approximation par projections et simulations de monte ... · approximation par projections et...

Approximation par projections et simulations de

Monte-Carlo des equations differentielles stochastiques

retrogrades.

Jean-Philippe Lemor

To cite this version:

Jean-Philippe Lemor. Approximation par projections et simulations de Monte-Carlo desequations differentielles stochastiques retrogrades.. Mathematiques [math]. Ecole Polytech-nique X, 2005. Francais. <pastel-00001396>

HAL Id: pastel-00001396

https://pastel.archives-ouvertes.fr/pastel-00001396

Submitted on 27 Jul 2010

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E YNEE3D d7a

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E

(

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1

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_ E

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t + dt

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G

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=

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# '#(- $.% '

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¸ (! No@oKLV!-! S

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_ M

°7 ¸ ³δt

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YT = ξ,

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[0, T [·

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dYt = [rYt + δt(µ− r)]dt+ δtσdWt − (R− r)(Yt − δt)−dt

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3# EJ! Q¶QE

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∂xi∂xj+

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[0, T ]!5# E !"!

N!QEJ

(tk = kh)0≤k≤N−1

h = T

N

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T

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ME D/!gI@ ¤ ª¬E ª K6! KL·!%! K66!DV6D N

α0,k

α1,k

·Ytk ≈ α0,k.p0,k(Xtk)

Ztk ≈ α1,k.p1,k(Xtk)

p0,k

·p1,k

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∆Wk = Wtk+1

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tk!5QEd!aD·

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1

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))

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)

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Ntk

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tk (αM0,k, αM1,k)

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I!QEE KLEJ

Y N,I,I,Mtk

± EJ K6 D ¡ $EKLN¬! ! aD·¬!PD£EJNC!5E:E"´!D ± EJ K6!¡E:QEJ "´´ N EK6N%aK6" E!:G→→→ AWb[ , [aUQ]^[ .U), [ fW ¨"/

αM0,N .p0,N (·) = φ(·) →→→ A , JNhU, [^fW hKJ ,Ch f A<h U ¶SgSW ,CSVeg` . ] 2 [^W .,CUQW ,

tk < T ¨" KH

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= XN,mtk

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!

1

M

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!

1

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), αM1,k.p1,k(XN,mtk

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f

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W

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∫ T

t

f(s,Xs, Ys, Zs)ds−∫ T

t

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f : [0, T ] × Rd × R × Rq → REJ :MN

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dG

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∫ t

0

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∫ t

0

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σi

! Ei¤ KL3!FE¡KEJ D

σ!E

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> v`7f-, Z@ .S%A -# #

f+ +- , + #

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F + ! $ - $ #%$+-,

Φ

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x2 $ +

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|x1t − x2

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h = TN

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tk)0≤k≤N

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tN− PN,k0,x′

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tk

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¨=! Q D3D

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M

∑Mm=1 p

ml,k[p

ml,k]

∗ 0 ≤ l ≤ q

(hfW0U),K¶h S.¥bEFE) E !" EJ4K5V- !¬D MN!" E VQE J EJ JEJNG

E6FE ´ ! $c =QEJEL6!6 K6 5! D D KLNc! aD· !"JE d (ρNl,k)0≤l≤q,0≤k≤N−1

KL 7QE V¨6E ρNk (PN

tk) = [ρN0,k(P

Ntk

), · · · , ρNq,k(PNtk

)]∗

OPVD !%D)K6 $b! Q H! aD _E £EJ "!LD£EJρNl,k(x) =

ρNl,k(PNtk

)ξ(x/ρNl,k(PNtk

))ρN,ml,k (x) = ρNl,k(P

N,mtk

)ξ(x/ρNl,k(PN,mtk

))

ξ : R 7→ R :

aD· /!¡DEC2b

MNξ(x) = x

|x| ≤ 3/2 |ξ|∞ ≤ 2

|ξ′|∞ ≤ 1

B) #"V¨L ! N J3E K6EVD a´ ¤º´ N¯J3VD !DKH/D KE E!=oKL ^!@ EN

tN = T! EN

t0 = 07E K6:!- ) a EN

i = 1, · · · , I EJ)!/¥cD£EJ! ·5FE% !)K6P!)KL!D£EJ QE PM

E DPKH a EJN ¬E! aD¬D D !K6KLN ± E KE LD QE # EJ¤ K63J (Y N,I,I,M , ZN,I,I,M )

KLN¬MNDE KEJ '! !@!"FE! D EgK6b!KHL! EJ!-¥c D£E! aFE= D-!! I

MN ¡!QE "D QE ¡ ·3!=KH"!"E D· 3I@N·¤¦ª¬E M

<¨@E:K6 FE! !QEJD!E aD¬!"QEJ"¬"QEE! !QE£JENE!EJ ! !=DV VFEJ 7 3DKLK6ND 3 EJKLEJ (Y N,I,I,M , ZN,I,I,M )

→→→ AWb[ , [aUQ]^[ .U), [ fW

± EJ K53 EJ ECD

Y N,i,I,MtN

= ΦN(PNtN

)^ ! !QEJKLK6!

iI Q¥c $FEH

(Ytk , Z1,tk , · · · , Zq,tk)!:= EN

tk3 N VFEH!DV6DN3!¡D·

(αi,I,Ml,k )0≤l≤qQE

Y N,i,I,Mtk

= ρN0,k(αi,I,M0,k · p0,k),

√h ZN,i,I,M

l,tk= ρNl,k(

√h αi,I,Ml,k · pl,k)

ρN0,k

ρNl,k

N¶ vD£EJ¶ NV! vEEE£JE £° MV ! bDK6KL DDV6DN¶¶D£E D v¡ EvY £E E ! !QEN((PN,m

tk)0≤k≤N )1≤m≤M

((∆Wm

k )0≤k≤N−1)1≤m≤M

ª¬«OP¥7´ ³3° %ª O³3°« ¡¥d¨« ¸ ° ¸ °c O±\µ¨¡³´ «I@°

→→→ A , JNhU, [^fW hKJ3,£hf?A<h U ¶SuSVW ,CSVeg` . ]¦2 [_W.,CUQW ,tk < T

¨ MV ED 3E KEJ Y N,I,I,Mtk+1

:= ρN0,k+1(αI,I,M0,k+1 · p0,k+1)

3=JY N,I,I,M,mtk+1

=

ρN,m0,k+1(αI,I,M0,k+1 · pm0,k+1)

@ ·JE QEJ -FEm

¤ºK6"KHFEJ →→→ ¥bY# EJ EJ

i = 0!\ EJ v!¥c D£E!£

Y N,0,I,Mtk

= 0ZN,0,I,Mtk

=

0#

α0,I,Ml,k = 0

0 ≤ l ≤ q

→→→ ¥b

i = 1, · · · , I T PDV6D αi,I,Mk = (αi,I,Ml,k )0≤l≤q

" EJ K6NVDKLK6" EJK6 '(α0, · · · , αq)

!¡FEHMVEN 1

M

M∑

m=1

(

Y N,I,I,M,mtk+1

− α0 · pm0,k + hfmk (αi−1,I,Mk ) −

q∑

l=1

αl · pml,k ∆Wml,k

)2

.^´

¸ 3@ K6@!%K6 !LDE D4¤ !-EgL ^ dFE KLEJD=! KL MNL :E)¶DMN E E£D5Ê5E 6 MNM

!VNE! -QE KLP!FE !K6PK6K6E ¸ D N@P a ! µ ± M ¶ !6MNMNPEJ 3TD P!QE EO °cÊJ DD - PE E EJE5E! D!D¡3 EJQE V!: ±Y:QEEJKL·:!/DD=!/D6E4K6L5'E:!'K6

hh → 0

QE"! aD HKHH!HKHFEJ M

M → +∞ v± KLEDP!:D QEMN5EE¤K6 7EQE V !QE b bVD QE D QE $JD!EN ! E! · EG! D E/=KL3! °7 ¸ ³ QD 3!3QE!%aD!QE

L2(Ω,P)JDHK6 MVH EH )E DHKH $ ©3- KLEJ!-

QE DK6KL !QE E4K5! D D ¤ !! D EH!D·K6!FEDKHQEJ ¤%!3 K63´ ·¤ ´ ¤ ´ ) Q¯ $EKLNMV ¶P)6EP! TD·I = 3 EJ ¬!¥cDE! a¬ K6¡´ ± N MVDED !LD L!L K6!LK6!D£EJ a´ " KL ¸ /! K6 D"MNvE Ê ¬ ! EKLN¬ECD¡/KH )QT! KHFE ! EJ!¥c D£E!=·3!%aD!¡QE¡MN

(Y Ntk, ZN

tk) = arg inf

(Y,Z)∈L2(Ftk)E(Y N

tk+1− Y + hf(tk, X

Ntk, Y, Z) − Z∆Wk)

2

L2(Ftk)

N%:JEFE 5E EJ^ ·K6N6KH ! K6 )!D£EJ " E· Ftk

¤ K6EJ$ <ª ¡DE ! £E 3E KE '!"# °d ¸ ³ ^´ /MNY$ D

Ytk+1+

∫ tk+1

tk

f(s,Xs, Ys, Zs)ds = Ytk +

∫ tk+1

tk

ZsdWs

# JE d! K6[tk, tk+1]

(Y N

tk)k

E · DK6K6 °7 ¸ ³ !K6 !D·aD QEJ " JE

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³

ksyj.pt=l=|

t@syj | kpY-~ l t@pY-st@pvmw |w t=|r j.

ª K6K63!MN 3 QEJVD! ! E!!D EJ6K6! MNQE ^´ / ^´

X¥b%!D ·% MNQEJ a´ uD KLE! °c$ ª gD KLE® (XN

tk)0≤k≤N

3 ·-!¡FEHDKLENi

3! Q¡QE"G

XNi,0 = Xi,0

XNi,tk+1

= XNi,tk

+ bi(tk, XNtk

)h+ σi(tk, XNtk

)∆Wk.

^´ /± V) EE:DND)!DD KLE5

X

#/[.$0hKJ , [ .U), [afW ¶S]aU 0fTW \[ , [afW ,CSNhJeg[_W¶UQ]aS¯EC! ¦!: MN MN¡EVD

Φ(X)/ EJ

ΦN(PNtN

)

ΦN aD· /! KL "

(PNtk

)0≤k≤ND EF¡!)I%E -!!K6

d′ ≥ d!NK6DKLEN3! ED !

(XNtk

)0≤k≤N °c! EJ:KL$ 5E: · K6N:!HJEFE )! E5 KLNEED

XNtk

H!- ' K6KEJ C KEJ LFE% D=! -D! K6QEJ 3! !QE!PPEE D!X

OMN P K6KLE J V N¶ · E ¥bDFEJ¬D $QC3MNMN3K6 DD·ECDd = 1

q = 1

G $5` , [^fW UQW\[_]^]aS

±YEC )5aD! ·K6 5!E/JEJ QQEJ6!¦¤EDNLG

Φ(X) = φ(XT ) ¶´ 7 E=QE!LKL5!QEDLD£EJ$\D K6K6N

PNtk

= XNtk

ΦN(PN

tN) = φ(PN

tN) d′

$EJ !1 <± V´ 5E)E MN

φ±YD ¬# 'MNQE # £E /EVD EN¬FED! LK6QE G

E|ΦN(PNtN

) − Φ(X)|2 ≤ Ch

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³ $5` , [^fWU-.[Û), [ ¶S

±¶QEC ! !®D· a !E:E D¡!

XGΦ(X) =

φ(XT ,∫ T

0Xtdt)

¶¨ ED PNtk

= (XNtk, h∑k−1

i=0 XNti

)

ΦN(PNtN

) = φ(PNtN

)

d′ ¬ $E !

2 ¥b! aD

φ !QEJ DPD # ! EJKLEJ !:FE'D! ®K6QEJ )E/D

E|ΦN(PNtN

) − Φ(X)|2 ≤ Ch ¶¨ E K6ND ! ¡!E KLEJ ¡ D !6E-K6 6!

X¬QE"K6 5D ! D 3!QEJ ±V

$5` , [^fW ]afvfbU0 Φ(X) = φ(XT ,mint∈[0,T ]Xt,maxt∈[0,T ]Xt)

¨-D a D ¤ ΦN(PN

tN) = φ(PN

tN)

ECDPNtk

= (XNtk,mini≤kX

Nti,maxi≤kX

Nti

) d′

L $E !3 7°c

E# ¬! E KE !FED! LK6QE E|ΦN(PN

tN)−Φ(X)|2 ·¤

K6 KL\ MV¬ D !K6KLNc· ¬KLE ! KLEbE √h log(1/h)

aQE¡K6 QE b¨g) EK6 EJ!5FE-V L!LDND √h

D ! EN VKLEED¬!D KLE)! °c D V adQEJK6 O§ ³ M ³KLEMNMND3 K6$Q# NV´ :3EJ Ê

¨¡E!QE¶D7D EJVD ^´ ! QE\7 MNQE E!(Y N , ZN )! D! Q "EJ E !! D EJ

tk = kh0 ≤ k ≤ N

¨' EJ 3QEJ ¤K6 D MNQE ) E!!D· HEY NtN

= ΦN(PNtN

) J¥c

(Y Ntk, ZN

tk)0≤k≤N−1N3! QE

ZNl,tk

=1

hEk(Y

Ntk+1

∆Wl,k),^´

Y Ntk

= Ek(YNtk+1

) + hf(tk, XNtk, Y N

tk, ZN

tk).

^´ ³KLEMN%MN

Y Ntk

® ! Q!QE ^´ 3D£E@ E DEJ Y 7→ Ek(Y

Ntk+1

) +

hf(tk, XNtk, Y, ZN

tk)

3¡DEJD /!QEJL2(Ftk)

h

)6EK6K6N ö 6!D@MN !D QEMN@ EL!@!D EJtkY Ntk

ZNtk

NL! QLE6! EDD! ¨-E KE d!QE FE P -VD ! ! EJ¤ KLEJ ! EDD! 7 EVD Y Ntk

·ZNtk

JE D· QE ¡ED6 N¡!5V6E " J "LE5D£EJED· EJ !=D(Y N

tk, ZN

tk) °cT E MNQEP# $E 4 !5ª¬ED ¤ ¸ D ¬EJ 5D! ! ^´ T®! !

