exp
σBt–σ
t
By Young’s inequality [], for anyε> ,
xβ–yβ≤βxβ–+yβ–|x–y|
≤β
β
ε
β
|x–y|β+β–
β ε
xβ–+yβ–β/(β–)
≤
ε
β
|x–y|β+ (β– )(ε
)β/(β–)
xβ+yβ, x,y≥.
Hence, for anyε> , we chooseCε> such that
xβ–yβ≤Cε|x–y|β+ε +xβ+yβ, x,y≥.
Therefore, by (.) and (.), we have
E|Xt–Yt|=Exβt –y
β
t
≤CεE
|xt–yt|β+εE +xβt +y
β
t
≤Cε|x–y|+εE[ +Xt+Yt]
≤Cε|x–y|+ε + βtβ+x+ βtβ+y,
which implies (.).
Remark . The uniqueness for (.) is violated ifx= andctis deterministic since and the limit processχt:=limx→+Xtxsatisfy (.) withX= , and
Eχt–γ=E
t
( –γ)
–
cs+σ
γ
χs–γ
ds
= . (.)
3 Viscosity solutions of the HJB equation
Definition . Letu∈C(,∞). Thenuis called a viscosity solution of (.) if the following
relations are satisfied:
αu(x)≤ σ
xq+xγ
p+U(x,˜ p), ∀(p,q)∈J,+u(x),∀x> ,
αu(x)≥ σ
xq+xγp+U(x,˜ p), ∀(p,q)∈J,–u(x),∀x> ,
whereJ,+u(x) andJ,–u(x) are the second-order superjets and subjets [].
Define the value functionuby
u(x) =sup
c∈AE
∞
e–αtU(c tXt)dt
, (.)
where the supremum is taken over all systems (,F,P,{Ft};{Bt},{ct}).
Lemma . We assume(.).Then we have
≤u(x)≤ζ(x) :=x+ζ, x> (.)
for some constantζ> ,and there exists Cρ> for anyρ> such that
u(x) –u(y)≤Cρ|x–y|+ρ( +x+y), x,y> . (.)
Proof Clearly,uis nonnegative. By concavity, there isC¯ > such that
U(x)≤α–/(–γ)x+C,¯ x≥. By (.) and (.), we have
E
∞
e–αtU(ctXt)dt
≤E
∞
e–αtα–/(–γ)Xt+C¯dt
≤
∞
e–αt αt/(–γ)+x+C¯dt
=x+α
∞
e–αtt/(–γ)dt+C/¯ α,
which implies (.).
Now, letρ> be arbitrary. By (.), there isδ> such thatU(x)≤ρfor allx∈[,δ]. Furthermore,
U(x) –U(y)≤U(δ)|x–y|, x,y≥δ.
Thus, we obtain a constantCρ> , depending onρ> , such that
U(x) –U(y)≤Cρ|x–y|+ρ, ∀x,y≥. (.)
By (.), (.) and (.), we get
u(x) –u(y)≤sup
c∈AE
∞
e–αtU(ctXt) –U(ctYt)dt
≤sup
c∈AE
∞
e–αt Cρ|Xt–Yt|+ρdt
≤Cρ
∞
e–αt Cε|x–y|+ε +t/(–γ)+x+ydt+ρ/α ≤C CρCε|x–y|+ (ε+ρ)( +x+y)
, x,y> , (.)
where the constantC> is independent ofε,ρ> . Replacingρbyρ/Cand choosing
sufficiently smallε> , we deduce (.).
Remark . The continuity ofuis immediate from (.).
