Homogenization of reflected semilinear PDE with nonlinear Neumann boundary condition

0.27 MB·English

by  Auguste Aman

Homogenization of re(cid:29)e
ted semilinear PDE with
nonlinear Neumann boundary
ondition.
9
0
0
2 ∗ †
Auguste Aman and Modeste N'Zi

n
a UFR de Mathématiques et Informatique,
J
22 BP 582 Abidjan 22, Cte d'Ivoire
5
1

]
R
P
. Abstra
t
h
t
a
We study the homogenization problem of semi linear re(cid:29)e
ted partial di(cid:27)erential equa-
m
tions (re(cid:29)e
ted PDEs for short) with nonlinear Neumann
onditions. The non-linear term
[
is a fun
tion of the solution but not of its gradient. The proof are fully probabilisti
and

3
uses weak
onvergen
e of asso
iated re(cid:29)e
ted generalized ba
kward di(cid:27)erential sto
hasti
v
6 equations (re(cid:29)e
ted GBSDEs in short).
8
9
2
MSC Subje
t Classi(cid:28)
ation: 60H30; 60H20; 60H05
.
2 Key Words: Re(cid:29)e
ted ba
kward sto
hasti
di(cid:27)erential equations; homogenization of PDEs;
1
7 vis
osity solution of PDEs; obsta
le problem.
0
:
v
i
X
1 Introdu
tion
r
a
Lε
Let x beanuniformlyellipti
se
ond-orderpartialdi(cid:27)erentialoperatorindexed byaparameter
ε > 0 ε 0
. The homogenization problem
onsists in analyzing the behavior as ↓ , of the
u Lεu = f Θ d
solution ε for the partial di(cid:27)erential equations (PDEs): x ε in a domain of IR .
Lε
The
oe(cid:30)
ients of the operator x are subje
t to an appropriate
onditions. For example,
periodi
ity by Freidlin [5℄, almost periodi
ity by Krylov [9℄. In this paper we shall
onsider
the
ase of lo
ally periodi

oe(cid:30)
ients introdu
ed by Ouknine et al. [13℄. In the probabilisti
approa
h the problem be
omes the following. What is the limit of the laws of the di(cid:27)usion
Xε Lε ε 0
pro
esses withgenerator x, as ↓ ? Thisproblemhasbeen studied fordi(cid:27)usionpro
esses
d
in the whole of IR by Krylov[9℄, Freidlin[5℄ and Bensoussan[1℄. In the
ase of the presen
e of
∗
E-mail address: augusteaman5yahoo.fr(Corresponding author)
†
E-mail address: modestenziyahoo.fr
1
boundary
onditions, we
an see the work of Tanaka [18℄ and re
ently this one of Ouknine et
al.[13℄. In [15℄, Pardoux has
ombined the probabilisti
approa
h of
hapter 3 of [1℄ to derive
homogenization results for semi linear paraboli
partial di(cid:27)erential equation involving a se
ond
order di(cid:27)erential operator of paraboli
type in the whole spa
e with highly os
illating drift
and nonlinear term. Furthermore Pardoux and Ouknine [12℄ pursue the same program for a
semi-linear ellipti
PDE with nonlinear Neumann boundary
onditions.
This paper is devoted to prove a similar result for obsta
le problem of semi-linear ellipti
g
partialdi(cid:27)erential equations (PDEs in short) with a nonlinearNeumann boundary
onditions
satisfyingmoreoverlineargrowth
onditions. Weuseforthispurposethelinkbetweenba
kward
sto
hasti
di(cid:27)erential equations (BSDEs in short) and PDEs.
The
on
rete formulation of BSDEs in the nonlinear
ase was (cid:28)rst introdu
ed by Pardoux
[ ]
and Peng 14 who proved existen
e and uniqueness of adapted solutions for these equations,
under suitable
onditions on the
oe(cid:30)
ient and the terminal value. They provide probabilisti
formulas for solutions of semi linear PDEs whi
h generalized the well-known Feynman-Ka
formula. The interest of this kind of sto
hasti
equations has in
reased steadily, sin
e it has
been widely re
ognized that they provide a useful framework for the formulation of many
[ ]) [ ]
problemsin mathemati
al(cid:28)nan
e (see 6 , insto
hasti