MN |ZNl,tk

| ≤ 1√h

√

Ek(Y Ntk+1

)2 c´ ¬ EJ ÊJD 'E5 E K65QE: DD=!

K6N¬MVY Ntk

·ZNtk

EQEN!L2(Ftk)

O E) (Y N

tk, ZN

tk)

!QEgD£EJ!L2 \HE ¡KEMNMNDDL

-! QDK6KL MN'K6K5 ^h

)6EK6K65· !EL2(Ftk)

!'FEMNQE E(Y N

tk+1− Y + hf(tk, X

Ntk, Y, Z) − Z∆Wk)

2.^´

ª¬«OP¥7´ ³3° @°7\OP¥d°5G<P´ ¸ ª¬³c´ ¸ Oc´ ¨"¯y°7¯u°I@¥ ¸∑N−1

k=0 E∫ tk+1

tk|Zs − Ztk |2ds

· EdKLE ¬EcK6KH! ED ! $E ¬D ¤º!DD¶DDFEH! K6 EJ ¤

,'SVegUh ¶S%A $ + +-$ , + $ + $ !# , # "+-! $ ,

+-E|Φ(X) − ΦN(PN

tN)|2 $ # - h , # ∑N−1

k=0 E∫ tk+1

tk|Zs − Ztk |2ds ##$ # # #$ ' # 0

, N #$ # , %$ $ +-'$ ! $ # & #$ ! +- %$'& +-, +-$

L2 %$# E|Φ(X)−ΦN (PNtN

)|2 #$ # 0 $ +

! $ # & #$ ! , +# + $ (Y N , ZN ) #

(Y, Z) %, + $ ', + +# $ $ # '$ ! $ # & #$ !+- +-$ , !)+ ! $ + !)+-, ! , +, $ + + ' ! $ #"& #$ ! , +# + $ #$

\ ! E!Q ^ED !3 d Ed3DEEJDKEJ C L!PN MN

Y Ntk

·ZNtkN3! aD· ! KL !

PNtk

G8hf<`cf .£[ ,C[afW A

$ + +- h + #$

$ +

Y Ntk

= yNk (PNtk

), ZNl,tk

= zNl,k(PNtk

) 0 ≤ k ≤ N

1 ≤ l ≤ q,

a´ C (yNk (·))k

(zNl,k(·))k,l

$ $ ! $ ' +,

¨ ¡E # K5TEDH! EC"EVD 5FE'D! ®K6 QE )QE" aD· ! ·¤K6 !¡FE)JE QQE "! D QE¡!I%EJ C °c6 5dK6 MV3 aDyNk (·)

zNl,k(·)±Y D ª D<! D!D£EJED·:±Y D !HELD! @KL QE ·"!HELD QE)!:I%EJ

PNtk

QEPE !5E5D! - E

8hf<`cf .£[ ,C[afW AF $ , $ + +- '

h+ $ +

|yNk0(x) − yNk0(x′)| +

√h|zNk0(x) − zNk0(x

′)| ≤ C|x− x′| a´ '$ #$#$

k0 ≤ N − 1

9M

ksyj.pt=l=|

t@syj | sjkjkl@monqpYr sut@pvmxw q| 7mwk~t@pvmxwk

O9# E=YgE/VgMV !L °7 ¸ ³! Q6QE^´ "· a´ ¶·VN )(D5!FE% ´ !! KL H aD· yNk (·)

zNl,k(·))

k:

l 7¨JE5EVD DaD!:# E !P!"D· !QEJDD ¬!"! KL

Q 7¥c6ED K6 ! D QEMN@ Etk0 ≤ k ≤ N − 1

7JE D D E KE -!yNk (·) !QE QEJD¡D·Y! )EQE!aD·

p0,k(·)$VD QEJMV

lVPE KEJ 5!

zNl,k(·)!QE QEJD3D<! PQE¡QE!%aD J

pl,k(·)

OkD QEMN E/!! D Etk¬ EV MN

Y Ntk

· E -! "QE' N a ^´ <ED! K6 · E <E ¬EVD 3! )EKLE¤ ® KLNDEDE! D)

(Y, Z) ·¡¡EJ !DH!· DQDKLK6¡ $¡:KL:! %)!/D QEMN6 E!5!D · EJ Y¥bDFELJE%TD:MN KH5Q! EJ!'¥cD£EJ!

I!%D QEMN' E)!! D E

tk

##

¨%JE! QHE KEJ =!(Y N , ZN )

!)KLE ·E! ¨"®)D:EV¤ KLEJ (Y N,I,I , ZN,I,I)

±¶KL I

ELMN !® ENH!'!D EJtk%TD·

I EJ3!H¥cDE!%P! Q

Y N,I,Itk

D!I

EJ MN ECJEJN!! QY N,I,Itk

®E! Qb!"E KEJ (Y N,I,I

tj )j≥k+1

®TD·QEJN%!D QEMN aI

EJ !¡¥c D£E! ±¶EH EJYN,i,

N−k−1 ︷︸︸︷

I, · · · , Itk

E ¬\"KLE Ev£EJD= ! ± EJKLEJ '3 E QEJ K6N¬=EJN3MV @E:¡ %aD -!QEp0,N (·) = ΦN(·) !:# E

tN·3=E

αI,I0,N = 1

¥c !:= Etk

0 ≤ k ≤ N − 1G

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³ ¨= EJ

α0,Il,k = 0

0 ≤ l ≤ q

¥b0 ≤ i ≤ I − 1

Q= G(αi,I0,k, · · · , αi,Iq,k)

= arg inf(α0,··· ,αq)

E(|αI,I0,k+1.p0,k+1 + hfk(αi−1,I0,k , · · · , αi−1,I

q,k ) − α0.p0,k −q∑

l=1

αl.pl,k∆Wl,k|2).^´ ) /

¯JE KLEJ Y N,I,Itk

EJ ! Q EY N,I,Itk

= αI,I0,k.p0,kZN,I,Il,tk

QEZN,I,Il,tk

=

αI,Il,k .pl,k ¨g)KEJMN# EQEJ 5E£D ^´ bª K6K66 E£! !@! Y) !QE¶D7QE D·7D£EJED· EJ "!

(Y N , ZN )Y! Y7E KLEJ ¯3!QEEHQEJ ´´DK6KLN # EDEED· EJ '!

(Y N , ZN )DK6KLP¦¤ EDD!

± N

(Y N , ZN )·# E KLE D %!H# E¡

(Y N,I,I , ZN,I,I) ¦¤ !QE3 KL" JEN¡G

()ZJNf<h @VegS@AF $ , $ + +- h + #$ $ +

max0≤k≤N

E|Y N,I,Itk

− Y Ntk|2 + h

N−1∑

k=0

q∑

l=1

E|ZN,I,Il,tk

− ZNl,tk

|2

≤ Ch2I−2[1 + |X0|2 + E|ΦN(PN

tN)|2]

+ CN−1∑

k=0

E|Rp0,k(Y N

tk)|2 + Ch

N−1∑

k=0

q∑

l=1

E|Rpl,k(ZN

l,tk)|2.

ª K6K63MN !DLQEpl,k

0 ≤ l ≤ q

DKHNE a!'KL:!" EJ E!$ v±Y:K6ENP¡MN:D¡"aMNHHKLK6PE£DHHDE

CMV HEPMN

N!%" )Q ª ¡KLEJ EJKL D ! K6 5!E § c"! E v\! E! 'D KLN:# HDK6KL L

Y N KLE $EJKLNHZN E5QEJ !QE¡ H K6 v H!QEJ¡:DN VY# " EQE V :K6!

E|Rp0,k(Y N,I,I

tk)|2 ·

E|Rpl,k(ZN,I,I

l,tk)|2 G\E#v# & D6!gD !)QEJ!

aD· 3QEDFEJP MN "3D£EJD =K6N!QE !!¡JD·¤ Rpk

KEJd EKLN!QE EE EJ £E αI,I0,k.p0,k

αI,Il,k .pl,k

¯J3 KL

ª¬«3O¥c´ ³3° %°c\O¥7° -GO¥7¥7³¨»´ I@O ´ ¨¡¯yP° \¨"¯ª ´ ¨"¯ ¸KL·3! EQEJ V!DK63 & D"! aD!¡QEJ ª DQE :KHH! EJ!5¥cD£EJ!

I E !-D QEMN5QE"!HK6v MN c )6EJN)MN

I ¡ $EJ !

2MN6HKL

h2I−2[1 + |X0|2 + E|ΦN(PN

tN)|2]

!· $ £E EE 3E/3DK6KL QEE $

± ! EJ:!5FE'6E 6KHFEJ !-D :! K65´ Y±Y! DLV!=# EJ !@E /!@DNED· !@FED· o(pl,k)

·3!¡FEH D !¡EJKL=D QEMN Etk

E/)!a´ ) / ¨uV! @! E!u J EJ QE%FE G AN(X0) = 1 + |X0|2 +

E|ΦN(PNtN

)|2 Q±¶¡ K6"!¡K6!D£EJ ^´ ) /(αi,I0,k, · · · , αi,Iq,k) =

arg inf(α0,··· ,αq)

E(|αI,I0,k+1.p0,k+1 + hfk(αi−1,I0,k , · · · , αi−1,I

q,k ) − α0.p0,k −q∑

l=1

αl.pl,k∆Wl,k|2)

¡ D θi,Ik = arg inf

θE(Y N,I,I

tk+1+ hfk(α

i−1,Ik ) − θ · vk)2

DEN6-D QE)K66!-JEFEθ∗ = (α∗

0,√hα∗

1, · · · ,√hα∗

q) dª KLK6- ! @ K6"!¡K6 !D£EJ $

θi,Ik Q5G

E(vkv∗k)θ

i,Ik = E(vkY N,I,I

tk+1+ hfk(α

i−1,Ik )).

∆Wk EDN 3 ! !QEJN!

(pl,k)0≤l≤qN ^EJD !cMNE"KEJ D

E(vkv∗k)!@E aK6" VD·¤º!FEQEJ P JE5G

E(vkv∗k) =

E(p0,kp∗0,k) 0 0

0

0

0 0 E(pq,kp∗q,k)

OP =N

E(p0,kp∗0,k)α

i,I0,k = E(p0,kY N,I,I

tk+1+ hfk(α

i−1,Ik )),

E(pl,kp∗l,k)α

i,Il,k = 1

hE(pl,kY

N,I,Itk+1

∆Wl,k),

1 ≤ l ≤ q,

D¡MNY MN E$QV=FEH! Q -!Y N,i,Itk

ZN,i,Il,tk

!/G

ZN,i,Il,tk

=1

hPpl,k

(Y N,I,Itk+1

∆Wl,k

),

^´ ) Y N,i,Itk

= Pp0,k

(Y N,I,Itk+1

+ hf(tk, XNtk, Y N,i−1,I

tk, ZN,i−1,I

tk)).

^´ )

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³±¶E5!/ K6¡´ : ¡! ¡= E *",CU`7S/G)) K6 QEJ ¡

E|ZN,i,Il,tk

|2 T³KLEMNMV)V a´ ZN,i,Il,tk

DENi ≥ 1

¶°cYLE D£EJ !5 $EJ H!Lª¬ED N¤ ¸ D ¬E L!|Ek(Y

N,I,Itk+1

∆Wl,k)|2 = |Ek([YN,I,Itk+1

− Ek(YN,I,Itk+1

)]∆Wl,k)|2 ≤ h(Ek[Y

N,I,Itk+1

]2 − [Ek(YN,I,Itk+1

)]2).¥cMN

(pl,k)l Ftk

¤ºKLE ¬- EFE · !DNED· ! EJ!¡JD V G

E|ZN,i,Il,tk

|2 =1

h2E[Ppl,k

(Ek[Y

N,I,Itk+1

∆Wl,k])]2 ≤ 1

h2E(Ek[Y

N,I,Itk+1

∆Wl,k])2

≤ 1

h

(E[Y N,I,I

tk+1]2 − E[Ek(Y

N,I,Itk+1

)]2).

^´ )

ª K6K6® E'EPEL T KLE[Ek(Y

N,I,Itk+1

)]2!QEJ*a´ ) ) DDEMN KE !H ¡QE MN

h!=

0

*",CU`7S F G¶L2

Y N,i,Itk

√hZN,i,I

l,tk

\´ d ÊD 5!K6MN ¡JEE E £EJ

Y N,i,Itk

· √hZN,i,I

l,tk

:!'DE ' N $E b¯3) Ê )K6DN L! KL

L2 aK65i, I, k,N

\O D·5Q\)JχN,Ik : Y ∈

L2(Ftk) 7→ Pp0,k

(Y N,I,Itk+1

+ hf(tk, XNtk, Y, ZN,i−1,I

tk))∈ L2(Ftk).