Proof According to [], the viscosity property ofufollows from the dynamic program-ming principle foru, that is,
u(x) =sup
c∈AE
τ
e–αtU(ctXt)dt+e–ατu(Xτ)
, x> (.)
for any bounded stopping time τ ≥, where the supremum is taken over all systems (,F,P,{Ft};{Bt},{ct}). Letu(x) be the right-hand side of (.). Let¯ Xt˜ =Xt+r andBt˜ = Bt+r–Br, whenτ=ris non-random. Then we have
dXt˜ =X˜tγ–˜ctXt˜
dt+σXt˜ dBt˜ , X˜=Xr
for the shifted processc˜={˜ct}ofcbyr,i.e.,ct˜ =ct+r. It is easy to see that
eαrE
∞
r
e–αtU(ctXt)dtFr
=E
∞
e–αtU(˜ctXt)˜ dtFr
=JXr(˜c) a.s.
with respect to the conditional probabilityP(·|Fr). We takeζ> such thatxγ ≤αx+ζ
and sufficiently largeζ> to obtain
–αζ+ σ
xζ +xγζ ≤
–αζ+ζ≤.
By (.) in Lemma ., Ito’s formula and Doob’s inequality, we have
E
sup
≤t≤Te
–αtJXt(c)˜≤E sup
≤t≤Te
–αtζ(Xt)≤ζ(x) +C, T>
for some constantC> . Hence, approximatingτ by a sequence of countably valued stop-ping times, we see that
Ee–ατJXτ(c)˜
=E
∞
τ
e–αtU(ctXt)dt
.
Thus
Jx(c) =E
τ
e–αtU(ctXt)dt+
∞
τ
e–αtU(ctXt)dt
≤E
τ
e–αtU(ctXt)dt+e–ατu(Xτ)
.
Taking the supremum, we deduceu≤ ¯u.
We shall show the reverse inequality in the case thatτ=ris constant. For anyε> , we consider a sequence{Sj:j= , . . . ,n+ }of disjoint subsets of (,∞) such that
diam(Sj) <δ, n
j=
Sj= (,R) and Sn+= [R,∞)
forδ,R> chosen later. We takexj∈Sjandc(j)∈Asuch that
By the same argument as (.), we note that
Jxc(j)–Jyc(j)+u(x) –u(y)≤Cε|x–y|+ε
( +x+y), x,y>
for some constantCε> . We choose <δ< such thatCεδ<ε/. Then we have
Jxc(j)–Jyc(j)+u(x) –u(y)≤ε( +x), x,y∈Sj,j= , , . . . ,n. (.)
Hence, by (.) and (.),
JXr
c(j)=JXr
c(j)–Jxj
c(j)+Jxj
c(j)
≥–ε( +Xr) +u(xj) –ε
≥–ε( +Xr) +u(Xr) –ε
≥–ε( +Xr) +u(Xr) ifXr∈Sj,j= , . . . ,n. (.)
By definition, we findc∈Asuch that
¯
u(x) –ε≤E
r
e–αtU(c
tXt)dt+e–αru(Xr)
.
In view of [, Chapter , Theorem .], we can takec,c(j)on the same probability space.
Define
crt=ct{t<r}+ct(j–)r{r≤t} ifXr∈Sj,j= , . . . ,n+ ,
where {·}denotes the indicator function. Then{crt}belongs toA. Let{Xtr}be the solution of
dXtr=Xrtγ –crtXtrdt+σXrtdBt, Xr=x> . Clearly,Xr
t=Xta.s. ift<r. Further, for eachj= , . . . ,n+ , we have on{Xr∈Sj}
Xtr+r=Xr+ t+r
r
Xsrγ –crsXsrds+ t+r
r
σXsrdBs
=Xr+ t
Xsr+rγ–cs(j)Xsr+rds+ t
σXsr+rdB˜s a.s.