ontrol and di(cid:27)erential games (see 7 ).
[ ]
A
lass of BSDEs
alled generalized BSDEs has been introdu
ed by Pardoux and Zhang in 16 .
This kind of BSDEs has an additional integral with respe
t to a
ontinuous in
reasing pro
ess
and is used to provide probabilisti
formulas for solutions of semi linear PDEs with nonlinear
Neumann boundary
onditions. Further, in order to provide probabilisti
formula for solutions
of obsta
le problem for PDEs with nonlinear Neumann boundary
onditions, Ren and Xia [17℄

onsidered re(cid:29)e
ted generalized BSDEs. Our goal in this paper is to give a homogenization
result of obsta
le problem for semi-linear PDEs with nonlinear Neumann boundary
onditions
via a weak
onvergen
e of re(cid:29)e
ted generalized BSDEs.
To des
ribe our result more pre
isely, we (cid:28)rst re
all some
lassi
al notations that will be
1( d)
used in the sequel of this paper. Let Cb IR be the spa
e of real
ontinuously di(cid:27)erentiable
Θ
fun
tdions su
h that thederivativesarρeboun1(dedd)and be a rρeg=ula0r
onΘv¯exρan>db0oundedddΘ¯omain
ρin(xI)R=. (Wd(xe,iΘ¯nt)r)o2du
ed a fun
tion ∈ CbΘ¯IR su
h that in , inΘIR \ and
in the neighborhood of . Let us note that, sin
e the domain is smooth,
ψ 2( d) d(x,∂Θ)
it is possible to
onsider an extension ∈ Cb IR of the fun
tion de(cid:28)ned on the
Θ ∂Θ Θ ∂Θ
restri
tion to of the neighborhood of su
h that and are
hara
terized by
Θ = x d : ψ(x) > 0 ∂Θ = x d : ψ(x) = 0
∈ IR and ∈ IR
(cid:8) (cid:9) (cid:8) (cid:9)
x ∂Θ ψ(x) Θ
and for all ∈ , ∇
oin
ide with the unit normal pointing toward the interior of
[ ] 3.1 ρ ψ
(see for example 10 , Remark ). In parti
ular we may and do
hoose and su
h that
ψ(x),δ(x) 0, x d δ(x) = ρ(x)
h∇ i ≤ for all ∈ IR , where ∇ whi
h we
all the penalization term.
Now let us
onsider the following re(cid:29)e
ted semi-linear partial di(cid:27)erential equations with Neu-
ε > 0)
mann boundary
ondition for :
2
min(uε(t,x) h(t,x),
−
 ∂uε(t,x) Lεuε f(x,uε(t,x)) = 0, if (t,x) Θ,
 − ∂t − x − ∈ IR+ ×


 (cid:1)
 Γεuε(t,x)+g(x,uε(t,x)) = 0, if (t,x) ∂Θ,
 x ∈ IR+ ×
 uε(0,x) = l(x), x Θ,

 ∈

 Γε ( )
where x is de(cid:28)ned in 2.2 . Under suitable linear growth, monotoni
and Lips
hitz
ondi-
f, g l uε(t,x)
tions on the
oe(cid:30)
ients and , we will show that the vis
osity solution of the
ε 0 u(t,x)
previous re(cid:29)e
ted PDEs
onverge, as goes to , to a fun
tion , the vis
osity solution to
the following re(cid:29)e
ted PDEs with nonlinear Neumann boundary
onditions:
min(u(t,x) h(t,x),
−

∂u(t,x) L0u f(x,u(t,x)) = 0, if (t,x) Θ,
 −∂t − x − ∈ IR+ ×


 (cid:1)
 Γ0u(t,x)+g(x,u(t,x)) = 0, if (t,x) ∂Θ,
 x ∈ IR+ ×

 u(0,x) = l(x), x Θ,

 ∈

 L0 Γ0 ( )
where operators x and x are in the form 2.4 .
2
The rest of this paper is organized as follows. In se
tion we re
all without proof, some
results due to Ouknine etal.[13℄ on the homogenizationof the re(cid:29)e
ted di(cid:27)usioninthe bounded
domain with lo
ally periodi