ª FE KLNE|χN,Ik (Y2) −

χN,Ik (Y1)|2 ≤ (Cfh)2E|Y2 − Y1|2

Cf

E)D E"!¡±Y D !f ¥ E¬D MVN

h)6EK6KLN· $v# E D£E

χN,Ik

DNED· E5E- MN5 N%Y N,∞,Itk

∈ L2(Ftk)ÊMN

ZN,i,Il,tk

! !%QEJ3!i ≥ 1

<¨"=E/GY N,∞,Itk

= Pp0,k

(Y N,I,Itk+1

+ hf(tk, XNtk, Y N,∞,I

tk, ZN,I,I

tk)),

^´ ) E|Y N,∞,I

tk− Y N,i,I

tk|2 ≤ (Cfh)

2iE|Y N,∞,Itk

|2 ^´ )

MNY N,0,Itk

= 0 OP E ! d6K63! D ¡

i ≥ 1

E|Y N,i,Itk

|2 ≤(1 +1

h)E|Y N,∞,I

tk− Y N,i,I

tk|2 + (1 + h)E|Y N,∞,I

tk|2

≤(1 + Ch)E|Y N,∞,Itk

|2. ^´ ) E± $E 7D ¤º! E K6\VE d

i = 0D£E

Y N,0,Itk

= 0 ¯ !3 KL

E|Y N,∞,Itk

|2 !)QE ! a´ °c/DKHEN $E ! d'^QEEKL·γEHD E!- E Pp0,k

(Y N,I,Itk+1

) = Pp0,k(Ek[Y

N,I,Itk+1

])EDEJD! Pp0,k

VFE ¡!±Y D "!fG

E|Y N,∞,Itk

|2 ≤(1 + γh)E|Ek[YN,I,Itk+1

]|2

+ Ch(h+1

γ)[Ef 2

k (0, · · · , 0) + E|Y N,∞,Itk

|2 + E|ZN,I,Itk

|2].

^´ )

ª¬«3O¥c´ ³3° %°c\O¥7° -GO¥7¥7³¨»´ I@O ´ ¨¡¯yP° \¨"¯ª ´ ¨"¯ ¸°c' EE ^´ ) c%a´ ) KEc KL

E|Y N,∞,Itk

−Y Ntk|2 V QE K6N

^´ ) <ª D¶DK6 "E:!= K6¡´ ¤

E

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³

ksyj.pt=l=|

t@syj | r t=qm q| q| mxw1t=|6 syl@vm

ª : EJ:"¡EQEJ !- · E5!'DDc"MN5 KLFEDH!QE¡E H!KLH!-K6!HDE $E

QEH/K6 -KL MNQE /M

KHEJ ! !QE!(PN

tk)0≤k≤N , (∆Wk)0≤k≤N−1

¶ª DcD! !- EJ4K6) N 6!QEJFE6 ¤ºD TEPD·)VD !T!"aD!D£EρNl,k

ρN,ml,k

$ Q¯3!"!: NT DQ ª ¡D£E VN! KL E/ Y N,i,Itk

, ZN,i,Il,tk

:!*aDDE! !QEJ

Y N,i,I,M,mtk

, ZN,i,I,M,ml,tk

!LE ^E ¡KKL K6 ª PK6 ELN3! !QEJ" EJ EN a @@ E! 8hf<`cf .£[ ,C[afW A

, '$ ! $ +-$ C0

+ #$&- +-$ , + ' $ ! $ (ρNl,k(·) = max(1, C0|pl,k(·)|) : 0 ≤ l ≤ q, 0 ≤ k ≤ N − 1)

#,%,

|Y N,i,Itk

| ≤ ρN0,k(PNtk

),√h|ZN,i,I

l,tk| ≤ ρNl,k(P

Ntk

), p.s.,

' i ≥ 0 I ≥ 0

0 ≤ k ≤ N − 1 O\D J EJ!PE ¤ºD ® VFE! L! aD· _E £EJ c!¤D£EρNl,k

^ ρN,ml,k

¬=! D <°c ¬ MN6G "EN"NEEN

αi,I0,k · p0,k = Y N,i,Itk

l = 0

√hαi,Il,k · pl,k =

√hZN,i,I

l,tk

l ≥ 1^

αi,I0,k · pm0,kl = 0

√hαi,Il,k · pml,k

l ≥ 1

3 3QEJ

2ρNl,k(PNtk

)^

2ρNl,k(PN,mtk

)

! K6" QEJ1

! D!3 aK6 K6N, l, k,m

O N$¶EJPMNQENQ"# "N(Y N,I,I,M , ZN,I,I,M )

·(Y N,I,I , ZN,I,I)

aD· ! KH®!®KHFEJ

M !-QE-! aD -! QE'!®K6

h

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³± EJQE d D \DK6 MN 7MN!E § VMN7\ ¶ EJ\! $N¬D EKK63KH P!E DKH ¥b ! KLEJ cEJMN$ aE¡ VMV3D QEMN3QEJ!aD·

pl,k KE FE !

PNtk

E(pl,kp

∗l,k) = Id

§ $CDdNV E TD·dQE\ 7D KE3NK6 MN MN%FE D· ®E%! aD·L - NJEJEN=QEL@EJ aKE £EJ !E:E °c@ Q=! # K6AMk = ∀ j ∈ k, · · · , N − 1 :‖V M

j − Id‖ ≤ h, ‖PM0,j − Id‖ ≤ h ‖PM

l,j − Id‖ ≤ 1

1 ≤ l ≤ q ^´ a) E!'E%¦¤ D 'HE@! g!:KLEJD

V Mj

·PMl,j

¸ %# NV! KLE @D EMNQEJpl,k

3@KE D(V M

k )0≤k≤N−1

(PM

l,k )0≤l≤q,0≤k≤N−1

DNN# !N :E£D6E -6 MVM → ∞ YO#

limM→∞ P(AMk ) = 1

<¯!"!5 NP¡ EJ3 D QEvMNQEJN"!6 & D!=KH"!¡KHEJ $ ()ZJNf<h @VegS@A

$ , $ + +- I ≥ 3 ! + + $ ! $ pl,k $ +-,

E|pl,k|4 <∞ ' k, l h + #$ $ + ' 0 ≤ k ≤ N − 1

E|Y N,I,Itk

− Y N,I,I,Mtk

|2 + h

N−1∑

j=k

E|ZI,Itj − ZI,I,M

tj |2

≤ 9N−1∑

j=k

E(|ρNj (PNtj

)|21[AMk

]c) + ChI−1

N−1∑

j=k

[1 + |X0|2 + E|ρNj (PN

tj)|2]

+C

hM

N−1∑

j=k

(

E‖vjv∗j − Id‖2F E|ρNj (PN

tj)|2 + E(|vj|2|p0,j+1|2)E|ρN0,j(PN

tj)|2

+ h2E[|vj|2(1 + |XN

tj|2 + |p0,j|2E|ρN0,j(PN

tj)|2 +

1

h

q∑

l=1

|pl,j|2E|ρNl,j(PNtj

)|2)])

.

¯"E£PKLN )QEMNH KLK6 MNQE[AM

k ]cDNH

0 MV

M → ∞ KLE ): E£5E: : D ! KLEJ )DDQEN5D'K6-!E# D H!® KL)´ ) T°c ·T# D )! H) KEJ ¡PD)KL! !! NV@ KLNE--'KLK6'! aD·-! QEJ ¥ E-KL ¬ρNj (PN

tj)

E5!3K6KLN! !¡ (!2Q VN¡G

P([AMk ]c) ≤∑N−1

j=k

[P(‖V M

j −Id‖ > h)+P(‖PM0,j−Id‖ > h)+

∑ql=1 P(‖PM

l,j −Id‖ > 1)] C¯EC

P(‖V Mk −Id‖ > h) ≤ h−2E‖V M

k −Id‖2 ≤ h−2E‖V Mk −Id‖2

F = (Mh2)−1E‖vkv∗k−Id‖2F

³KLEMN¡MN5) K65´ -VD555! ¤¦EJVKLMN vOPVD E KL´ ·)´ \ 7KL· a "!LD KEJ K6N¡6QEJ!

ª¬«OP¥c´ ³° @°cbOP¥7° /GI 7«P¨"° ° I®¨"¯3° ¤ ª O³3±\¨KL

h<EP! aD3¡KH¡! KHFE EJ !¡ D ! ¯3V!3@@ E!/D¡D -D N!¡QEEKL·$ ´ ¬ 6 $EKLN: '!-KLH EJ6EJVK6MN-!- - K6-!-FE K64DEJ

()ZJNf<h @VegS@AL $ , $ + +- , -

#f !#, +

C1 +- +# (y, z)+ *! ## #$ '$ #$ ,

# #$ $ E|pl,k|2+ε < ∞ k, l ε > 0 , , *! '[√M(αi,I,Mk −

αi,Ik )]i≤I,k≤N−1

! $ #"& +- ,#$ # '$ *! '& + # #$ !#$ , M #$ # , %$ $ 8hS S ¶S]aU-`\h f<`7f .£[ , [afTWA

76E :´ 7NEN7! FE !I%E J :d! E 6K6cMVE" :´ ¡ 3! ^´ ) / D 3E£D3 7 EDD!

Ek

QI@E DE K6¬KLED PE¬D£EJ¬E !QE¬ED· "EPE6YD! Ek

KLEE0

T±Y)®EKLNE ) EN$ ¨" D Y N,i,Itk

= αi,I0,k · p0,k(PNtk

) ®QEJ7QE ^´ ) /:E

CAN(X0) ≥ E|Y N,i,Itk

|2 =

αi,I0,k · E[p0,kp∗0,k]α

i,I0,k ≥ |αi,I0,k|2λmin(E[p0,kp

∗0,k])

EQEJ |Y N,i,Itk

| ≤ |αi,I0,k||p0,k(PNtk

)| ≤|p0,k|

√

CAN(X0)/λmin(E[p0,kp∗0,k])

OP -!ρN0,k(x) = max(1, |p0,k(x)|

√

CAN(X0)/λmin(E[p0,kp∗0,k])) P"KK6

√h|ZN,i,I

l,tk| LE

ρNl,k(x) = max(1, |pl,k(x)|√

CAN(X0)/λmin(E[pl,kp∗l,k])) N³KLEMN

MNHpl,k

)QEH KLE <®Eλmin(E[pl,kp

∗l,k]) = 1

·P D !¡ KL QN$ ¤

8hS S , ZJNf<hK@VegS%A

PE3E: =

AN,Mk =

1

M

M∑

m=1

|ρN0,k(PN,mtk

)|2, BN,Mk =1

M

M∑

m=1

|fmk (0, · · · , 0)|2.

P¬KLE · ! 5EE(AN,M

k ) = E|ρN0,k(PNtk

)|2 E(BN,Mk ) ≤ C(1+|X0|2)

JO N$bE ¶FE 7 E!E!!dDNED!E\7DE\! )K67!dK6!D£EJ !QERM EQE EJ/DE

L2(Ω,P) Q¨-D !"" "!"

(xm)1≤m≤M·3

(vm)1≤m≤M!D· !

Rn NEVD !FE"KLEJ DV M = 1

M

∑Mm=1 v

m[vm]∗MN " :

λmin(VM) > 0

TO$# MND·"!Rn θx = arg infθ |x −

θ · v|2M ! ¡QE

θx =[V M ]−1

M

M∑

m=1

vmxm.^´

± EJ D£E x 7→ θx

E P3!¡ $@E:# E 4 λmin(V

M)|θx|2 ≤ |θx.v|2M ≤ |x|2M .^´

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³¥bFEH ¡!3D£EJD E5EMN"!

(θi,I,Mk )∗ =((αi,I,M0,k )∗,

√h(αi,I,M1,k )∗, · · · ,

√h(αi,I,Mq,k )∗

)

!5E:FED¡!αi,I,Mk

OP EJ¬!¥c D£E!=!a´ N¡ D

θi+1,I,Mk = arg inf

θ

1

M

M∑

m=1

(ρN,m0,k+1(α

I,I,M0,k+1.p

m0,k+1) + hfmk (αi,I,Mk ) − θ.vmk

)2.

^´

°c NV! EL# K6NAMk

dQE- DK6@FE %!±Y D @!aD·

ρNl,k3@ E# KLE !

pl,kQ=

E|Y N,I,Itk

− Y N,I,I,Mtk

|2 + hN−1∑

j=k

E|ZN,I,Itj − ZN,I,I,M

tj |2 ≤ 9N−1∑

j=k

E(|ρNj (PNtj

)|21[AMk

]c)

+ E(1AMk|αI,I,M0,k − αI,I0,k|2) + h

N−1∑

j=k

q∑

l=1

E(1AMk|αI,I,Ml,j − αI,Il,j |2).

^´ ¥bP E K6H´ ) ®JE'K6 |θI,I,Mk − θI,Ik |2 ·KLN

AMk

Yª D E = E *",CU`7S G\ · !DNED·gFEJ !@FE= (θi,I,Mk )i≥0

b°c H K6 !QE3 "KLK6 JEJN 07SegegS@A

' h + #$ ' AM

k , ' +-$ $ +- + |θi+1,I,M

k − θi,I,Mk |2 ≤ Ch|θi,I,Mk − θi−1,I,Mk |2.

, '$ '$ *! 'θ∞,I,Mk

#,

θ∞,I,Mk = arg inf

θ

1

M

M∑

m=1

(ρN,m0,k+1(α

I,I,M0,k+1.p

m0,k+1) + hfmk (α∞,I,M

k ) − θ.vmk)2.

! $ + |θ∞,I,Mk − θI,I,Mk |2 ≤ [Ch]I |θ∞,I,M

k |2.8hS S

I@E <¥cMN

1 − h ≤ λmin(VMk )

λmax(P

Ml,k ) ≤ 2

0 ≤ l ≤ q

¬AMk

E/V=!5^´ ¬=MN(1 − h)|θi+1,I,M

k − θi,I,Mk |2 ¡QEh2

M

M∑

m=1

(fmk (αi,I,Mk ) − fmk (αi−1,I,M

k ))2

≤Ch2

q∑

l=0

|αi,I,Ml,k − αi−1,I,Ml,k |2λmax(P

Ml,k ) ≤ Ch|θi,I,Mk − θi−1,I,M

k |2.

O E 3 3DE$ ¥bDQ@EMN"E ¬@EEMVθ0,I,Mk = 0

¤

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³OP vQE5a´ ¤º´ P ! DKL

θ∞,Ik − θ∞,I,M

k = B1 + B2 + B3 + B4

YD6MN!AMk

|θ∞,Ik − θ∞,I,M

k |2 ≤ 3(1 +1

h)(|B1|2 + |B2|2 + |B3|2) + (1 + h)|B4|2.

a´ C*",CU`7S OG KE Y E !