Hence,Xr
t+r coincides with the solutionX
(j)
t of (.) for (˜,F˜,P,˜ { ˜Ft};{ ˜Bt},{c
(j)
t }) a.s. on
{Xr∈Sj}withX(j)=Xr. Thus, we get
JXr˜cr=EP˜
∞
e–αtUcrt+rXrt+rdtF˜r
=EP˜
∞
e–αtUc(tj)Xt(j)dtF˜r
Now, we fixx> and chooseR> , by (.), (.) and (.), such that
sup
c∈AE
u(Xr){Xr≥R}≤sup c∈AE
ζ(Xr){Xr≥R}
≤sup
c∈A
RE
Xr+ζXr
≤ C
R
+x+x<ε, (.)
where the constantC> depends only onr,ζ. By (.), (.) and (.), we have
E
∞
r
e–αtUcr tXtr
dt =E E ∞ r
e–αtUcr tXtr
dtFr
=Ee–αrJXr˜cr =E
n+
j=
e–αrJ Xr
c(j){Xr∈Sj}
≥E
n
j=
e–αr u(Xr) – ε( +Xr){Xr∈Sj}
≥Ee–αr u(X
r) –u(Xr){Xr≥R}
– εE[ +Xr]
≥Ee–αru(Xr)–ε– εC( +x)
for some constantC> independent ofε. Thus
u(x)≥E
r
e–αtUcrtXtrdt+
∞
r
e–αtUcrtXtrdt
≥E
r
e–αtU(ctXt)dt+e–αru(Xr)
–ε– εC( +x)
≥ ¯u(x) – ε– εC( +x).
Lettingε→, we getu¯≤u.
In the general case, by the above argument, we note that
u(Xr) =u(X˜)≥E
s
e–αtU(˜c
tX˜t)dt+e–αsu(X˜s)Fr
=E
s
e–αtU(ct+rXt+r)dt+e–αsu(Xs+r)Fr
a.s.s,r≥.
Hence{e–αsu(Xs) +s
e–
αtU(ctXt)dt}is a supermartingale. By the optional sampling
the-orem,
u(X)≥E
τ
e–αtU(c
tXt)dt+e–ατu(Xτ)F
a.s.
Taking the expectation and then the supremum overA, we conclude thatu¯≤u. Noting
4 Classical solutions
In this section, using the viscosity solutions technique, we show theC-regularity of the viscosity solutionuof (.). For any fixed <a<b, we consider the boundary value prob-lem
αw= σ
xw +xγ
w +U˜x,w in (a,b), (.)
with boundary condition
w(a) =u(a), w(b) =u(b), (.)
given byu.
Proposition . Let wi∈C[a,b],i= , ,be two viscosity solutions of(.), (.).Then, under(.),we have
w=w.
Proof It is sufficient to show thatw≤w. Suppose that there existsx∈[a,b] such that
w(x) –w(x) > . Clearly, by (.),x=a,b, and we findx¯∈(a,b) such that
:= sup
x∈[a,b]
w(x) –w(x)
=w(x) –¯ w(x) > .¯
Define
k(x,y) =w(x) –w(y) –
k |x–y|
fork> . Then there exists (xk,yk)∈[a,b]such that
k(xk,yk) = sup
(x,y)∈[a,b]
k(x,y)≥k(x,¯ x) =¯ , (.)
from which
k
|xk–yk|
<w
(xk) –w(yk). Thus
|xk–yk| → ask→ ∞. (.)
Furthermore, by the definition of (xk,yk),
k(xk,yk)≥k(xk,xk).
Hence, by uniform continuity
k
|xk–yk|
≤w
(xk) –w(yk)≤ sup
|x–y|≤ρ
w(x) –w(y)
By (.), (.) and (.), extracting a subsequence, we have
(xk,yk)→(x,˜ x)˜ ∈(a,b) ask→ ∞. (.)
Now, we may consider that (xk,yk)∈(a,b)for sufficiently largek. Applying Ishii’s lemma
[] tok(x,y), we obtainX,Y∈Rsuch that
k(xk–yk),X
∈ ¯J,+w (xk),
k(xk–yk),Y∈ ¯J,–w(yk), (.)
X
–Y
≤k
–
–
.
