One Kind of Multiple Dimensional Markovian BSDEs with Stochastic Linear Growth Generators

0.19 MB·English

by  Rui Mu

MuandWu
RESEARCH
One Kind of Multiple Dimensional Markovian
BSDEs with Stochastic Linear Growth Generators
Rui Mu1† and Zhen Wu2*ˆ
5
1
0 *Correspondence:
Abstract
2 [email protected]
2SchoolofMathematics, In this article, we deal with a multiple dimensional coupled Markovian BSDEs
n ShandongUniversity,250100
a Jinan,P.R.China system with stochastic linear growth generators with respect to volatility
J Fulllistofauthorinformationis processes. An existence result is provided by using approximation techniques.
availableattheendofthearticle
3
†SupportedinpartbytheNatural Keywords: Backward Stochastic Differential Equations; Stochastic Differential
1
ScienceFoundationforYoung Equations; Approximation Techniques

ScientistsofJiangsuProvince,
] P.R.China(No.BK20140299) AMS Subject Classification: 39A05; 60H10
R
ˆSupportedinpartbyNational
P NaturalScienceFoundationof
. China(11221061and61174092),
h 111project(B12023),the
t NationalNaturalScienceFundfor
a 1 Introduction
DistinguishedYoungScholarsof
m
China(11125102). Backwardstochasticdifferentialequations(BSDEs)wasproposedfirstlybyBismut
[
(1973) [3] in linear case to solve the optimal control problems. Later this notion

1
was generalized by Pardoux and Peng (1990) [21] into the general nonlinear form
v
7 and the existence and uniqueness results were proved under the classical Lipschitz
8 condition. A class of BSDE is also introduced by Duffie and Epstein (1992) [8] in
9
pointofviewofrecursiveutilityineconomics.Duringthepasttwentyyears,BSDEs
2
0 theoryattractsmanyresearchers’interestsandhasbeenfullydevelopedintovarious
.
1 directions. Among the abundant literature, we refer readers the florilegium book
0
editedbyEl-KarouiandMazliark(1997)[9]fortheearlyworksbefore1996.Surveys
5
1 on BSDEs theory also includes [10] which is written by El-Karoui, Hamad`ene and
: Matoussi collected in book (2009) [7] (see Chapter 8) and the book by Yong and
v
i Zhou (1999)[23] (see Chapter 7). Some applications on optimization problems can
X
be found in [10]. About Other applications such as in field of economics, we refer
r
a to El-Kaoui,Peng and Quenez (1997)[11] . Recently, a complete review on BSDEs
theory as well as some new results on nonlinear expectation are introduced in a
survey paper by Peng (2011) [22].
One possible extension to the pioneer work of [21] is to relax as much as possible
theuniformLipschitzconditiononthecoefficient.Aweakerhypothesisispresented
by Mao (1995) [19] which we translate as follows: for all y,y¯,z,z¯ and t ∈ [0,T],
the generator of the BSDE satisfies |f(t,y,z)−f(t,y¯,z¯)|2 ≤ κ|y−y¯|2+c|y−y¯|2
a.s. where c > 0 and κ is a concave non-decreasing function from R+ to R+ such
that κ(0) = 0, κ(u) > 0 for u > 0 and 1/κ(u)du = ∞. An existence and
0+
uniqueness result is proved under such condition in [19]. Hamad`ene introduced in
R
(1996)[13]a one-dimensionalBSDE with localLipschitz generator.LaterLepeltier
and San Martin (1997) [18] provided an existence result of minimum solution for
one dimensional BSDE where the generator function f is continuous and of linear
growthintermsof(y,z).Whenf isuniformlycontinuousinz withrespectto(ω,t)
MuandWu Page2of14
and independent of y, a uniqueness result was obtained by Jia [17]. BSDEs with
polynomialgrowthgeneratoris studied by Briandin [4]. The case of 1-dimensional
BSDEs with coefficient whichis monotonic in y and non-Lipschitz onz is shownin
work[5].AbouttheBSDE withcontinuousandquadraticgrowthdriver,aclassical
research should be the one by Kobylanski [12] (2000) which investigated a one-
dimensional BSDE with driver |f(t,y,z)≤C(1+|y|+|z|2) and bounded terminal
value. This result was generated by Briand and Hu into the unbounded terminal
value case in [6] (2006).