B1

B2

B3

B4

AMk

J¨"E 77 EJbDFEMN µ "± M ¡G ¸ ‖Id − F‖ < 1

F−1 − Id =

∑∞k=1[Id − F ]k

‖Id − F−1‖ ≤ ‖F−Id‖1−‖F−Id‖ .

¥ ED MVN

F = V Mk

NE(1AM

k‖Id− (V M

k )−1‖2) ≤ (1−h)−2E‖Id−V Mk ‖2 ≤

(1 − h)−2E‖V Mk − Id‖2

F = (M(1 − h)2)−1E‖vkv∗k − Id‖2F .

O#@EE(|B1|21AM

k) ≤ C

ME‖vkv∗k − Id‖2

F E|ρNk (PNtk

)|2.¥cMN

AMk

‖(V Mk )−1‖ ≤ 2

Q v E(|B2|21AM

k) ≤ C

ME(|vk|2|p0,k+1|2)E|ρN0,k(PN

tk)|2,

E(|B3|21AMk

) ≤ Ch2

ME[|vk|2(1 + |XN

tk|2 + |p0,k|2E|ρN0,k(PN

tk)|2

+1

h

q∑

l=1

|pl,k|2E|ρNl,k(PNtk

)|2)].

ª K6K6!QEFEH!/KLK6P´ ·= EJN ‖PM0,k+1‖ ≤ 1 + h

AMk

/ ^ED K6

(1 − h)|B4|2 ≤ (1 + h)(1 + γh)|αI,I0,k+1 − αI,I,M0,k+1|2 + Ch(h+1

γ)

q∑

l=0

|α∞,Il,k − α∞,I,M

l,k |2.

*",CU`7S G KLEJ QQE ¸ εk = E‖vkv∗k−Id‖2

F E|ρNk (PNtk

)|2+E(|vk|2|p0,k+1|2)E|ρN0,k(PNtk

)|2+h2E[|vk|2(1+|XN

tk|2+

|p0,k|2E|ρN0,k(PNtk

)|2 + 1h

∑ql=1 |pl,k|2E|ρNl,k(PN

tk)|2)].¨ D:" K6 "D ¤º!¡!

B1, B2, B3, B4

!QE ^´ £ v D γ = 3C

hVD ¡!53E3MN

Ch+ Cγ≤ 1

2

QEK6 QDEJ $=NE(1AM

k|θ∞,I,Mk − θ∞,I

k |2) ≤ CεkhM

+ (1 + Ch)E(1AMk|αI,I0,k+1 − αI,I,M0,k+1|2).

I%E %E Vu!u K6K6 ´ ¤ D=·®!%K6 ^´ E ¤ ´ NE(1AM

k|αI,I0,k+1 −

αI,I,M0,k+1|2) ≤ (1 + h)E(1AMk|α∞,I

0,k+1 − α∞,I,M0,k+1 |2) + ChI−1

(1 + |X0|2 + E|ρN0,k+1(P

Ntk+1

)|2 +

E|ρN0,k+2(PNtk+2

)|2).bQEJK6E£K6N

E(1AMk|θ∞,I,Mk − θ∞,I

k |2) ≤ CεkhM

+ ChI−1(1 + |X0|2 + E|ρN0,k+1(P

Ntk+1

)|2

+ E|ρN0,k+2(PNtk+2

)|2)

+ (1 + Ch)E(1AMk|α∞,I,M

0,k+1 − α∞,I0,k+1|2).

ª¬«OP¥c´ ³° @°cbOP¥7° /GI 7«P¨"° ° I®¨"¯3° ¤ ª O³3±\¨°c' Ed# EKLNd!DEJD LDK6K63!QE ^´ M N# ! D ∞ ·KLED QE

IdELD QEJ $E _E£D@@D EJN

CJ K6N6!T N ª D ·¡ D

E(1AMk|αI,I,M0,k − αI,I0,k|2) + h

q∑

l=1

E(1AMk|αI,I,Ml,k − αI,Il,k |2)

≤ CεkhM

+ ChI−1(1 + |X0|2 + E|ρN0,k+1(P

Ntk+1

)|2 + E|ρN0,k+2(PNtk+2

)|2)

+ (1 + Ch)E(1AMk|αI,I,M0,k+1 − αI,I0,k+1|2).

¨=DKL "FEH= EN¬" K6KL!)µ"¬EJ ! D3O ) ¤

,'SVegUh ¶S%AF $ +- # , '+-$ !

h $ , "!# +-$

, ! ' , & #$ - I ≥ 3

+ , I ≥ 2

+ +-$ $ $ $ + %, # # +-$ !! $ - $ $ #,%, +-)+# + # , + $ ! $ ' ^´ ) +- , $ +-$ , + !# $ + ,#$ %, , + !' $ ! $ $ !)+ ' ρNl,k +-$

ρN,ml,k

8hS S , ZJVf<hK@VegS:AL±YcEKLN Nc E!E! c V)EN KLN !c!F¤

QD c!J EJ ±Yc! 7K6 !f

QE E !y

zl

NcD K6 ∂0f

·∂lf

±YEEK6·β ∈]0, 1]

3c ! D¬!3« ! ¸ dEJ ! E 3MVε < β

¬MND EMN"QEJ!aD· pl,k

KEJ V¥bk < N − 1

=! Q 3MNQE

AMl,k(α) =1

M

M∑

m=1

vmk ∂lf(tk, XN,mtk

, α0 · pm0,k, · · · , αq · pmq,k)[pml,k]∗,

BMk =

1

M

M∑

m=1

vmk [pm0,k+1]∗, DM

k =√M(Id − V M

k ),

CMk (α) =

M∑

m=1

vmk [αI,I0,k+1 · pm0,k+1 + hfmk (α)] − E

(vk[α

I,I0,k+1 · p0,k+1 + hfk(α)]

)

√M

.

¥bk = N −1

N6BMk = 0

!ECMk (α)

N 7KLαI,I0,k+1 ·pm0,k+1

·αI,I0,k+1 ·p0,k+1N"!5K6FEDD· K6NQE

ΦN(PN,mtN

)

ΦN(PNtN

) <±Y3! !

AMl,k(α)·

DMk

N"PJE !$ v¥b" K6$ D XM w→ E' aN K6%!JEJD /KLEJD

(XM)MDN%^EK5N 3"JEFE "E £E

E HD MNM

!®# )Q# ¥bPFELDND:QE ¡

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³LD E b

XM P→ \¥c MNLKHEJ ¡N! !QEb DN ¤D EN3NVE¡G

(AMl,k(αi,Ik ), BM

k , VMk )i≤I−1,l≤q,k≤N−1

P→,

GM = (CMk (αi,Ik ), DM

k )i≤I−1,l≤q,k≤N−1w→ .

^´ /³KLEMN¬MV

limM→∞ V Mk

p.s.= Id

N °c/ £E E aD· f

·ρN,m0,k+1!QE bc!

θi,Ik = E(vk[αI,I0,k+1.p0,k+1+hfk(α

i−1,I0,k , · · · , αi−1,I

q,k )])·θi,I,Mk

! QE ^´ =N|V Mk

√M(θi,I,Mk − θi,Ik ) −DM

k θi,Ik − CM

k (αi−1,Ik )

−BMk

√M(αI,I,M0,k+1 − αI,I0,k+1) − h

q∑

l=0

AMl,k(αi−1,Ik )

√M(αi−1,I,M

l,k − αi−1,Il,k )|

≤ 1k<N−1C√M

|αI,I,M0,k+1 − αI,I0,k+1|2M∑

m=1

|vmk ||pm0,k+1|2

+C√M

|αi−1,I,Mk − αi−1,I

k |1+βM∑

m=1

|vmk ||pmk |1+β.a´

¥b K6¡´ Q=K6"QE DDk

MN([√M(θi,I,Mj − θi,Ij )]j≥k,i≤I ,GM)

w→ <³3EMNθ0,I,Mj = θ0,I

j = 0

j ª ! =! E!

k = N − 1/MN

BMk = 0

·i = 1

OP V! a´ ¤´ CDFE KLN([√M(θi,I,MN−1 − θi,IN−1)]i≤1,GM)

w→ ¥bi = 2

-VMN' KK6PEKLN - E ^´ ¤º´ c·¬'N([√M(θi,I,MN−1 − θi,IN−1)]i≤2,GM)

w→ VMNE¡3 3!QE"^´ DN3 QEJ ¥bdDFEQ= MM = M−1−β/2∑Mm=1 |vmN−1||pmN−1|1+β

= D

1√M

|α1,I,MN−1 − α1,I

N−1|1+βM∑

m=1

|vmN−1||pmN−1|1+β = |√M(α1,I,M

N−1 − α1,IN−1)|1+βMM .

¥cMN[√M(α1,I,M

N−1 −α1,IN−1)]M

d!VJ3ELdE MM D 3 MNM → ∞ c³3KEMN5MN |vN−1||pN−1|1+β ∈ L 2+ε

2+β(P)

OP FE® a -!E!KH$!QEJPD£EJP!EEE EJ# # ! E£D)K6 ) )Q <D! !∑M

m=1 |vmN−1||pmN−1|1+β = O(M2+β2+ε

+r) $ ¬

r > 0 Q¥ EJD MN<=D EN

rLEK6KLN3· $TVN MM → 0

°c@ · EDEKLN$=N([√M(θi,I,MN−1 − θi,IN−1)]i≤I,GM)

w→ ¥b# !D 'MNk < N − 1

@EJ MN"KK6¬D MN$ <´ v/E:¡D ¤=EJ!! P! !BMk

MNY "EJ "DKLK6¡E£JE ¤

ª¬«OP¥c´ ³° @°cbOP¥7° /GI 7«P¨"° ° I®¨"¯3° ¤ ª O³3±\¨ # #

±Y@ K6%´ % & ® & D%!%FE ! D EK6'E 6MN® = KL´ MNQEQ=!QE6# 6DK6KL -QE6# E4K5 # & D=!o V%·!KH! aD· !)QE °c®VH! )D D:!(Y N,I,I , ZN,I,I)

(Y, Z)

T DNVN !D! Ê c!N

)Q\! Eb7 V · 7KH! aD·!¡QE!" ¡MN=K63# ! E KLEJ !5# EJNtk

q∑

l=1

E|Rpl,k(√hZN

l,tk)|2 + E|Rp0,k

(Y Ntk

)|2

!¡0

KEJ¬ $EKLN)6EJKLK6V "MNN−1∑

k=0

( q∑

l=1

E|Rpl,k(√hZN

l,tk)|2 + E|Rp0,k

(Y Ntk

)|2) a´ C

!@0 ¬ª DEg@DEL'E£!' )aKE 6 (

yNk (·),√hzNk (·)

)

MNN6 L!QE5E® ´ c°7 QEJ D :HQE¬! DEED·±YD H!:D%aD DV$v E5E"!QEJ"FEQE HNKL MN!QEVD£EJ¡!N! VDH! E δ \§ $¶

aE-!δ

0

!5KLE !'EEJN MN6E-K6KL'^´ £ ! 0

MNN!=3 )Q ³ 7I=N·¤¦ª¬E °c:TcEcMN N

(Y N,I,I,M , ZN,I,I,M )·

(Y N,I,I , ZN,I,I)!:

0E bMV

N!:7 )QV

δ

0V Ê7E)K6N6KHL!'KHEJ

M °cT$bLKLE E! !QE)5 KL´ %DE K63D EN

NKLE $EKLN3DEP!E¡KH! aD·3!E ! D£E ^N®EJ 5!KL

E‖vjv∗j − Id‖2F

E(|vj|2|p0,j+1|2)

°cT$"QE K6H K6!:EQEJ:!H NVDTMN E!EFEQE VK6 MN"!)' EJN ! tk

)d = q = d′ = 1

ª · KLN $ K6EFE aK61[a,b](x)

b− a = δ

Q¨$ DN P!" ¬!QEJ P K6´ H ÊJ !¡ EJ!¬!FEQE

vk KLE EJ :7FE!

PNtk

!FEQE! NVD$ EKL E"aD·5KE D!QE !1[a,b](x)

E 1q

P(PNtk

∈[a,b])1[a,b](x)

ª K6K6\KL

E‖vjv∗j−Id‖2F

a b! EDc E! MN2

! DaD· ¡KEJ ¡JEYD c!d# ¶!QE\ KL7´ ! EJ! ! ED3! 1

P(PNtk

∈[a,b])

<¨/KLE PE!KLN " K6MN¡DFEH MNδ → 0

·3MN /! K6!3K6E¬!¡E:QE PMVPNtk

¡¡!QEJ NJEJ !¡δ P KLE ® E ¬" -! D= K6! QE/! !D£E D'!T·

E

´ QOP±\µ¨"³´¦«I%°¥d¨"©P³±\° ¸ °7 ¸ ³NVD< E:!LD ! !¡KLE P D ¡ ·JEEJ -!/K6KH!@! /!QEL / K6@´ ·!D=L!6D -!@E£5DK6K6 ^E -E

M aD !KH-! aD:!-QE-5!N

5E5/DND=!# E4K6 ©'!PKE EJ d D "^EJ !DPDE K6Nd ·\ PEQE JEJN

¯¬EC !QEJ FEQE ćE · !(Y N

tk, ZN

tk)

! ·PKL K6 !EL2 ! LDE5MVEN " MNQEJ ^´ ¶¯E !/ N " N E@ aK6"! T ED)D! *a´ ¤º´ 3E 63 EJH!'D D= 5!QE:L K6'´ ·: :)!5KLE E ¤ P! "MN KL·N"! Ê JEKH EJ K6QEEK6P!# E4K6 ^QE¬!PKL "·KHP!"aD·¬!"QEJKH! K:FEJ PPE'DD \¨ E' T"DEJ HMV: KLEJ! ¡!QE" K6´ )K6 MNQE N KLKLN ! ! !5! aD !"EJ¤KL PPEL\!