oe(cid:30)
ients and some basi
notations on generalized BSDEs. Se
-
3
tion is devoted to the statement and proof of our main result. Finally, in Se
tion 4, we give
a homogenization result for some PDEs with nonlinear Neumann boundary
onditions.
2 Lo
ally periodi
homogenization of re(cid:29)e
ted di(cid:27)usion
In this se
tion, we state some results whi
h were proved in Ouknine et al. [13℄, and whi
h we
Θ
will use in the rest of the paper. Let us re
all that is a regular
onvex and bounded domain
d;(d 1)
in IR ≥ de(cid:28)ned as previous and let
d d d
x 1 x x
Lε = a (x, )∂ ∂ + b (x, )∂ + c (x, )∂
x ij ε i j ε i ε i i ε i (2.1)
i,j i=1 i=1
X X X
and
d
x
Γε = ∂ ψ(x, )∂
x i ε i (2.2)
i=1
X
3
ε > 0 ∂ = ∂/∂x
i i
be given (with ), where . We also set
d d
L = a (x,y)∂ ∂ + b (x,y)∂ , x,y d
x,y ij i j i i
∈ IR
i,j i=1
X X
and require the following:
L a(x) = [a ] σ(x,y)σ∗(x,y)/2
x,y i,j
(H.1) is uniformly ellipti
and the matrix is fa
tored as
σ : d Θ d, b : d Θ d, c : d Θ d
(H.2) The fun
tions IR × → IR IR × → IR IR × → IR are lo
ally
1 Θ
periodi
(i.e, periodi
with respe
t to se
ond variable; of period in ea
h dire
tion in ).
(i) C ζ = σ,b,c,
Global Lips
hitz
ondition: there exists a
onstant su
h that for any
ζ(x,y) ζ(x′,y′) C( x x′ + y y′ ), x,x′ d, y,y′ Θ,
 k − k ≤ k − k k − k ∀ ∈ IR ∈


 (ii) ∂ ζ(x,y) ∂2 ζ(x,y)
 The partial derivatives x as well as the mixed derivatives xy
 , x d,y Θ ζ = a,b,c

 exist and are
ontinuous ∀ ∈ IR ∈ for
(iii) C

 ζ = a,Tb,hce
oe(cid:30)ζ
(ixe,nyt)s areCb,ouxnded,dt,hyat isΘ,.there exists a
onstant su
h that for any

 k k ≤ ∈ IR ∈



 Lε Θ Γεu = 0 ∂Θ
The di(cid:27)erential operator x insideXε togΘ¯ether with the bound(aLrεy,Γ
oεn)dition x on d
x x
determine a unique di(cid:27)usion pro
ess in whi
h we
all the -di(cid:27)usion. Given a -
(W : t 0) (Ω, , )
t
dimensional Brownian motion ≥ de(cid:28)ned on a
omplete probability spa
e F IP ,
(Xε,Gε) Θ
+
let be the unique solution with values in ×IR of the following re(cid:29)e
ted SDE:
dXε = 1b(Xε, Xtε)dt+c(Xε, Xtε)dt+σ(Xε, Xtε)dW + ψ(Xε)dGε, t 0
t ε t ε t ε t ε t ∇ t t ≥

Xε Θ, Gε ,
 t ∈ is
ontinuous and in
reasing su
h that


 0t∇ψ(Xsε)1{Xsε∈Θ}dGεs = 0, t ≥ 0 (2.3)
 R
 Xε = x.
 0



 L d L y
x,y x,y
By requirement there exists a -di(cid:27)usion on IR with generator and by -periodi
ity
Ux d
assumTptdion on the
oe(cid:30)
ients this pro
ess indu
es di(cid:27)usion pro
esms (x,.)on the -dimensional
torus , moreover this di(cid:27)usion pro
ess is ergodi
. We denote by its unique invariant
Lε ε 0
measure. In order for the pro
ess with generator x to have a limit in law as ↓ , we need
the following
ondition to be in for
e.
(H.3) x
Centering
ondition: for all ,
b(x,u)m(x,du) = 0.
ZΘ
4
Let us denote
b
1
b
 2 
b.
b =  b. 
 
 
.
 
 
b .
 
 
 b 
d
 
 
∞
bk(x,u) = bk(x,Ux) dt b , Ux u
with IEu{ t } , where under IPu starts from .
Z0
b
We put
d d d
L0 = Aij(x)∂ + Ci(x)∂ , Γ0 = γ0(x)∂ ,
x 0 xixj 0 xi x i xi (2.4)
i,j=1 i=1 i=1
X X X
where the
oe(cid:30)
ients are respe
tively de(cid:28)ned by
1
C (x,y) = ∂ bb+(I +∂ b)c+ Tr∂2 bσσ∗ (x,y),
0 x y 2 xy
(cid:20) (cid:21)
b b b
S (x,y) = (I +∂ b)σ , A (x,y) = S S∗(x,y),
0 y 0 0 0
h i
C (x) = C (x,u)m(x,du), Ab(x) = A (x,u)m(x,du),
0 0 0 0
Td Td
Z Z
γ0(x) = (I +∂ b) ψ (x,y)dm(x,dy).
y
∇
Td−1
Z
h i
b
e
Let us
onsider the following SDE:
dX = A1/2(X )dW +C (X )dt+γ0(X )dG , t 0,
t 0 t t 0 t t t ≥