There are plenty works on one-dimensional BSDE. However,limited results have
beenobtainedaboutthemulti-dimensionalcase.WereferHamad`ene,Lepeltierand
Peng [14] 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 [1] [2] for high-dimension BSDE with local Lipschitz coefficient.
In the present article, we consider a high dimensional BSDE under Markovian
framework as follows:
T T
Yi =gi(X )+ H (s,X ,Y1,...,Yn,Z1,...,Zn)ds− ZidB .
t T i s s s s s s s
Zt Zt
for i = 1,2,...,n, with process X as a solution of a stochastic differential equa-
tion (SDE for short). For each i = 1,2,...,n, the coefficient H is continuous on
i
(y1,...,yn,z1,...,zn) and satisfies:
|H (t,x,y1,...,yn,z1,...,zn)|≤C(1+|x|)|zi|+C(1+|x|γ +|yi|), γ >0.
i
AsthegeneratoroftheBSDE,actually,H isofstochasticlineargrowthonZi,orin
i
anotherword,it is oflinear growthω by ω.Similar situationwasconsideredin [15]
in the background of nonzero-sum stochastic differential game problem. However,
in[15],thegeneratorH isindependenton(y1,...,yn).Accordingtoourknowledge,
i
this generalformof highdimensionalcoupledBSDEs systemwith stochastic linear
growthgeneratorhasnotbeenconsideredinliterature.Thisisthemainmotivation
of the present work.
The rest of this article is organized as follows: in Section 2, we give some nota-
tionsansassumptionsonthecoefficient.ThepropertiesoftheforwardSDEarealso
provided. The main existence result of BSDEs is provedin Section 3 where a mea-
sure domination result plays an important role. This domination result holds true
when we assume that the diffusion process of the SDE satisfies the uniform elliptic
condition.For the proofofthe main result,we adoptanapproximationscheme fol-
lowingthe wellknowmollify technique.Theirregularcoefficientsareapproximated
by a sequence of Lipschiz functions. Then, we obtain the uniform estimates of the
sequence of solutions as well as the convergence result in some appropriate spaces.
Finally,weverifythatthelimitofthesolutionsisexactlythesolutiontotheoriginal
BSDE which completes the proof.
2 Notations and assumptions
In this section, we will give some basic notations, the preliminary assumptions
throughoutthispaper,aswellassomeusefulresults.Let(Ω,F,P)beaprobability
MuandWu Page3of14
space on which we define a m-dimensional Brownian motion B = (B ) with
t 0≤t≤T
integer m ≥ 1. Let us denote by F = {F ,0 ≤ t ≤ T} for fixed T > 0, the
t
natural filtration generated by process B and augmented by NP the P-null sets,
i.e. Ft =σ{Bs, s≤t}∨NP.
Let P be the σ-algebra on [0,T]× Ω of F -progressively measurable sets. Let
t
p ∈ [1,∞) be real constant and t ∈ [0,T] be fixed, We then define the following
spaces:Lp ={ξ :F -measurableandRm-valuedrandomvariables.t. E[|ξ|p]<∞};
t
Sp = {ϕ=(ϕ ) : P-measurableandRm-valued s.t. E[sup |ϕ |p]<∞}
t,T s t≤s≤T s∈[t,T] s
andHtp,T ={ϕ=(ϕs)t≤s≤T :P-measurableandRm-valueds.t.E[( tT |ϕs|2ds)p2]<
∞}. Hereafter, Sp and Hp are simply denoted by Sp and Hp.
0,T 0,T T T R
The following assumptions are in force throughout this paper. Let σ be the func-
tion defined as:
σ :[0,T]×Rm −→Rm×m
which satisfies the following assumption.
Assumption 2.1 (i) σ is uniformly Lipschitz w.r.t x. i.e. there exists a constant
C such that, ∀t∈[0,T], ∀x,x′ ∈Rm, |σ(t,x)−σ(t,x′)|≤C |x−x′|.
1 1
(ii) σ is invertible and bounded and its inverse is bounded, i.e., there exits a con-
stant C such that ∀(t,x)∈[0,T]×Rm, |σ(t,x)|+ σ−1(t,x) ≤C .