PN MNHD"K6K6 · E NP!6D K6D£E DE 3® E ¯@E -D"@ D=! /!@KLEJ ÊEN/ N/ K6@KH"!%aD!¡QE¡ 3EVD

(Ytk , Ztk)3=KLK6 ¯¬EJ 7!QEJ DPQE 3KLV!QdE! D E LK6MNP¬E£ TD !QEFELQE ´"MNH!·V"VD£EJ D5°7 ¸ ³x!D·H!HKLE !EMN3!# °7 ¸ ³ ¥c $<3! D3 E4K5"EVD D°7 ¸ ³1! DLVDE ¶¯ E EJ¡! DKLK6 !QEJ:# E4K5·5)QH!D:!-# EJ gEJMN'!'D ª E"3# V! @ED@D!¡ !QED

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³

E

ksyj.pt=l=|

|-~ l@pvj t@pvmw q| sxvymxl@pt@kr |

#

±¶E'KEJ)!N"¡E£"K6 : EE VH!H# E K6!:EQE)´3 D E "D KEH K6 D43 ! Y N ´ D#VDE)EHQE¬ DE Q¥b¬DE V! 3D

(Y N , ZN)!DKLE :G

Y Ntk

= Etk(YNtk+1

) + hEtkf(tk, XNtk, Y N

tk+1, ZN

tk),

^´´ /hZN

tk= Etk(Y

Ntk+1

∆Wk).^´´

ECDY NtN

= φN(PNtN

) ±¶EL!T DHE£D ^´ ·*a´ PMVH %K6ED !EP !

fY Ntk

QEY Ntk+1

V! EJNE D KLE D d¥bELDN dDZNtk

!QEL/! f

: QEZNtk+1

G 0S ,£SVhegSoS ., SWxS 7S , 2ºfW \UQeSW ,CUQ])`7f ¶hh SVW ¶h S ,Cf ,CS . ]aS .S ., [^egU), [ fTW.W\fW S9NY`b]af .[ S .'`\Uh@h U`\`7f<h , N v°cTD: D6E:!E=D£EJ!" EJYFE)JE ! P!¡FEHKLEMV¡´ ´ <¬E ^ED!K67MVFE"DD6!6 K63´ 3VE E£DDD KE ¥b!E MNEJQEE v)E !@ HVDE ¡ °7 ¸ ³ ! ¤D ^´´ ·¤ ´´ ¥bHDE)b NV! )LD

RT = (Ri)1≤i≤d′ ¸ :DV6DN"D! !! P!)DEJHELD! =KL QE φN

P!4¤f

a ¤º !6® EJ MN Tª D£ENPKL·! EMNH]aU.f]` , [afW ¶S]¦2 J\U, [^fWohKJ ,Ch f A<h U ¶S®]^f 0U]_[ . JVS=S ., 7f<hWJVS ¨=V! ¬=QE D G

fR(t, x, y, z) = f(t,−R1 ∨ x1 ∧R1, · · · ,−Rd ∨ xd ∧Rd, y, z)

φN,R(x) = φN(−R1 ∨ x1 ∧R1, · · · ,−Rd′ ∨ xd′ ∧Rd′).^´´

¨/! ¡E ‖φN,R‖∞· ‖fR‖∞

MNQE supx |φN,R(x)| ·

supt supx |fR(t, x, 0, 0)|MNV!N!QEJ! 'MN !E 7O QE 6!%D

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³

-30 -20 -10 0 10 20 30-1

0

1

2

3

4

5

6

7

φN,R φN R1 = 10

!#"$%&(' !")"$ *+*,-" .$/0 !1&2 "$%.43516789)"$ *1&)*:);!"=<#'+#>)<#!"$0 *@?Y N,Rtk

= Etk(YN,Rtk+1

) + hEtkfR(tk, X

Ntk, Y N,R

tk+1, ZN,R

tk), ACB#B $EDF

hZN,Rtk

= Etk(YN,Rtk+1

∆Wk), ACB#B $HGFI+' Y N,R

tN= φN,R(PN

tN)

JLK)M NPORQTSVUXWY[ZTW\^] O;Z_][\`ZXa ZTW\b]cQd Ie <f^*1+<# g'+,h,- ;!!#'ji!+ (Y N,R, ZN,R)

8k +'"l9^<*nmo"$*p 10.=! !+<mC 'rq)"$ < yN,R,Mk (·) / (zN,R,Ml,k (·))1≤l≤q

&s !!#'ji8 Y N,Rtk

/ (ZN,Rl,tk

)1≤l≤q8 yN,R,Mk (PN

tk) /

(zN,R,Ml,k (PNtk

))1≤l≤q

k ,-,-@8 <R.=b8#*"= B /8.$+<;mC '*"= < yN,R,Mk (·) / (zN,R,Ml,k (·))1≤l≤q<* R!1&2 "=<pft8u)"$;!

.=!*<'+&vh'"= )< (αMl,k)0≤l≤q<*R+<Rw89<*+<R!@mC ')"$ < (pl,k)0≤l≤q

x y.zs,-+,-|,h "$>+*h67!y8 <b.zs89#)"$ B /X <#"=,b!.$ M ))~#+'&)"=*<^! (PN ,∆W ) " <#j )<L!|"=<*'*1&)"=<*9)"$ (tk)0≤k≤N

X mo9"$/!**h!.=!<L!*1'+"=</ 9s!.$)L<*!!0q<*@6735 <#"=,b!.$ M )*~#+'*"=#+< (PN

tk,∆Wk,k+1)k

∆Wk,k+1<#` '*+!L.$10)"$*

! %.$+<!*,h"$>+#+<',-<) )<<* %'+ <u)")!1+<%!<%',h<) )+<!;.435" '+#1+,- w*I "= ∆Wk

l<*" PN = XN /"=.<#;<#v-<) !p'+ ! ")*9/fL.3E" <uj tk / ∆Wk<*"=,b!.$+

PNtk+1

! "$<<#"l!&,h.= (PNtk

)∗ =

(

(XNtk

)∗,max0≤i≤kXN,1ti

) y ) XN,1 .z`!*,h"$>+#'+,h<* )`! XN / nI"=*'+*<+/!g<*"$,b!.= PN L,| "=>*@!.$!<gv-'+"$ )/f35**+<p99*"z9w!.=<n.$10)"$*< A '+,h,-L8p&,h!.$/.z-<*"$,b!.z0)" !<*3E nw!*I;q "$ <#!X.3E" )+u9.$.$%!V)+,-!< [tk, tk+1]

Fr~ " <*"4/~) "=',-)V!'+<X9*"=w!.$+<X9.=10)"$*<!"$)"$ ! .$.$+</ <#!!<# 673E <*"=,b!.$RfL'ri!67!n" <uj tk M *1+.="$<)9*"$ <(" !1 ! *+<

D

ª¬«O¥c´ ³3°u %P° ¸ ª¬³3´¦¥c´¨"¯yP° ± O±\µ¨¡³´ «I@°! oDE £EJ

(∆Wmk,k+1)1≤m≤M

D·MNDN EJoKL K)K D K6 ∆Wm

k

± EJ K6:MN6# EQEJ V5!QED·6QE : D E/KHEJ !6!QE£JE E5!JEJFEJE £EJ MN)!EFE5QE ¡´ D!QEJN! 3D QEJ !D·QE ¤ )-E EJ g!'D·-NV c°7 QE D $¶ KH \SN 2 fT[ .`b]` .¶S UhJ[aUb]^S .UQ]`JNU),£fT[ah S . !D QEMN: EJN

tk O#®:MN !'D QEMN: EN

tk KH

MhKJVU]_[ .U), [afW .


> \SgeVegS]afT[ ¶S(∆Wm

k,k+1)1≤m≤M

U S 0o[_W JN`cSVW ¶UQW0S \S . Uh[aU ]aS .(∆Wm

i,i+1,∆Wmi,i+1 : 1 ≤ i ≤ N − 1,1 ≤ m ≤ M)/ ¶¨ JE/ D¡JEJFEJ¡E EJ¡D

Mh JNUQ]^[ .U), [afW . ¶S

PNtk+1

W¶fX, JNS .

(PN,mtk+1

)1≤m≤M

%J EPN,mtk+1

= Tk(PN,mtk

,∆Wmk,k+1)

Tk

! FE:E /!# EJNtk

!5# EJNtk+1

!¡E:D QE ¡!I%EJ CPN

P0

t0 t1 t2 t3 t4 t5

PN,1t1

PN,1t1

PN,2t1

PN,2t1

PN,3t1

PN,3t1

¶ ¸ KHEJ ¬! D KLNEJ FEJ

¨6D c!6 7 aCy(R)

·Cz(R)

MNQd !¡E" VD£E E :! D !QE"D QEJ " D !P3MNY D EE: O QEJ ¬! aD / ψ=! Q 3 aD

[ψ]y·

[ψ]zE¡G

[ψ]y(·) = −Cy(R) ∨ (ψ(·) ∧ Cy(R)),

[ψ]z(·) = −Cz(R) ∨ (ψ(·) ∧ Cz(R)).

± EJ K6!¡DQEJ "¡ N¡ 3%aK6" E!:G→→→ ¨= E

k = NQE

yN,R,MN (·) = φN,R(·) →→→ Ok EJN/! !D EJ

tk

(αMl,k)1≤l≤q =! Q¡DK6KL® K6 K5 aEO H "D -!=K6 K5 ¬=D£EJ!¡KLEJD!¡ /! $ !

E

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³KL!¡KL!DE ¡G

infαl

1

M

M∑

m=1

|yN,R,Mk+1 (PN,mtk+1

)∆Wm

l,k

h− αl.p

ml,k|2.

a´´

¨! Q "EzN,R,Ml,k (·) QEJ

zN,R,Ml,k (·) = [αM

l,k.pl,k]z(·) ¥c

αM0,k"! Q DK6KLK6 K6 ¬!=K6"!K6 !DE G

infα0

1

M

M∑

m=1


) + hf(tk, XN,mtk

, yN,R,Mk+1 (PN,mtk+1

), zN,R,Ml,k (PN,mtk

)) − α0.pm0,k|2.a´´ E

¨=! Q 3Ey

N,R,Mk (·) E

yN,R,Mk (·) = [αM

0,k.p0,k]y(·) →→→ ¨= MN !6# E

t0

¸ T'DK6QEE£DP# E K63! D !QE FEQE´ V'¬ ! D JEJNG O D QEMN EN

tkQ"KL"!K6!DE a´ ^E JEE E £EJ

(PN,mtk

, PN,mtk+1

,∆Wmk )1≤m≤M

E\MN \KLb!K6!\D£E ^´´ ¤´´ E aN N ¬JEE E EJ

(PN,mtk

, PN,mtk+1

,∆Wmk )1≤m≤M

± EJ K6"!D·E ¡ D QE3! EJ !)¥c D£E!@D£EJ3@ ¡!QEJ# MNQE ^´´

Y N,Rtk+1

!QE"! f

E E4K5!FEQE ´ uJE K K6KL DV6DN(αi,I,Ml,k )0≤l≤q

JEH KLL!'KL!D£EJ '^´ ¥bH!HJ¤ K6¶!dMNQE4¤ D E !dD¶!7FEKLEJDc!7 $ ·b¶!\KL!"DK6 !"FEH '! =KL!¡K6!DE Q !q + 1

KL !K6 ! D£EJ MN ^´´ ¤º´´ EbEQEEJ ^ED MV! ! ¡KLH7MN ^´ ¶°c)Q¶¡MNL ¡KLEJD"!6 !^´´ ¶b!

l!T N N N bKK6\MVE KLNbbKK6\"D ®!5E! aDMN5¡D! ¡!QE¡E'QE HVK6 MN vOP # E K6'!@D@QE /K:=6E E ! K6/FED=NKL MVK6N$ ª !QE¶ EJ4K6)!5E'QE H´3K::!¡!¡ E¡ECD5!"JEJFEJD · ¬QEVK6 MN

E' L! EJ K67 =!LD£E$ ±¶EEJ-E ®! N!QEP!%D£EJ$ PEPE6QE ´ D£EJPPP!)PVD!DN dd# K6[AM

k ]cdMN dKLEJD7! $L 7! K6 3MN

M → ∞ PE¬ EJ4K53!PDENFED£EJ ^' FE-VDE EJ P NV EJMVLFE- !6# °7 ¸ ³! D6! Q 6QE^´´ ¤ ´´ d "·EJMN "DEPDE"! aD ª !E5 MN ¡

E

ª¬«O¥c´ ³3°u %P° ¸ ª¬³3´¦¥c´¨"¯yP° ± O±\µ¨¡³´ «I@° °c)Q\!gE K5¡D! !@!gEVD )!T N!g K6 ±¶E K6K5 EJ QE ^´ LD D !EJVD /E! K KL%KL! K6 EJ ^´ ) / =K6FE £E# ED¡QE¡K6C¡KL MN

M ¤KHFE ¬!I@·¤¦ª¬E °7=D¡$< v PEVD ! K6! K6EJ VD E MN cPE ¬¼ LMN3!QEdJD£EN K6 7QE ¤DKLN KLP!'^EJ ¬!=D "K6 !¬D£E $ <± EJ K6P!¡D¡QE P 5E£JE E/!/# !