 X Θ, G ,

 ∈ is
ontinuous and in
reasing



 0tγ0(Xs)1{Xs∈Θ}dGs = 0, t ≥ 0,
 R
 X0 = 0.



 L0 Γ0 x
The operators and are a
ting on .
We
an now state the main result of Ouknine et al. [13℄ (see Theorem 3.1), whi
h is a
generalization of a result of Tanaka [18℄ (see Theorem 2.2).
5
(H1) (H3) (Lε,Γε) Xε
x x
Theorem 2.1 Under assumptions - , the -di(cid:27)usion pro
ess
onverges in
(L0,Γ0) X ε 0
law to an x x -di(cid:27)usion as → .
Moreover,
(Xε,MXε,Gε) = (X,MX,G),
⇒
MX MXε X Xε G Gε
where (resp., ) is the martingale part of (resp. ), (resp., ) is the lo
al time
ψ(X) ψ(Xε) 0
of (resp., ) at and
= ([0,T], d) ([0,T], d) ([0,T], d)
" ⇒" designates the weak
onvergen
e in the spa
e C IR ×D IR ×C IR
([0,T], d)
endowed with the produ
t topology of the uniform norm for the spa
e C IR and the
S ([0,T], d)
-topology for D IR .
It follows moreover easily from the result of Tanaka [18℄ that
(H1) (H3),
Lemma 2.1 Under assumptions −
sup sup Xε p +Gε < ,
IE | s| t ∞
ε (cid:18)0≤s≤t (cid:19)
p 1
for any ≥ .
3 Re(cid:29)e
ted generalized BSDEs and weak
onvergen
e
3.1 Preliminaries and statement of the problem
Let us introdu
e some spa
es:
H2,d(0,t) = ψ ; 0 s t : d
s
{ ≤ ≤ } IR -valued predi
table pro
ess
t
(cid:8) ψ(s) 2ds < ,
su
h thatIE | | ∞
Z0 (cid:27)
S2(0,t) = ψ ; 0 s t :
s
{{ ≤ ≤ } real-valued, progressively measurable pro
ess
sup ψ(s) 2 < ,
su
h thatIE | | ∞
(cid:20)0≤s≤t (cid:21) (cid:27)
A2(0,t) = K ; 0 s t ,
s
{{ ≤ ≤ }real-valued adapted,
ontinuous,
K(0) = 0, K(t) 2 < .
and in
reasing pro
ess su
h that IE| | ∞
(cid:9)
In addition, we give the following assumptions:
f,g : Θ
Let ×IR →CIR> b0,epa
o1n,tiµnuous fun
tiβon<s0atisfying the followixng aΘ¯ss,uym,pyti′ons: ,
There exist
onstants ≥ ∈ IR and , su
h that for all ∈ ∈ IR
6
(f.i) (y y′)(f(x,y) f(x,y′)) µ y y′ 2,
− − ≤ | − |

(f.ii) f(x,y) C( x p + y 2),

 | | ≤ | | | |
( )

H.4  (g.i) (y y′)(g(x,y) g(x,y′)) β y y′ 2,

− − ≤ | − |
 (g.ii) g(x,y) C( x p + y 2).

 | | ≤ | | | |
Moreover, let l : Θ → IR be a
ontinuous fun
tion and h ∈ C1,2(IR+ ×Θ) for whi
h there exist
C
a
onstant su
h that
(i) l(x) C(1+ x p),
≤ | |

H.5 (ii) h(t,x) C(1+ x p),

 ≤ | |



(iii) h(0,x) l(x).
p ≤ (H4)

where is obtained in .
(t,x) Θ (Yε,Zε,Kε)
For ea
h (cid:28)xed ∈ IR+ × , we
onsider s s s 0≤s≤t, the unique solution of
re(cid:29)e
ted generalized BSDE
Yε = l(Xε)+ tf(Xε,Yε)dr+ tg(Xε,Yε)dGε +Kε Kε tZεdMXε,
s t s r r s r r r t − s − s r r
 R R R
Yε h(s,Xε),
 s ≥ s (3.1)