σ σ
(cid:12) (cid:12)
Remark 2.1 Uniform elliptic condition. (cid:12) (cid:12)
Under Assumptions 2.1, we can verify that, there exists a real constant ǫ> 0 such
that for any (t,x)∈[0,T]×Rm,
ǫ.I ≤σ(t,x).σ⊤(t,x)≤ǫ−1.I (2.1)
where I is the identity matrix of dimension m.
Suppose that we have a system whose dynamic is described by a stochastic dif-
ferential equation as follows: for (t,x)∈[0,T]×Rm,
s
Xt,x =x+ σ(u,Xt,x)dB , s∈[t,T];
s u u
 Zt (2.2)
Xst,x =x, s∈[0,t].
Thesolution X = (Xt,x) exists and is unique under Assumption 2.1. (cf.
s s≤T
Karatzas and Shreve 1991 [20], p.289). We recall a well-known result associates to
integrabilityof the solution.For any fixed (t,x)∈[0,T]×Rm, p≥2,it holds that,
P-a.s.,
E sup |Xt,x|p ≤C(1+|x|p), (2.3)
s
(cid:20)0≤s≤T (cid:21)
where the constant C is only depend on the Lipschitz coefficient and the bound
of σ. In addition, property (2.3) holds true, as well, for the expectation under the
probability which is equivalent to P.
MuandWu Page4of14
For integern≥1,wefirst presentthe followingBorelianfunction asthe terminal
coefficient of the n-dimensional BSDE that we considering:
gi :Rm −→R, i=1,2,...,n.
which satisfy
Assumption 2.2 The function gi, i = 1,2,...,n, are of polynomial growth with
respect to x, i.e. there exist constants C and γ ≥0 such that
g
|gi(x)|≤C (1+|x|γ),∀x∈Rm, for i=1,2,...,n.
g
Now,weconsiderBorelianfunctionsH ,i=1,2,...,n,from[0,T]×Rm×Rn×Rnm
i
into R as follows:
H (t,x,y1,...,yn,z1,...,zn), i=1,2,...,n
i
which satisfy the following hypothesis:
Assumption 2.3 (i) For each (t,x,y1,...,yn,z1,...,zn) ∈ [0,T]×Rm ×Rn ×
Rnm,thereexistsconstantsC ,C andγ >0,suchthat,foreachi=1,2...,n,
2 h
|H (t,x,y1,...,yn,z1,...,zn)|≤C (1+|x|)|zi|+C (1+|x|γ+|yi|); (2.4)
i 2 h
(ii) themapping(y1,...,yn,z1,...,zn)∈Rn×Rnm 7−→H (t,x,y1,...,yn,z1,...,zn)∈
i
R is continuous for any fixed (t,x)∈[0,T]×Rm.
The BSDE that we concern in this work is the following:
T T
Yi =gi(X0,x)+ H (s,X0,x,Y1,...,Yn,Z1,...,Zn)ds− ZidB . (2.5)
t T i s s s s s s s
Zt Zt
for i = 1,2,...,n. From Assumptions 2.2 and 2.3, we know that this is a multiple
dimensional coupled BSDEs system under Markovian framework, with unbounded
terminal value and the generator of the i equation is of linear growth on Zi
th
component ω by ω, or in another word, it is of stochastic linear growth.
3 Existence of solutions for the multiple dimensional coupled
BSDEs system
In this section, we will provide an existence result of BSDEs (2.5) when n = 2 as
anexample.Actually,the caseforn>2canbedealtwithinthesamewaywithout
any difficulties.
3.1 Measure domination
Before we state our main theorem, let us first recall a result related to measure
domination.
MuandWu Page5of14
Definition 3.1 : Lq-Domination condition
Let q ∈ (1,∞) be fixed. For a given t ∈ [0,T], a family of probability measures
1
{ν (s,dx),s∈[t ,T]} defined on Rm is said to be Lq- dominated by another family
1 1
of probability measures {ν (s,dx),s∈[t ,T]}, if for any δ ∈(0,T −t ], there exists
0 1 1
an application φδ :[t +δ,T]×Rm →R+ such that:
t1 1
(i) ν (s, dx)ds= φδ (s, x)ν (s, dx)ds on [t +δ,T]× Rm.