(Y N , ZN) aKL'! ED£6D! !D MN3DFEJ MN! E KEJ ! EDD! aQEKL µ²¡²¼

±Y3D(Y N

tk, ZN

tk)

! QQEJ"a´´ /c· ^´´ c QEJd V± N ·! E£cVDE # MNQE E!HMN5 :D (Y N,R

tk, ZN,R

tk)

! cQE a´´ P5^´´ P ª" PDMN¡3E ¬D ¤º!$

= I #&- '-&'! # 1:<*#2 # Q-!D MN

(Y N,R, ZN,R)= ¥b-DE JE ! K6-E JEJN5G

8hf<`cf .£[ ,C[afW A A ' h ≤ 1 ∀k : 0 ≤ k ≤ N ∀l : 1 ≤ l ≤ q |Y N,R

tk| ≤ Cy(R)

|ZN,Rl,tk

| ≤ Cy(R)√h

= Cz(R)

Cy(R) = exp(2γ∗ +1 + γ∗

qT )‖φN,R‖2

∞ + 2T (1 +1

γ∗)‖fR‖2

∞ a´´

+ *!γ∗ = 4qC2

f

¯33! E MN3 ±YK6KL"!)µ¡ Ev! D3O 6!D! |Y N,Rtk

| ! |Y N,Rtk+1

| ¥bDE)E MN3 $E ¡! d ^O !L# MNQEJ ^´´ ´ ¶VN$3γ > 0

G

|Y N,Rtk

|2 ≤ (1+γh)|Etk(YN,Rtk+1

)|2+2(h2+h

γ)|fR(tk, X

Ntk, 0, 0)|2+2C2

f (h2+h

γ)Etk|Y N,R

tk+1|+|ZN,R

tk|2,

E E

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³

hE

N−1∑

k=0

|ZNtk− ZN,R

tk|2

≤CN−1∑

k=0

[E|Y N

tk+1− Y N,R

tk+1|2 − E(|Etk(Y

Ntk+1

− Y N,Rtk+1

)|2)]

≤CE|φN(PNtN

) − φN,R(PNtN

)|2 + CN−1∑

k=0

[E|Y N

tk− Y N,R

tk|2 − E(|Etk(Y

Ntk+1

− Y N,Rtk+1

)|2)]

≤CE|φN(PNtN

) − φN,R(PNtN

)|2 + Cγh+ Ch2(1 +1

γh)

N−1∑

k=0

E|Y Ntk+1

− Y N,Rtk+1

|2

+ Ch2(1 +1

γh)E

N−1∑

k=0

|f(tk, XNtk, Y N

tk+1, ZN

tk) − fR(tk, X

Ntk, Y N

tk+1, ZN

tk)|2

+ Ch2(1 +1

γh)E

N−1∑

k=0

|ZNtk− ZN,R

tk|2

EJN ^´´ C3EL!) E T°c E E! γ = 2C

·"hLEK6N3 E !¡E: -! ! ^EDK6 ¤± N ·/ =MN ! N^EJD K6N'/ KLE EJN'! E D4¤ !' ENL!'KLEEJ 6!%MN-!%! !

XN PN OP o K6 7 6MN /!=6E != MNQEJ E!/VDE E

φNf ³3KLEMN@EJ-MN ! FE KKLKLE MN !QE/FEQE ´

Y N,Rtk

= yN,Rk (PNtk

)·ZN,Rl,tk

= zN,Rl,k (PNtk

)!"aD 3KLE

yN,Rk (·) ·zN,Rl,k (·)

1 ≤ l ≤ q 3± N ·@! / ¤ ! %MN "K6@! D %!

aD· yN,Rk (·) ·

zN,Rl,k (·) MN ¡±YD )aK6 K6NPR

D£EJ aD·φN,R

fRN±¶ D aKL KLN

RG/ ·FE)! K6EJ'!"E) ´ QE 9M $ EJ MN !# !N MN E£D¬! D EJNc!¬±YD

yN,Rk (·) · √hzN,Rl,k (·)! !QE!

R ¨ E PQE !:D5 · H ¡ )¦V5·¡HK:H! aD·!¡QE !: $

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³ :D+\ #*/-'$');+:? #¨=¡QEDY! EJ EJ $

0 ≤ l ≤ q

0 ≤ k ≤ N − 1G

pml,k =pl,k(PN,mtk

),

pml,k+1 =pl,k+1(PN,mtk+1

),

PMl,k =

1

M

M∑

m=1

pml,k(pml,k)

∗.

¨=PE ¬! Q=D EJ/KH¡!¡DV6D 3!¡D !5 Etk

G

PM0,kα

M0,k =

1

M

M∑

m=1

pm0,k

(

[αM0,k+1.pm0,k+1]y + hfR(tk, X

N,mtk

, [αM0,k+1.pm0,k+1]y, [α

Ml,k.p

ml,k]z)

)

,

a´´ PM

0,kα1,M0,k =

1

M

M∑

m=1

pm0,k[αM0,k+1.p

m0,k+1]y,

a´´

PM0,kα

2,M0,k =

h

M

M∑

m=1

pm0,kfR(tk, X

N,mtk

, [αM0,k+1.pm0,k+1]y, [α

Ml,k.p

ml,k]z),

^´´

hPMl,kα

Ml,k =

1

M

M∑

m=1

pml,k∆Wml,k[α

M0,k+1.p

m0,k+1]y,

a´´

PM0,kβ

M0,k =

1

M

M∑

m=1

pm0,kyN,Rk+1 (PN,mtk+1

) + hfR(tk, XN,mtk

, yN,Rk+1 (PN,mtk+1

), zN,Rl,k (PN,mtk

)), ^´´ E

PM0,kβ

1,M0,k =

1

M

M∑

m=1

pm0,kyN,Rk+1 (PN,m

tk+1),

^´´

PM0,kβ

2,M0,k =

h

M

M∑

m=1

pm0,kfR(tk, X

N,mtk

, yN,Rk+1 (PN,mtk+1

), zN,Rl,k (PN,mtk

)),^´´ M

hPMl,kβ

Ml,k =

1

M

M∑

m=1

pml,k∆Wml,ky

N,Rk+1 (PN,m

tk+1).

^´´ C ¸ V b MVEJ bD4¤ ! ! Q N QEbbDVLDN !D ! KLE d MN^ VT KE D

PMl,k

N ! $ D d!P! Q DV6DN¬!D·!¡EHKLE MV ¡!QEJ# E ¡O <¨/" $EKLN-D·!¡DV:¤D !¡JDα

βEk+1(α) = α

Ek+1(β) = β

³KLEMN MV D4¤ ! E! FE EJ fR(t, x, y, zl) = fR(t, x, y, (zl)1≤l≤q)

! KLE ! · /! ·=q

DK6EN!z Q¯MN¡!QEFEH! -!

αMl,N−1

=α0,N .p0,N (·) = φN,R(·)

ª¬«O¥c´ ³3° @OP¯OP± ¸ ° P° ± O±\µ¨¡³´ «I@° `= $)+*¥baD

ψ=! ¡G

‖ψ‖2k,M =

1

M

M∑

m=1

|ψ(PN,mtk

)|2,

‖ψ‖2k,M

=1

M

M∑

m=1

|ψ(PN,mtk

)|2.

'': #*/- V*¨o! =QE

Kl,k

/D£EJ! QE!pl,k

0 ≤ l ≤ q

d¨o Pl,k

QEJD/D·!aD· \)! QE\FEQEJ

pl,k

[Pl,k]y dK KL7QED ! aD \KLEvMN cE

Cy(R)#

[Pl,k]y = [α.pl,k(·)]y, α ∈ RKl,k ¨H QEKl,k(ω)

0 ≤ k ≤ N−1, 0 ≤

l ≤ q "EJ6!E:KLEJD!¡ =!3 ¬

(pl,k(PN,mtk

)∗)1≤m≤M

+'+7 *¸ :EEK6

βMN

1 < β ≤ 2 ª EEJKL· EFE3V !¬DN D aD5!LEd!K6

h!3 E4K5 ¨LV!Ec E!Ld N EJ !

βKLEd! !¬KEJNQEJNMV

β > 1 DE" MN# E K63D

aV®! MN)E ·MNβ = 2

K6P!)!H !'# E K6L! K KL!'! E!:MN-# H!'!D · EJg K6H! K6¡´ ©a β) <3E=!¡ ·KLN

AMk

! ¶QEJ¡GAMk = ∀ψ ∈ [P0,k+1]y − yN,Rk+1 (·) : ‖ψ‖k+1,M − ‖ψ‖k+1,M ≤ h

β+12 . ^´´ /

`^ )+& )*°c)QvV!K6"! ¡ JEYMN N "!E¡ EQE H!:FEQEE! ! EJ4K6C· b MNb)KLE! DN \ aD!M

h

·3!=KH¡!aD!E5GT1,k,M = ‖yN,Rk (·) − β

M

0,k.p0,k(·)‖2k,M ,

^´´ CT2,k,M = ‖α1,M

0,k − α1,M0,k .p0,k(·)‖2

k,M ,^´´ £

T3,k,M = ‖β2,M0,k − β

2,M

0,k .p0,k(·)‖2k,M ,

^´´ T4,k,l,M = ‖βMl,k.pl,k(·) − zN,Rl,k (·)‖2

k,M ,^´´ C

T5,k,l,M = ‖αMl,k − αMl,k.pl,k(·)‖2k,M .

^´´

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³ ¥bdEJQE V7# ^´´ 5EEJ MNc6KLK6¬!Pµ"¬E !D· ´ QdE5!!!FE= JELMV5# !@ EN

tk!@ MN JE !:# E

tk+1

8hf<`cf .£[ ,C[afW A A

E‖yN,Rk (·) − [αM0,k.p0,k]y(·)‖2k,M ≤ET1,k,M + (1 + Ch)E(T2,k,M )

+ (1 + Ch)E‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1]y(·)‖2k+1,M

+C

hET3,k,M + ChE

q∑

l=1

T4,k,l,M + T5,k,l,M.

)bc! E! MN[yN,Rk ]y = yN,Rk

^ )´´ /J:N: ENMN[·]y

1¤ ±¶ D

1

M

M∑

m=1

|yN,Rk (PN,mtk

) − [αM0,k.pm0,k]y|2 ≤

1

M

M∑

m=1

|yN,Rk (PN,mtk

) − αM0,k.pm0,k|2.

¥cQDKLK6*a ^´´

Ek+1

(

yN,Rk+1 (PN,mtk+1

) + hfR(tk, XN,mtk

, yN,Rk+1 (PN,mtk+1

), zN,Rl,k (PN,mtk

))

)

= yN,Rk (PN,mtk

),

βM

0,k

a! Q<QE(^´´ E βM

0,k = Ek+1(βM0,k)

d :!6KL!3K6 !cD£EJ 3G

infα

1

M

M∑

m=1

|yN,Rk (PN,mtk

) − α.pm0,k|2.

P¡D·¡ · "! KLE / @E MNQE KL"!¥ VQE5G1

M

M∑

m=1

|yN,Rk (PN,mtk

) − αM0,k.pm0,k|2 =

1

M

M∑

m=1

|yN,Rk (PN,mtk

) − βM

0,k.pm0,k|2

+1

M

M∑

m=1

|βM0,k − αM0,k.pm0,k|2

=T1,k,M +1

M

M∑

m=1

|βM0,k − αM0,k.pm0,k|2.

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³¨ LE !@E E@DED· g!E=D·g (

(pm0,k)∗)

1≤m≤ME dMN:DEED6±YD :!

fR^aK6 KLN

R" D a DK6QEE5^´´ ·

^´´ M PG

1

M

M∑

m=1

|(β2,M0,k − α2,M

0,k ).pm0,k|2 ≤Ch2

M

M∑

m=1

|yN,Rk+1 (PN,mtk+1

) − [αM0,k+1.pm0,k+1]y|2

+Ch2

M

q∑

l=1

M∑

m=1

|zN,Rl,k (PN,mtk

) − [αMl,k.pml,k]z|2.

³ D£EJ $ ¯3E£V=FEHKE EJ G

E1

M

M∑

m=1

|yN,Rk (PN,mtk

) − yN,R,Mk (PN,mtk

)|2

≤(1 + γh)1

ME

M∑

m=1


) − [αM0,k+1.pm0,k+1]y|2 + ET1,k,M + (1 + γh)E(T2,k,M )

+ 2(1 +1

γh)ET3,k,M + C(h2 +

h

γ)E

1

M

M∑

m=1

|yN,Rk+1 (PN,mtk+1

) − [αM0,k+1.pm0,k+1]y|2

+ C(h2 +h

γ)E

1

M

q∑

l=1

M∑

m=1

|zN,Rl,k (PN,mtk


^´´

¯3E ¬K6 Q D!¡ MNQEJ==3 N EJNE/K6:G

E1

M

q∑

l=1

M∑

m=1

|zN,Rl,k (PN,mtk


³KLEMN! E!=MN¡D"KL"¡KE "E

E1

M

q∑

l=1

M∑

m=1

|zN,Rl,k (PN,mtk

) − αMl,k.pml,k|2

ª¬«O¥c´ ³3° @OP¯OP± ¸ ° P° ± O±\µ¨¡³´ «I@°H EN\FE3 ´´ · ^E bMN

[·]z

1¤ ±YD J¥cJ: Eb# E 4

(a+ b+ c)2 ≤ 3(a2 + b2 + c2)Q V N¡G

1

M

M∑

m=1

|zN,Rl,k (PN,mtk

) − αMl,k.pml,k|2 ≤

3

M

M∑

m=1

|zN,Rl,k (PN,mtk

) − βM

l,k.pml,k|2

+3

M

M∑

m=1

|(αMl,k − αMl,k).pml,k|2

+3

M

M∑

m=1

|(βMl,k − αMl,k).pml,k|2

=3T4,k,l,M + 3T5,k,l,M +3

M

M∑

m=1

|(βMl,k − αMl,k).pml,k|2.^´´ M

°c' EJNdEDNED· L!FE¡D· L ((pml,k)

∗)1≤m≤M

· DK6QEJE ^´´ 7·^´´ C Q VN¡G

h

M

M∑

m=1

|(βMl,k − αMl,k).pml,k|2 ≤

1

M

M∑

m=1

|Ek+1(yN,Rk+1 (PN,mtk+1

) − [αM0,k+1.pm0,k+1]y)

∆Wml,k√h

|2

≤ 1

M

M∑

m=1

Ek+1|yN,Rk+1 (PN,mtk+1

) − [αM0,k+1.pm0,k+1]y|2

− 1

M

M∑

m=1


) − [αM0,k+1.pm0,k+1]y|2

^´´ J

@ E $E "!Hª ED N¤ ¸ D ¬E )3E:! $EJ Qª" PD·! KLEEJ -MV¶JE63TK6¡!K6 Q^´´ <°c%T$<% D EJN3D·KE ¤EJ -!QEJ a´´ =!D¡KE

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2

E

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³QE"G

ET1,k,M + (1 + γh)E(T2,k,M ) + (1 + γh)1

ME

M∑

m=1


) − [αM0,k+1.pm0,k+1]y|2

^´´ V/+2(1 +

1

γh)ET3,k,M + C(h2 +

h

γ)E

1

M

M∑

m=1

|yN,Rk+1 (PN,mtk+1

) − [αM0,k+1.pm0,k+1]y|2

+C(h2 +h

γ)E

q∑

l=1

T4,k,l,M + T5,k,l,M

+C(h+1

γ)E

1

M

M∑

m=1

|yN,Rk+1 (PN,mtk+1

) − [αM0,k+1.pm0,k+1]y|2

−C(h+1

γ)E

1

M

M∑

m=1


) − [αM0,k+1.pm0,k+1]y|2.