Kε Kε = 0 t(Yε h(s,Xε))dKε = 0.
 is nonde
reasing su
h that 0 and 0 s − s s


 R ( )
Let us note that existen
e and uniqueness of a solution to 3.1 is proved in [17℄.
(Y ,Z ,K )
s s s 0≤s≤t
Furthermore, we also
onsider the unique solution of the re(cid:29)e
ted BSDE
Y = l(X )+ tf(X ,Y )dr + tg(X ,Y )dG tZ dMX +K K
s t s r r s r r r − s r r t − s
 R R R
Y h(s,X )
 s s
 ≥ (3.2)


t
K K = 0 (Y h(s,X ))dK = 0.
 is nonde
reasing su
h that 0 and 0 s − s s


 R
Throughout the rest of this paper, we put:
s s
Mε = ZεdMXε, M = Z dMX; 0 s t
s r s r r ≤ ≤
Z0 Z0
(Y,M,K) (Yε,Mε,Kε) D([0,t]; )
and we
onsider (resp., ) as a randomelement of the spa
e IR ×
D([0,t]; ) C([0,t]; ) D([0,t]; )
+
IR × IR . IR denotes the Skorohod spa
e (the spa
e of "
adlag"
S C([0,t]; )
+
fun
tions), endowed with the so-
alled -topologyof Jakubowski [8℄ and IR the spa
e
[0,t] d
of fun
tions of with values in IR equipped with the topology of uniform
onvergen
e.
7
3.2 Weak
onvergen
e
The goal of this se
tion is to prove the following result
(H1) (H4) (Yε,Mε,Kε)
ε>0
Theorem 3.1 Under
onditions - the family of pro
esses
onverge
(Y,M,K) D([0,t], ) D([0,t], ) C([0,t]; ). ε 0
+
in law to on the spa
e IR × IR × IR as → .
Moreover,
limYε = Y .
ε→0 0 0 in IR
The proof ofthis theoremrelies on fourlemmas. The (cid:28)rst oneisa simpli(cid:28)edversion ofTheorem
25)
4.2 in (Billingsley [2℄, p. so we omit the proof. We note also that the
onvergen
e is taken
in law.
(Uε)
Lemma 3.1 Let ε>0 be a family of random variables de(cid:28)ned on the same probability spa
e.
ε 0 (Uε,n)
For ea
h ≥ , we assume the existen
e of a family random variables n≤1, su
h that:
(i) n 1, Uε,n = U0,n ε
For ea
h ≥ ⇒ as goes to zéro and
(ii) U0,n = U0 n +
⇒ as −→ ∞.
(iii) Uε,n = Uε n + ε
We suppose further that ⇒ as −→ ∞, uniformly in
Uε U0
Then,
onverges in law to .
For the last third lenmmNa∗s,, let us introdu
e the penalization method to
onstru
t a sequen
e of
GBSDE. For ea
h ∈ we set
f (x,y) = f(x,y)+n(y h(s,x))−
n
−
n ∗
Thanks to the result of Pardoux and Zhang [16℄, for ea
h ∈ IN , there exists a unique pair of
(Yε,n,Zε,n) d
t
F −progressively measurable pro
esses with values in IR×IR satisfying
t
sup Yε,n 2 + Zε,n 2ds <
IE | | | s | ∞
(cid:20)0≤s≤t Z0 (cid:21)
and
t t
Yε,n = l(Xε)+ f (Xε,Yε,n)dr+ g(Xε,Yε,n)dGε
s t n r r r r r
Zs Zs
(Mε,n Mε,n),
− t − s (3.3)
where s
Mε,n = Zε,ndMXε.
s r r
Z0
C
From now on, is a generi