1 t1 0 1
(ii) ∀k ≥1, φδ (s,x)∈Lq([t +δ,T]×[−k,k]m; ν (s,dx)ds).
t1 1 0
Lemma 3.1 Let x ∈ Rm, (t,x) ∈ [0,T]×Rm, s ∈ (t,T] and µ(t,x;s,dy) the
0
law of Xt,x, i.e.,
s
∀A∈B(Rm), µ(t,x;s,A)=P(Xt,x ∈A).
s
UnderAssumption2.1onσ,foranyq ∈(1,∞),thefamilyoflaws{µ(t,x;s,dy),s∈
[t,T]} is Lq-dominated by {µ(0,x ;s,dy),s∈[t,T]} for fixed x ∈Rm.
0 0
Proof See [15], Lemma 4.3 and Corollary 4.4, pp. 14-15.
3.2 High dimensional coupled BSDEs system
Our main result in this section is the following theorem.
Theorem 3.1 Let x ∈ Rm be fixed. Then under Assumptions 2.1, 2.2 and 2.3,
there exist two pairs of P-measurable processes (Yi,Zi) with values in R1+m, i =
1,2, and two deterministic functions ςi(t,x) which are of polynomial growth, i.e.
|ςi(t,x)|≤C(1+|x|γ) with γ ≥0, i=1,2 such that,
P-a.s.,∀t≤T,Yi =ςi(t,X0,x) and Ziis dt-square integrable P-a.s.;
t t
 T T
Yt1 =g1(Xt0,x)+ZtT H1(s,Xs0,x,Ys1,Ys2,Zs1,Zs2)ds−ZtT Zs1dBs; (3.1)
Y2 =g2(X0,x)+ H (s,X0,x,Y1,Y2,Z1,Z2)ds− Z2dB .
t t 2 s s s s s s s
 Zt Zt
The result holds true as well for case i=n>2 following the same way.
Proof Thestructureofthisproofisasfollows.Wefirstusethemollifytechniqueon
the generator H , to construct a sequence of BSDEs with generators which are of
i
Lipschitz continuous. 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 (3.1).
Step 1. Approximation.
Let ξ be an element of C∞(R2+2m,R) with compact support and satisfies:
ξ(y1,y2,z1,z2)dy1dy2dz1dz2 =1.
ZR2+2m
MuandWu Page6of14
For (t,x,y1,y2,z1,z2)∈[0,T]×Rm×R2+2m, we set,
H˜ (t,x,y1,y2,z1,z2)
1n
= n4H (s,ϕ (x),p1,p2,q1,q2)·
1 n
ZR2+2m
ξ n(y1−p1),n(y2−p2),n(z1−q1),n(z2−q2) dp1dp2dq1dq2,
(cid:0) (cid:1)
where the truncation function ϕ (x) = ((x ∨ (−n)) ∧ n) , for x =
n j j=1,2,...,m
(x ) ∈Rm. We next define ψ ∈C∞(R2+2m,R) by,
j j=1,2,...,m
1, |y1|2+|y2|2+|z1|2+|z2|2 ≤1,
ψ(y1,y2,z1,z2)=
(0, |y1|2+|y2|2+|z1|2+|z2|2 ≥4.
Then, we define the measurable function sequence (H ) as follows:
1n n≥1
∀(t,x,y1,y2,z1,z2)∈[0,T]×Rm×R2+2m,
y1 y2 z1 z2
H (t,x,y1,y2,z1,z2)=ψ( , , , )H˜ (t,x,y1,y2,z1,z2)
1n 1n
n n n n
We have the following properties:
(a) H is uniformly lipschitz w.r.t (y1,y2,z1,z2);
1n
 (b) |H (t,x,y1,y2,z1,z2)|≤C (1+|ϕ (x)|)|z1|+C (1+|ϕ (x)|γ +|y1|);
 ((dc))|FHo11rnna(nt,yx(,ty,1x,)y∈2,z[01,,Tz2])×|≤Rcmn2,,afnordaKnnya(ct,oxm,pya1,cyt2s,uhzb1s,ezt2)o;f Rn 2+2m,
sup H (t,x,y1,y2,z1,z2)−H (t,x,y1,y2,z1,z2) →0,
 a(ys1,ny2→,z1,∞z2).∈K(cid:12)(cid:12) 1n 1 (cid:12)(cid:12) (3.2)
The construction of the approximating sequence (Hn) is carried out in the
2 n≥1
same way.