°c®D E !6 γ = C

!QEJP MNQEJ @D ¤º!TYV N$3)D EC! 5G

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2

≤(1 + Ch)E1

M

M∑

m=1

|yN,Rk+1 (PN,mtk+1

) − [αM0,k+1.pm0,k+1]y|2

+ ET1,k,M + (1 + Ch)ET2,k,M +C

hET3,k,M + ChE

q∑

l=1

T4,k,l,M + T5,k,l,M. ^´´ ¤¥b3E MN¡ K6K6"!)µ"¬E v!D·Q^E!E MN!QE3E5 /´´ )<E

E‖yN,Rk+1 (·)− [αM0,k+1.p0,k+1(·)]y‖2k+1,M

!)EFEJD!E‖yN,Rk+1 (·)− [αM0,k+1.p0,k+1(·)]y‖2

k+1,M

±YPQEE!P !# E"^EJ !# E !3!P# KLN

AMk·¬# Ed EJ d K65G

()ZJNf<h @VegS@ACA

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2

≤CE

N−1∑

i=k

T1,i,M + T2,i,M + h−1T3,i,M + Chβ−1

+ CCy(R)2

h

N−1∑

i=k

P([AMi ]c) + ChE

N−1∑

i=k

q∑

l=1

T4,i,l,M + T5,i,l,M.

ª¬«O¥c´ ³3° @OP¯OP± ¸ ° P° ± O±\µ¨¡³´ «I@°)ª K6KLED bQE$JD K6 cED MND!D· !H K6KL7!µ"¬E ! D E aK6 !QE E ´´ K6E‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖2

k+1,M

E‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖2

k+1,M

¬¥b-DFE= D ¡GE‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖2

k+1,M

=E

(

‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖k+1,M − ‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖k+1,M+

‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖k+1,M

)2

≤(1 + h−1)E

(

‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖k+1,M − ‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖k+1,M

)2

+

+ (1 + h)E‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖2k+1,M .

°c ! EN !' ENMV:# ®)HAMk

[AMk ]c

·" ENFEH! Q -!AMk

VN¡GE‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖2

k+1,M

≤Chβ +C

hCy(R)2P([AMk ]c) + (1 + h)E‖yN,Rk+1 (·) − [αM0,k+1.p0,k+1(·)]y‖2

k+1,M .

OP ^´´ !V NG

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2 ^´´ J ≤ET1,k,M + (1 + Ch)E(T2,k,M ) + Chβ

+ (1 + Ch)E1

M

M∑

m=1

|yN,Rk+1 (PN,mtk+1

) − yN,R,Mk+1 (PN,mtk+1

)|2

+ CCy(R)2

hP([AMk ]c) +

C

hET3,k,M + ChE

q∑

l=1

T4,k,l,M + T5,k,l,M.

©E D£EJ '!=K6K6!µ¡ E K63E!DD ¤

±Y K6´´ ^E 3 N%DE=KH!K63! PMV)PE (! NKLE¬aD· /!

M,h·3!=KH¡!aD !¡QEJ

M

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³ = ;)2 #Y-'

T1,k,M07SegegS@A A +# #, $ Kl,k(ω)

# &-$ , +-$'& , + + "! &- # $ $ ,, &-$ $ (pl,k(P

N,mtk

)∗)1≤m≤M

ET1,k,M ≤ CCy(R)2 + ‖fR‖2

∞M

E(K0,k(ω)) + infα

E|α.p0,k(PNtk

) − yN,Rk (PNtk

)|2.,'SVegUh ¶S%ACA

+-$ , #$ $ ! , !# $ + $ , +'!#

E(K0,k(ω)) +-K0,k

"+-$ $ $ K0,k(ω) ≤ K0,k #$ +-$ ! $ # #

E(K0,k(ω))+ '$ %$ #

!# !# +-, !)+- +-$ #$ !)+ K0,k(ω) ¿ K0,k $# # $ +- , , + *+ ! $ %$ -"!)+ "! #*! +- δ $ + $ +-$ , + *! $

, + +- $ *!# % $ ! , #$ +-$ , + +- $ ## $ $ + $ ! # δ & #$ # +-$ +-%$# '$ &-)+-$ $ # ! $ ! '$$

K0,k

#, , , #K0,k(ω)

$ %$ - $ %$"+-$ $ $ tk ,,$ # ! ! $ #$+-$ # ', + $ PN,mtk

+! # +-$ , + $ # $ , # ! $ ! $ #$+-$ + # ', + $ $ $ $ # ! , $ $ $',%, +-$ , + + "! "&-# $ $ ! # $ + +-$ , + $ +-$'&

K0,k(ω)

! + #"!

g°c= ENEk+1(β

M0,k) = β

M

0,k

T

ET1,k,M =E1

M

M∑

m=1

|yN,Rk (PN,mtk

) − βM

0,k.pm0,k|2

=E1

M

M∑

m=1

|Ek+1yN,Rk (PN,mtk

) − βM0,k.pm0,k|2

≤E1

M

M∑

m=1

|yN,Rk (PN,mtk

) − βM0,k.pm0,k|2.

¨= PEJ!QE3 "D£E!¡!/ KL¡O ECD

Y = yN,Rk+1 (PNtk+1

) + hfR(tk, XNtk, yNk+1(P

Ntk+1

), zNl,k(PNtk

)),

X = PNtk.

³KLEMNMNE(Y 2|X = x)

KLE QEJ ^@ EJNXNtk

= xPNtk

= x′G

2Cy(R)2 + 2h2E

(

|fR(tk, XNtk, yN,Rk+1 (PN

tk+1), zN,Rl,k (PN

tk))|2∣∣∣∣PNtk

= x′)

≤2Cy(R)2 + 8C2fh

2Cy(R)2 + 8qhC2fCy(R)2 + 4h2|fR(tk, x, 0, 0)|2

≤C(Cy(R)2 + ‖fR‖2∞).

¤

M

ª¬«O¥c´ ³3° @OP¯OP± ¸ ° P° ± O±\µ¨¡³´ «I@° = ;)2 #Y-'

T2,k,M07SegegS@A AF

ET2,k,M ≤ Cy(R)2

ME(K0,k(ω)).

c=FEH!D Ê !QE# E ¡O /EJ MN @EB

EHKE DP!M

(pm0,k)

∗ bH)HMN B∗BM

= Id\D ! EN)MV

BE@KH"!¡D $EJ !

K0,k(ω) QED¡D£E

α1,M0,k

! QYQE

α1,M0,k =

1

MB∗V

V

! ¡D3!RM !¡DV!

Vm = [αM0,k+1.pm0,k+1]y

<¨@E6E "G

|pm0,k.(α1,M0,k − α1,M

0,k )|2 = (pm0,k)∗B

∗

MV − Ek+1(V )V − Ek+1(V )∗ B

Mpm0,k,

¡G

Ek+1|pm0,k.(α1,M0,k − α1,M

0,k )|2 = (pm0,k)∗B

∗

MEk+1

(

V − Ek+1(V )V − Ek+1(V )∗)B

Mpm0,k.

¨ K6(m,m′)

!¡E:KLEJDEk+1

(V − Ek+1(V )V − Ek+1(V )∗

) JEGEk+1(Vm − Ek+1(Vm)Vm′ − Ek+1(Vm′)) =Ek+1([α

M0,k+1.p

m0,k+1]y[α

M0,k+1.p

m′

0,k+1]y)

− Ek+1([αM0,k+1.p

m0,k+1]y)E

∗k+1([α

M0,k+1.p

m′

0,k+1]y).

¨[αM0,k+1.p

m0,k+1]y = ψ(αM0,k+1, P

N,mtk

,∆Wmk,k+1)

)'D E 5aD gK6Eψ ª K6K6

αM0,k+1

PN,mtk

Fk+1¤ºKLE MNH

m 6= m′ ∆Wmk,k+1

·∆Wm′

k,k+1N3! !QE3! Fk+1

3 ! !QEJN3N¡ <=E/GEk+1([α

M0,k+1.p

m0,k+1]y[α

M0,k+1.p

m′

0,k+1]y) = Ek+1([αM0,k+1.p

m0,k+1]y)Ek+1([α

M0,k+1.p

m′

0,k+1]y)

)!DEk+1(Vm − Ek+1(Vm)Vm′ − Ek+1(Vm′)) = 0

m 6= m′ \OP dK6V¤ !EE)!FEPKLEJD

Ek+1

(V − Ek+1(V )V − Ek+1(V )∗

) N7V$ V&L24S., Yh UQ[FXegSVW , ¶UQW .5] 24UQW\W\]Û, [^fW \S 0S . ,CSNhJeS . <\S@hKJ .£[ ¶S@] 2 [^W , JNh , ¶S%]Û .£[_e \]aU), [ fTW \S .Uh[Ûb]aS.


±YKL!EQEJ'KLE NQEEk+1([α

M0,k+1.p

m0,k+1]

2y) ≤ Cy(R)2

M

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³OP ‖Ek+1

(V − Ek+1(V )V − Ek+1(V )∗

)‖2 ≤ Cy(R)2 ·3T G

(pm0,k)∗B

∗

MEk+1

(V − Ek+1(V )V − Ek+1(V )∗

) B

Mpm0,k

≤‖Bpm0,kM

‖2‖Ek+1

(V − Ek+1(V )V − Ek+1(V )∗

) B

Mpm0,k‖2

≤‖Bpm0,kM

‖22‖Ek+1

(V − Ek+1(V )V − Ek+1(V )∗

)‖2

≤Cy(R)2

M(pm0,k)

∗B∗B

Mpm0,k

=Cy(R)2

M(pm0,k)

∗pm0,k

=Cy(R)2

M

K0,k(ω)∑

j=1

|pm0,k,j|2.

= E# NV B∗BM

= Id <°c@K6KEJN

mD·¡FEJ VN¡G

Ek+11

M

M∑

m=1

|pm0,k.(α1,M0,k − α1,M

0,k )|2 ≤ Cy(R)2

M

1

M

M∑

m=1

K0,k(ω)∑

j=1

|pm0,k,j|2 =Cy(R)2K0,k(ω)

M

/QE3DK6"! B∗BM

= Id Q´ Y@! D

E(T2,k,M ) ≤ Cy(R)2

ME(K0,k(ω))

¤

= `= ;)2 #Y-'T3,k,M07SegegS@A A

ET3,k,M ≤ C(Cy(R)2 + ‖fR‖∞)h

ME(K0,k(ω)).

g³KLEMVMNβ2,M

0,k − β2,M

0,k = hB∗

MV − Ek+1(V )

ECDFE¡KK63! Q L!B

MN EKLK6 D !V E£D# NV B∗BM

= Id3@ EV

D·!RM !¡DV! G

Vm =fR(tk, XN,mtk

, yN,Rk+1 (PN,mtk+1

), zN,Rl,k (PN,mtk

)).

³KLEMN:E)MNfR(tk, X

N,mtk

, yN,Rk+1 (PN,mtk+1

), zN,Rl,k (PN,mtk

)) = ψ(PN,mtk

,∆Wmk,k+1)

-D EJ aDψ

KLE ±¶5KK65EJK6HMV EK6K6 D ! ¤K6! E 6K66MNEk+1

(Vm − Ek+1(Vm)Vm′ − Ek+1(Vm′)

)= 0

m 6= m′

M

ª¬«O¥c´ ³3° @OP¯OP± ¸ ° P° ± O±\µ¨¡³´ «I@°OP DK6KL D !K6K6<@E/G

Ek+1

(

(pm0,k)∗B

∗

M

(V − Ek+1(V )V − Ek+1(V )∗

) B

Mpm0,k

)

≤maxm Ek+1(V2m)

M

K0,k(ω)∑

j=1

|pm0,k,j|2.

¨@E6EJ$m

GEk+1(V

2m) ≤Ek+1|fR(tk, X

N,mtk

, yN,Rk+1 (PN,mtk+1

), zN,Rl,k (PN,mtk

))|2

≤2|fR(tk, XN,mtk

, 0, 0)|2 + 4C2fCy(R)2 + 4C2

f

q

hCy(R)2 ≤ C

Cy(R)2 + ‖fR‖2∞

h.

OP v/DDDKLK6¡EJ=KLK6 D !NP "G

ET3,k,M ≤ C(Cy(R)2 + ‖fR‖2

∞)h

ME(K0,k(ω)).

¤

= ;)2 #Y-'T4,k,l,M07SegegS@A A

L

ET4,k,l,M ≤ Cy(R)2

MhE(Kl,k(ω)) +

1

hinfα

E|√hzN,Rl,k (PN

tk) − α.pl,k(P

Ntk

)|2.

¨@E-G

T4,k,l,M =1

M

M∑

m=1

|zN,Rl,k (PN,mtk

) − βM

l,k.pml,k|2 =

1

Mh

M∑

m=1

|√hzN,Rl,k (PN,m

tk) − (

√h)β

M

l,k.pml,k|2.