By Definition .,
αw(xk)≤
σ
x
kX+x
γ
kμ+U(xk,˜ μ),
αw(yk)≥ σ
y
kY+y
γ
kμ+U(y˜ k,μ),
whereμ=k(xk–yk). Putting these inequalities together, we get
α w(xk) –w(yk)
≤
σ
x
kX–ykY
+xγk–yγkμ+ U(xk,˜ μ) –U(yk,˜ μ)
≡I+I+I, say.
By (.) and (.), it is clear that
I=
σ
xkX–ykY≤σ
k(xk–yk)
→ ask→ ∞.
Also, by (.)
I=k
xγk –yγk(xk–yk)≤kγaγ–|xk–yk|→ ask→ ∞. By (.), (.), (.) and (.), we have
I ≤ max ≤c≤
U(cxk) –U(cyk)+|xk–yk||μ|
≤ Cρ|xk–yk|+ρ+k|xk–yk|
→ ask→ ∞and thenρ→.
Consequently, by (.), we deduce that
α≤α w(x) –˜ w(x)˜
≤,
which is a contradiction.
Proof For any <a<b, we recall the boundary value problem (.), (.). Since
U()≤U(x)( –x) +U(x), x> ,
we have
K:= sup
<x≤axU(x) <∞. Hence, by (.)
U(cx) –U(cx)≤cU(ca)|x–x|
≤K
a |x–x|, x,x∈[a,b], ≤c≤. Also, by (.)
U(x˜ ,y) –U(x˜ ,y)≤ max
≤c≤U(cx) –U(cx)+|xy–xy|
≤K
a|x–x|+|x–x||y|+b|y–y|, y,y> .
Thus the nonlinear term of (.) is Lipschitz. By uniform ellipticity, a standard theory of nonlinear elliptic equations yields that there exists a unique solutionw∈C(a,b)∩C[a,b] of (.), (.). For details, we refer to [, Theorem .] and [, Chapter , Theorem .]. Clearly, by Theorem .,uis a viscosity solution of (.), (.). Therefore, by Proposi-tion ., we havew=uanduis smooth. Sincea,bare arbitrary, we obtain the assertion.
5 Optimal consumption
In this section, we give a synthesis of the optimal policyc∗ ={c∗t}for the optimization problem (.) subject to (.). We consider the stochastic differential equation
dXt∗=Xt∗γ–ηXt∗X∗tdt+σXt∗dBt, X∗=x> , (.) whereη(x) =I(x,u(x)) andI(x,y) denotes the maximizer of (.) forx,y> ,i.e.,
I(x,y) =
(U)–(y)/x ifU(x)≤y,
otherwise. (.)
Our objective is to prove the following.
Theorem . We assume(.).Then the optimal consumption policy{c∗t}is given by
c∗t =ηXt∗. (.)
Lemma . Under(.),the value function u is concave.In addition,we have
u(x) > for x> , (.)
u(+) =∞. (.)
Proof Letxi> ,i= , . For anyε> , there existsc(i)∈Asuch that
u(xi) –ε<E
∞
e–αtUc(i)
t X
(i)
t
dt
,
where{X(ti)}is the solution of (.) corresponding toc(i)withX(i)=xi. Let ≤ξ≤, and
we set
¯
ct=ξc
()
t X
()
t + ( –ξ)c
()
t X
()
t
ξXt()+ ( –ξ)X()t ,
which belongs toA. Define{ ¯Xt}and{ ˜Xt}by
dXt¯ =(Xt)¯ γ–¯ctXt¯ dt+σXt¯ dBt, X¯=ξx+ ( –ξ)x,
˜
Xt=ξX()t + ( –ξ)X
()
t .
By concavity,
˜
Xt≤ξx+ ( –ξ)x+
t
(Xs)˜ γ–¯csXs˜ ds+ t
σXs˜ dBs a.s.
By the comparison theorem, we have
˜
Xt≤ ¯Xt for allt≥ a.s.