Foreachn≥1and(t,x)∈[0,T]×Rm,sinceH andH areuniformlylipschitz
1n 2n
w.r.t (y1,y2,z1,z2), by the result of Pardoux-Peng(see [21]), we know, there exist
two pairs of processes (Y¯in;(t,x),Z¯in;(t,x)) ∈ S2 (R)×H2 (Rm), i = 1,2, which
t,T t,T
satisfy, for s∈[t,T],
T
Y1n;(t,x) =g1(Xt,x)+ H (r,Xt,x,Y1n;(t,x),Y2n;(t,x),Z1n;(t,x),Z2n;(t,x))dr
s T 1n r r r r r
 Zs
Y2n;(t,x) =g2(Xt,x)−+ZsTTZHr1n;((rt,,xX)dtB,xr,;Y1n;(t,x),Y2n;(t,x),Z1n;(t,x),Z2n;(t,x))dr
s T 2n r r r r r
Zs
 −ZsT Zr2n;(t,x)dBr.
MuandWu Page7of14
(3.3)
Meanwhile, the properties (3.2)-(a),(c) and the result of El-karoui et al. (ref.
[11]) yield that, there exist two sequences of deterministic measurable applications
ς1n(resp. ς2n):[0,T]×Rm →R and z1n(resp. z2n):[0,T]×Rm →Rm such that
for any s∈[t,T],
Y1n;(t,x) =ς1n(s,Xt,x) (resp. Y2n;(t,x) =ς2n(s,Xt,x)) (3.4)
s s s s
and
Z1n;(t,x) =z1n(s,Xt,x) (resp. Z2n;(t,x) =z2n(s,Xt,x)).
s s s s
Besides, we have the following deterministic expression: for i=1,2, and n≥1,
T
ςin(t,x)=E gi(Xt,x)+ F (s,Xt,x)ds , ∀(t,x)∈[0,T]×Rm, (3.5)
T in s
h Zt i
where,
F (s,x)=H (s,x,ς1n(s,x),ς2n(s,x),z1n(s,x),z2n(s,x)).
in in
Step 2. Uniform integrability of (Y1n;(t,x),Z1n;(t,x)) for fixed (t,x) ∈ [0,T]×
n≥1
Rm.
For each n≥1, let us first consider the following BSDE:
T
Y¯1n =g1(Xt,x)+ C (1+|ϕ (Xt,x)|)|Z¯1n|
s T 2 n r r
Zt
T
+C (1+|ϕ (Xt,x)|γ +|Y¯1n|)dr− Z¯1ndB . (3.6)
h n r r r r
Zs
For any x ∈ Rm and n ≥ 1, the mapping (y1,z1) ∈ R × Rm 7→ C (1 +
2
|ϕ (Xt,x)|)|z1|+C (1+|ϕ (Xt,x)|γ +|y1|) is Lipschitz continuous, therefore, the
n r h n r
solution (Y¯1n,Z¯1n) ∈ S2 (R)×H2 (Rm) exists and is unique. Moreover, it fol-
t,T t,T
lows from the result of El-Karoui et al. (see [11]) that, Y¯1n can be characterized
throughadeterministic measurablefunctionς¯1n :[0,T]×Rm→R,thatis,forany
s∈[t,T],
Y¯1n =ς¯1n(s,Xt,x). (3.7)
s s
Next let us consider the process
s
Bn =B − 1 (r)C (1+|ϕ (Xt,x)|)sign(Z˜1n)dr, 0≤s≤T,
s s [t,T] 2 n r r
Z0
which is, thanks to Girsanov’s Theorem, a Brownian motion under the proba-
bility Pn on (Ω,F) whose density with respect to P is E := E ( T C (1 +
T T 0 2
R
MuandWu Page8of14
|ϕ (Xt,x)|)sign(Z¯1n)1 (s)dB ), where for any z = (zi) ∈ Rm, sign(z) =
n s s [t,T] s i=1,...,d
(1 zi ) and E (·) is defined by
[|zi|6=0]|zi| i=1,...,d t
E(M):=(exp{M −hMi /2}) , (3.8)
t t t≤T
for any (F ,P)-continuous local martingale M =(M ) . Then (3.6) becomes
t t t≤T
T T
Y¯1n =g1(Xt,x)+ C (1+|ϕ (Xt,x)|γ+|Y¯1n|)dr− Z¯1ndBn, s∈[t,T]. (3.9)
s T h n r r r r
Zt Zs
Then, by using Itoˆ’s formula to eChtY¯1n, t≤T, we know,
t
T T
eChtY¯t1n =eChTg1(XTt,x)− CheChr(1+|ϕn(Xrt,x)|γ)dr+ eChrZ¯r1ndBrn.