onstant that may vary from line to another.
8
(H1) (H4) n (Y0,n,M0,n)
Lemma 3.2 Assume - . Then for any , there exists the unique pro
ess
(Yε,n,Mε,n) D([0,T], ) D([0,T], )
su
h that the family of pro
esses ε>0
onverges to it in IR × IR ,
ε 0
as goes to .
ε n
Proof. Step 1. A priori estimate uniformly in and
Yε,n 2
Applying It's formula to the fun
tion | t | , we get
t t
Yε,n 2 + Zε,n 2dr = l(Xε) 2 +2 Yε,nf (Xε,Yε,n)dr
| s | | r | | t | r r r
Zs Zs
t
+2 Yε,ng(Xε,Yε,n)dGε
r r r r
Zs
t t
2 Yε,nZε,ndMXε +2 Yε,ndKε,n,
− r r r r r (3.4)
Zs Zs
where
s
Kε,n = n [Yε,n h(r,Xε,n)]−dr
s r − r (3.5)
Z0
(H3) (H4) tYε,ndKε,n th(r,Xε)dKε,n
By using assumptions and , the inequality s r r ≤ s r r and
γ > 0
Young's inequality, we get for every R R
t t
Yε,n 2 + Zε,n 2dr 1+C Xε 2p + (2µ+1) Yε,n 2 +C2 Xε 2q dr
IE| s | | r | ≤ IE| t| IE | r | | r|
Zs Zs
(cid:0) t (cid:1) (cid:8) 2C2 (cid:9)
+ (2β +γ) Yε,n 2 + Xε 2q dGε
IE | r | γ | r| r
Zs (cid:26) (cid:27)
t
+ h(r,Xε)dKε,n.
IE r r
Zs
(h.i) γ = β δ > 0
By virtue of assumption , Young's equality and
hoosing | | we obtain for every
t t t
Yε,n 2 + Zε,n 2dr + β Yε,n 2dGε C(1+ Yε,n 2dr)
IE | s | | r | | | | r | r ≤ IE | r |
(cid:18) Zs Zs (cid:19) Zs
1
+ sup h(s,Xε) 2
δIE | s |
(cid:18)0≤s≤t (cid:19)
+δ (Kε,n Kε,n)2.
IE t − s
But
t
(Kε,n Kε,n)2 C Yε,n 2 + l(Xε) 2 + ( Xε 2q + Yε,n 2)dr
IE t − s ≤ IE | s | | t | | r| | r |
(cid:18) Zs (cid:19)
t
+C ( Xε 2q + Yε,n 2)dGε.
IE | r| | r | r
Zs
9
1 β
δ = inf , | | ,
So, if 2C 2C
(cid:26) (cid:27)
t t t
Yε,n 2 + Zε,n 2dr + Yε,n 2dGε C 1+ Yε,n 2dr
IE| s | IE | r | IE | r | r ≤ IE | r |
Zs Zs (cid:18) Zs (cid:19)
and it follows from the Gronwall's lemma that
t t
Yε,n 2 + Zε,n 2dr+ Yε,n 2dGε + Kε,n 2 C, 0 s t, n ∗, ε > 0.
IE | s | | r | | r | r | t | ≤ ≤ ≤ ∀ ∈ IN ∀
(cid:18) Zs Zs (cid:19)
Finally, Burkhölder-Davis-Gundy inequality applying to (3.4) gives the following:
t t
sup sup sup Yε,n 2 + Zε,n 2dr+ Yε,n 2dGε + Kε,n 2 C.
n∈ ∗ ε>0 IE(cid:18)0≤s≤t| s | Z0 | r | Z0 | r | r | t | (cid:19) ≤
IN
Step 2: Tightness
S
We adopt the point of view of the -topology of Jakubovski [8℄. In fa
t we de(cid:28)ne the
Yε,n [0,t]

onditional variation of the pro
ess on the interval as the quantity
CV (Yε,n) = sup (Yε,n Yε,n/ ε)
t IE |IE ti+1 − ti Fti |
π !
i
X
π : 0 = t < t < ... < t = t
0 1 m
where the supremum is taken over all partitions of the interval
[0,t]
.
Clearly
t t
CV (Yε,n) f(Xε,Yε,n) dr+ g(Xε,Yε,n) dGε + Kε,n ,
t ≤ IE | r r | | r r | r | t |
(cid:18)Z0 Z0 (cid:19)
1 (H3)
and it follows from step and that
sup CV (Yε,n)+ sup Yε,n + sup Mε,n + Kε,n 2 < + .
t IE | s | | s | | t | ∞
ε (cid:18) (cid:18)0≤s≤t 0≤s≤t (cid:19) (cid:19)
Let us denote
s
Hε,n = g(Xε,Yε,n)+n[Yε,n h(r,Xε)]− dGε.
s r r r − r r
Z0
(cid:0) (cid:1)
(H3)
By virtue of step 1 and , we have
sup CV (Hε,n)+ sup Hε,n < + .
t IE | s | ∞
ε (cid:18) (cid:18)0≤s≤t (cid:19)(cid:19)
10