Thus, by (.)
uξx+ ( –ξ)x
≥E
∞
e–αtU(ct¯Xt¯ )dt
≥E
∞
e–αtU(ct¯Xt˜ )dt
=E
∞
e–αtUξc()
t X
()
t + ( –ξ)c
()
t X
()
t
dt
≥ξE
∞
e–αtUc()t X()t
dt
+ ( –ξ)E
∞
e–αtUc()t Xt()
dt
≥ξu(x) + ( –ξ)u(x) –ε.
Therefore, lettingε→, we obtain the concavity ofu.
To prove (.), by Theorem ., we recall thatuis smooth. Furthermore, we getu(x)≥ forx> . If not, thenu(a) < for somea> . By concavity,
which is a contradiction. Suppose thatu(z) = for somez> . Then, by concavity, we haveu(x) = for allx≥z. Hence, by (.) and (.),
αu(z) =αu(x) =U(x, ) =˜ U(x), x≥z.
This is contrary to (.). Thus, we obtain (.). Next, by definition, we have
<E
∞
e–αtU(Xˇ t)dt
≤u(x), x> ,
where{ ˇXt}is the solution of (.) corresponding toct= . As in (.), the limit process
ˇ
χt:=limx→+Xtˇ is different from . Hence
<E
∞
e–αtU(χˇt)dt
≤u(+).
Suppose thatu(+) <∞. By (.) and concavity, we getu(+) = , which is a contradiction.
This implies (.).
Lemma . Under(.),there exists a unique positive strong solution{Xt∗}of(.).
Proof Let{Nt}be the solution of (.) corresponding toct= . We can take the Brownian motion{Bt}on the canonical probability space [, p.]. Since ≤η≤, the probability measurePˆis defined by
dP/dPˆ =exp
–
t
η(Ns)/σdBs–
t
η(Ns)/σds
for everyt≥. Girsanov’s theorem yields that
ˆ
Bt:=Bt+ t
η(Ns)/σds is a Brownian motion underP.ˆ
Hence
dNt=
(Nt)γ–η(Nt)Nt
dt+σNtdBˆt underP.ˆ
Thus, (.) admits a weak solution. Now, by (.), we have
η(x)x=min U–◦u(x),x. Hence, by (.) and concavity,
d dx
U–◦u(x) = u (x)
U ◦(U)–◦u(x)≥.
Yamada-Watanabe theorem [], we deduce that (.) admits a unique strong solution
{Xt∗}.
Proof of Theorem. Since{c∗t}satisfies (.), it belongs toA. By Lemma ., we note that
<u(x)x≤u(x) –u(+) <u(x), x> .
Hence, by (.) and (.),
E
t
e–αsuXs∗Xs∗ds
≤E
t
e–αsuXs∗ds
≤E
t
e–αsζX∗ s
ds
<∞.
This yields that{te–αsu(X∗
s)Xs∗dBs}is a martingale. By (.), (.) and Ito’s formula,
Ee–αtuX∗t=u(x) +E
t
e–αs
–αuXs∗+Xs∗γuXs∗
–c∗sXs∗uXs∗+ σ
X∗ s
u X∗sds
=u(x) –E
t
e–αsUc∗sX∗sds
.
By (.) and (.), it is clear that
Ee–αtuX∗ t
≤Ee–αtζX∗ t
≤e–αt ( –γ)t+x(–γ)/(–γ)+e–αtζ→ ast→ ∞.
Lettingt→ ∞, we deduce
E
∞
e–αtUc∗tXt∗dt
=u(x).
By the same calculation as above, we obtain
E
∞
e–αtU(c tXt)dt
≤u(x)
for anyc∈A. The proof is complete.
Remark . From the proof of Theorem ., it follows that the solutionuof the HJB equa-tion (.) coincides with the value funcequa-tion. This implies that the uniqueness holds for (.).
Competing interests
Acknowledgements
I would like to thank Professor H Morimoto for his useful help. The research was supported by the National Natural Science Foundation of China (11171275) and the Fundamental Research Funds for the Central Universities (XDJK2012C045).
Received: 10 June 2014 Accepted: 24 September 2014 Published:13 Oct 2014
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