Zt Zt
Considering (3.7), we have,
T
ς¯1n(t,x)=e−ChtEn eChTg1(XTt,x)+ CheChs(1+|ϕn(Xst,x)|γ)ds Ft ,
" Zt (cid:12) #
(cid:12)
(cid:12)
where En is the expectation under the probability Pn. Taking the expectation on
both sides under the probability Pn and taking account of ς¯1n is deterministic, we
obtain,
T
ς¯1n(t,x)=e−ChtEn eChTg1(XTt,x)+ CheChs(1+|ϕn(Xst,x)|γ)ds .
" Zt #
Since, g1 is of polynomial growth, and e−Cht ≤ 1 for t ∈ [0,T], we infer that,
∀(t,x)∈[0,T]×Rm,
|ς¯1n(t,x)|≤CEn sup 1+|Xt,x|γ
s
(cid:20)t≤s≤T (cid:21)
(cid:0) (cid:1)
≤CE E · sup 1+|Xt,x|γ ,
T s
(cid:20) t≤s≤T (cid:21)
(cid:0) (cid:1)
where the constant C depends only on T, C and C . It follows from a result of
h g
Haussmann([16], p.14,see also[15], Lemma 3.1)that, there exists some p ∈(1,2)
0
and a constant C which is independent on n, such that, E[|ET|p0] ≤ C uniformly.
As a result of Young’s inequality and estimate (2.3), we have,
|ς¯1n(t,x)|≤Cp0{E[|ET|p0]+E sup 1+|Xst,x|pp00−γ1 },
(cid:20)t≤s≤T(cid:16) (cid:17)(cid:21)
which yields,
|ς¯1n(t,x)|≤C(1+|x|λ) with λ=p γ/(p −1)>2. (3.10)
0 0
MuandWu Page9of14
Next by the comparison theorem of BSDEs and property (3.2)-(b), we actually
have, for any s∈[t,T],
Y1n;(t,x) =ς1n(s,Xt,x)≤Y¯1n =ς¯1n(s,Xt,x),
s s s s
and by choosing s = t, we get that ς1n(t,x) ≤ C(1+|x|λ), (t,x) ∈ [0,T]×Rm.
In a similar way, we can also show that for any (t,x) ∈ [0,T]×Rm, ς1n(t,x) ≥
−C(1+|x|λ).As a conclusion,ς1n is ofpolynomialgrowthon(t,x) uniformly inn,
i.e. there exists constants C which is independent of n, and λ>2 such that,
|ς1n(t,x)|≤C(1+|x|λ). (3.11)
Combining (3.11) and (2.3), we deduce that, for any α>1, i=1,2,
E sup |Yin;(t,x)|α ≤C. (3.12)
s
(cid:20)t≤s≤T (cid:21)
On the other hand, by applying Itoˆ’s formula with (Yin;(t,x))2 and considering the
uniform estimate (3.12), we can infer in a regular way that, for any t ∈ [0,T],
i=1,2,
T
E |Zin;(t,x)|2ds ≤C. (3.13)
s
"Zt #
Step 3. There exists a subsequence of ((Y1n;(0,x),Z1n;(0,x)) ) which con-
s s 0≤s≤T n≥1
verges respectively to (Y1,Z1) , solution of the BSDE (3.1). Let us recall the
s s 0≤s≤T
expression (3.5) for case i = 1, and apply property (3.2)-(b) combined with the
uniform estimates (3.12), (3.13), (2.3) and the Young’s inequality to show that, for
1<q <2,
T
E |F (s,X0,x)|qds
1n s
(cid:20)Z0 (cid:21)
T
≤CE (1+|ϕ (X0,x)|)q|Z¯1n;(0,x)|q+(1+|ϕ (X0,x)|γq+|Y¯1n;(0,x)|q)ds
n s s n s s
"Z0 #
T
≤C{E |Z¯1n;(0,x)|2ds +E sup |Y¯1n;(0,x)|q +1}
s s
"Z0 # (cid:20)0≤s≤T (cid:21)
≤C.
(3.14)
Therefore, there exists a subsequence {n } (for notation simplification, we still
k
denoteitby{n})andaB([0,T])⊗B(Rm)-measurabledeterministicfunctionF (s,y)
1
such that:
F →F weakly in Lq([0,T]×Rm;µ(0,x;s,dy)ds). (3.15)
1n 1
MuandWu Page10of14
Next we aim to prove that (ς1n(t,x)) is a Cauchy sequence for each (t,x) ∈
n≥1
[0,T]×Rm. Now let (t,x) be fixed, η >0, k, n and m≥1 be integers. From (3.5),
we have,
T
|ς1n(t,x)−ς1m(t,x)|= E F (s,Xt,x)−F (s,Xt,x)ds
1n s 1m s
(cid:12) hZt i(cid:12)
(cid:12) t+η (cid:12)
≤(cid:12)E F (s,Xt,x)−F (s,Xt,x) d(cid:12)s
1n s 1m s
hZt T(cid:12) (cid:12) i
+ E (cid:12) F (s,Xt,x)−F (s,Xt,(cid:12)x) .1 ds
1n s 1m s {Xst,x|≤k}
(cid:12)(cid:12) hZt+Tη(cid:0) (cid:1) i(cid:12)(cid:12)
+(cid:12)E F (s,Xt,x)−F (s,Xt,x) .1 ds(cid:12),
1n s 1m s {|Xst,x|>k}
(cid:12)(cid:12) hZt+η(cid:0) (cid:1) i(cid:12)(cid:12)
(cid:12) (cid:12)
where on the right side, noticing (3.14), we obtain,
t+η
E |F (s,Xt,x)−F (s,Xt,x)|ds
1n s 1m s
hZt i
T
≤ηq−q1{E |F1n(s,Xst,x)−F1m(s,Xst,x)|qds }q1 ≤Cηq−q1.
hZ0 i
q
Atthesametime,Lemma3.1associateswiththeLq−1-dominationpropertyimplies:
T
E F (s,Xt,x)−F (s,Xt,x) .1 ds
(cid:12)(cid:12) hZt+η(cid:0) 1n sT 1m s (cid:1) {|Xst,x|≤k} i(cid:12)(cid:12)
(cid:12) = (F (s,η)−F (s,η)).1 µ(cid:12)(t,x;s,dη)ds
1n 1m {|η|≤k}
(cid:12)ZRmZt+η (cid:12)
(cid:12) T (cid:12)
=(cid:12) (F (s,η)−F (s,η)).1 φ (s,η)µ(0,x(cid:12);s,dη)ds .
1n 1m {|η|≤k} t,x
(cid:12)ZRmZt+η (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Since φt,x(s,η) ∈ Lq−q1([t+η,T]×[−k,k]m; µ(0,x;s,dη)ds), for k ≥ 1, it follows
from (3.15) that for each (t,x)∈[0,T]×Rm, we have,
T
E F (s,Xt,x)−F (s,Xt,x) ds →0 as n,m→∞.
hZt+η 1n s 1m s 1{|Xst,x|≤k} i
(cid:0) (cid:1)
Finally,
T
E F (s,Xt,x)−F (s,Xt,x) . ds
(cid:12)(cid:12) hZt+η(cid:0) 1nT s 1m s (cid:1) 1{|XTst,x|>k} i(cid:12)(cid:12)
(cid:12) ≤≤CC{kE−hq−qZ1t.+η1{|Xst,x|>k}dsi}q−q1{EhZt+η(cid:12)(cid:12)F1n(s,X(cid:12)st,x)−F1m(s,Xst,x)(cid:12)(cid:12)qdsi}q1
Therefore, for each (t,x) ∈ [0,T]×Rm, (ς1n(t,x)) is a Cauthy sequence and
n≥1
then there exists a borelian application ς1 on [0,T] × Rm, such that for each
(t,x)∈[0,T]×Rm, lim ς1n(t,x)=ς1(t,x), which indicates that for t∈[0,T],
n→∞