¨$EJ MNβM

l,k = Ek+1(βMl,k)

3MN √hzN,Rl,k (PN,m

tk)

Fk+1

¤ K6E OP G

E1

Mh

M∑

m=1

|√hzN,Rl,k (PN,m

tk) − (

√h)β

M

l,k.pml,k|2

=E1

Mh

M∑

m=1

|Ek+1√hzN,Rl,k (PN,m

tk) −

√hβMl,k.p

ml,k|2

^´´

≤E1

Mh

M∑

m=1

|√hzN,Rl,k (PN,m

tk) −

√hβMl,k.p

ml,k|2.

M

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³¨ )E "!E¡:D£EJ!5! EJ D£E ! K65O ECD

Y =∆Wl,k√

hyN,Rk+1 (PN

tk+1)

·X = PN

tk

<³KLEMN3E ¬MVE(Y 2|X) ≤ Cy(R)2.

OP @E MNQE K6¡O Q V N¡G


MhE(Kl,k(ω)) +

1

hinfα

E|√hzN,Rl,k (PN

tk) − α.pl,k(P

Ntk

)|2.

¤

= ;)2 #Y-'T5,k,l,M07SegegS@A A

O


MhE(Kl,k(ω)).

³EJ MNT5,k,l,M = 1

M

∑Mm=1 |(αMl,k − αMl,k).p

ml,k|2

TP PD a 3QEB

E:KLEJD"!M

(pml,k)

∗ 3QEJV

D3!RM !¡DV!

Vm =∆Wm

l,k√h

[αM0,k+1.pm0,k+1]y.

¨@E6EJ$ECDD3 EKLEJD G

Ek+1|(αMl,k − αMl,k).pml,k|2 =

1

h(pml,k)

∗B∗

MEk+1

(

V − Ek+1(V )V − Ek+1(V )∗)B

Mpml,k.

³KLEMN E \MN ∆Wml,k√h

[αM0,k+1.pm0,k+1]y = ψ(αM0,k+1, P

N,mtk

,∆Wmk,k+1)

OP# ! FE3K KL^E ¡MN7 D !KLK6N£vK6Y¤ !FEQEP!

Ek+1

(V−Ek+1(V )V−Ek+1(V )∗)N3V$ Q±YK6!EE'KLE NQE

Ek+1(|∆Wm

l,k√h

|2[αM0,k+1.pm0,k+1]

2y) ≤ Cy(R)2.

OP !¡FE:KKL ^E /MV D !K6KLN$<@E/G


MhE(Kl,k(ω)).

¤

M

ª¬«O¥c´ ³3° @OP¯OP± ¸ ° P° ± O±\µ¨¡³´ «I@° = `^ ;)2 #Y-'

P([AM

k]c)07SegegS@A A

*! , $ "+ $ , +-$ $

P([AMk ]c)

≤2E

(

N2

(hβ+1

2

3√

2, [P0,k+1]y − yN,Rk+1 (·), (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M))

exp

(

− Mhβ+1

144Cy(R)2

)

.

±Y6 EJ)DFEJ MN ¥\ ¡·)))QEKLLE/! ·¤K6EJ ! !QE µ¡²¡²"¼ KL@ \¥bK6 Q¡ J EJ¶! QEJ H FEDE!aD[P0,k+1]y−yN,Rk+1 (·) ´ NV!73 !JEFEE £EJ 5# # !

(Um)1≤m≤MJEEN

1 −1

E£D@QE 12

7! !QE!=E"EE "E EJ KH "!E# E K6 P QGZm =PN,m

tk+1

ZM+m =PN,mtk+1

Um = 1

Zm =PN,mtk+1

ZM+m =PN,mtk+1

Um = −1 OP = D:G

P([AMk ]c) =P(∃ψ ∈ H : ‖ψ‖k+1,M − ‖ψ‖k+1,M > h

β+12

)

=P(∃ψ ∈ H :

√√√√ 1

M

M∑

m=1

|ψ(PN,mtk+1

)|2 −

√√√√ 1

M

M∑

m=1

|ψ(PN,mtk+1

)|2 > hβ+1

2

)

=P(∃ψ ∈ H :

√√√√ 1

M

M∑

m=1

|ψ(Zm)|2 −

√√√√ 1

M

M∑

m=1

|ψ(ZM+m)|2 > hβ+1

2

)

D£EJPN,mtk+1

·PN,mtk+1

K KL: Y¨6DJE D6!6FE-! 6 $EJ :D! ¤ QEJNQE(Um)m

´ NV! E¬¡D G = g1, . . . , gn! H E=3 ¬

ψ ∈ H Q v gj ∈ G YMN

√√√√ 1

2M

M∑

m=1

|ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2 + |ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2 ≤ 1

3√

2h

β+12 .

±YcD£EJ! QE! G \ EE¡KH !cD N2

(

hβ+1

2

3√

2,H, (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M

)

ECD"¬EJ NV! !QE# E"O ³3KEJMN¬MN"DKLK6 P0,k+1

M

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³QEDPD·T!P! KL a!D!!K6 ª NDPDE! QE< Q a ± M @ KLO TP)$ K6! G E

2Cy(R)DFEL · ED£EJ KLN! H ¸

ψ ∈ H 3D! = K6gj ∈ G MNv Q"FED! /!DD ¤º! T¨@E5E$= EN¬# E 4 EFEJ HG

√√√√ 1

M

M∑

m=1

|ψ(Zm)|2 −

√√√√ 1

M

M∑

m=1

|ψ(ZM+m)|2

≤

√√√√ 1

M

M∑

m=1

|ψ(Zm) − gj(Zm)|2 +

√√√√ 1

M

M∑

m=1

|gj(Zm)|2 −

√√√√ 1

M

M∑

m=1

|gj(Zm+M )|2

+

√√√√ 1

M

M∑

m=1

|gj(ZM+m) − ψ(ZM+m)|2.

±Y¡KL K6¡!=K6KH"!¡! KLE QE"G√

2

√√√√ 1

2M

M∑

m=1

|ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2 + |ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2 ≤ 1

3h

β+12

3E:KK6" $E JEFEJ"! ¬K6¡!=K6KH"!¡! OP G

P(∃ψ ∈ H :

√√√√ 1

M

M∑

m=1

|ψ(Zm)|2 −

√√√√ 1

M

M∑

m=1

|ψ(ZM+m)|2 > hβ+1

2

)

≤P(∃g ∈ G :

√√√√ 1

M

M∑

m=1

|g(Zm)|2 −

√√√√ 1

M

M∑

m=1

|g(ZM+m)|2 > 1

3h

β+12

).

ª ! )QE(PN,m

tk+1, PN,m

tk+1)1≤m≤M

¯3:MN-E%DE G ! ! MNK6N:!DJEFEE EJ¬3E!(Um)m

<¨"/ ¡G

P

(

∃g ∈ G :

√√√√ 1

M

M∑

m=1

|g(Zm)|2 −

√√√√ 1

M

M∑

m=1

|g(ZM+m)|2 > 1

3h

β+12

∣∣∣∣(PN,m

tk+1, PN,m

tk+1)1≤m≤M

)

≤N2

(1

3√

2h

β+12 ,H, (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M

)

maxj

P

(√√√√ 1

M

M∑

m=1

|gj(Zm)|2 −

√√√√ 1

M

M∑

m=1

|gj(ZM+m)|2 > 1

3h

β+12

∣∣∣∣(PN,m

tk+1, PN,m

tk+1)1≤m≤M

)

.

^´´ M

ksyj.pt=l=|

s zmxl@sut@pvmxw q| |l=l@|ql ~ mr r pY'| ksywksxvymxl@pt@kr |

O EJ!!cKLE EJ b! KL ! c!:D E D !:N QE K6N K6¡ ENMV¶¡# DK6KL !QE3 EJ4K6()ZJNf<h @VegS@ACA

F

max0≤k≤N

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2

≤CCy(R)2 + ‖fR‖2∞

M

N−1∑

k=0

max0≤l≤q

E(Kl,k(ω)) + C

N−1∑

k=0

infα

E|α.p0,k(PNtk

) − yN,Rk (PNtk

)|2

+ C

N−1∑

k=0

q∑

l=1

infα

E|α.pl,k(PNtk

) −√hzN,Rl,k (PN

tk)|2 + Chβ−1

+C

hCy(R)2e

− Mhβ+1

144Cy(R)2

N−1∑

k=0

E

(

N2

(hβ+1

2

3√

2, [P0,k+1]y − yN,Rk+1 , (P

N,mtk+1

)1≤m≤M , (PN,mtk+1

)1≤m≤M))

.

¨- Q"!D¡MNβ

! " (!1

EJEHDND!"# E K6MNh → 0

HMNβ > 2

:N 6D£E) 6KLhβ−1 !V !E :E $£E!·JE

hMN¬ NV NH!QE:E%KE EJ g!-# H!-! D E K6 K6/´ d¨ V!E :E!!QE5D/D E -5# KE ! K:-!D N2

MN¶ NVND ¤º!3KE 3# EJN=E6$ TD!! !¡! K6¡´´ :!33 a´´ 3ZN,R

PNtk¨/JE5 · E K6

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³()ZJNf<h @VegS@ACA

max0≤k≤N

E

∫

|yN,Rk (x) − yN,R,Mk (x)|2µk(dx)

≤Chβ−1 + CCy(R)2 + ‖fR‖2

∞M

N−1∑

k=0

max0≤l≤q

E(Kl,k(ω)) + C

N−1∑

k=0

infα

E(|yN,Rk (PN

tk) − α.p0,k(P

Ntk

)|2)

+ C

N−1∑

k=0

q∑

l=1

infα

E(|α.pl,k(PN

tk) −

√hzN,Rl,k (PN

tk)|2)

+ CCy(R)2 max0≤k≤N−1K0,k

Mlog(M)

+C

hCy(R)2e

− Mhβ+1

144Cy(R)2

N−1∑

k=0

E

(

N2

(hβ+1

2

3√

2, [P0,k+1]y − yN,Rk+1 , (P

N,mtk+1

)1≤m≤M , (PN,mtk+1

)1≤m≤M))

.

O\EN!! K6ND K6<= MN %N3FE5KKL¡MN3 ^´´ /!:/KL¡E!!

CCy(R)2 max0≤k≤N−1K0,k

Mlog(M)

g¯33E!DE:J EJ

‖ψ‖2k =

∫

|ψ(x)|2µk(dx)

"*aDψ

K6E: ¶± E∫|yN,Rk (x) − yN,R,Mk (x)|2µk(dx)

6 ·¤ D EJE‖yN,Rk − yN,R,Mk ‖2

k

=E

(

‖yN,Rk − yN,R,Mk ‖k − 2‖yN,Rk − yN,R,Mk ‖k,M + 2‖yN,Rk − yN,R,Mk ‖k,M)2

≤E

(

max‖yN,Rk − yN,R,Mk ‖k − 2‖yN,Rk − yN,R,Mk ‖k,M , 0 + 2‖yN,Rk − yN,R,Mk ‖k,M)2

≤E(T1) + E(T2)

ECDT1 = 2 max‖yN,Rk −yN,R,Mk ‖k−2‖yN,Rk −yN,R,Mk ‖k,M , 02

T2 = 8‖yN,Rk −yN,R,Mk ‖2k,M

°coEMNQEJN: ' K6-´´ *^EJD !/E(T2)

EN5! /MN D¤QE 5 a´´ / 7¥\5KE E(T1)

coEJ!QE=E=! K6 !E µ²¡²¼ Y= D JEN¡G

P(T1 > u) =P

(

2 max‖yN,Rk − yN,R,Mk ‖k − 2‖yN,Rk − yN,R,Mk ‖k,M , 02 > u

)

≤P

(

∃ψ ∈ [P0,k]y − yN,Rk : ‖ψ‖k − 2‖ψ‖k,M >

√u

2

)

≤3EN2(

√u

24, [P0,k]y − yN,Rk , PN,1:2M

tk) exp(− Mu

576(2Cy(R))2)

a´´ ) /

C

ª¬«OP¥c´¦³° ) %I@O"¢¨"³Oc´ ¨"¯°± °7³3³°7©³ ª¨¡I®I@´ ¸ ° OP¯ ¸ ± O±\µ¨¡³´ «I@°- E KLO M E)! Q '! N2

^! Q O /dE)J EJPN,1:2Mtk

=

(PN,1tk

, · · · , PN,Mtk

, PN,M+1tk

, · · · , PN,2Mtk

)

(PN,M+mtk

)1≤m≤M 3D ! !QEJN!

(PN,mtk

)1≤m≤M <¨"=E !5 )G

N2(

√u

24, [P0,k]y − yN,Rk , PN,1:2M

tk) =N2(

√u

24, [P0,k]y, P

N,1:2Mtk

)

=N2(

√u

24, [P0,k]y + Cy(R), PN,1:2M

tk).

[P0,k]y +Cy(R)E 7PDFEP!aD· d ·¬ ¬E

2Cy(R) ¨!D6E MV) K6:O aDH KL:"FE' EJ

VG+

"5DE!%aD· G ¬ D ¡MN5G

N2(

√u

24, [P0,k]y + Cy(R), PN,1:2M

tk) ≤3

(2e(2Cy(R))2

u576

log3e(2Cy(R))2

u576

)V([P0,k]y+Cy(R))+

≤3

(1728e(2Cy(R))2

u

)2V([P0,k]y+Cy(R))+

= EMNlog(x) ≤ x

¯3EC3E GV([P0,k]y+Cy(R))+ = V([P0,k]y)+ .°c@ ·Q

n = V([P0,k]y+Cy(R))+·3D ! 3@¦¤ KH !

Rd′+1 EJVD (z1, t1), . . . , (zn, tn).

¥ Eb! Q )!V([P0,k]y+Cy(R))+

b ¤ºKH I

! 1, . . . , n HEMN V g ∈ [P0,k]y + Cy(R)

MNg(zi) ≥ ti

i ∈ I

g(zi) < ti

¨ E®EJ !/D ! ¡:¦¤ KH (z1, t1 − Cy(R)), . . . , (zn, tn − Cy(R)) ¡DD 5MNV([P0,k]y+Cy(R))+ ≤ V([P0,k]y)+

<± E P!QE# E"3$ E !¡FEHK KL"^E P¡K KL (^EJD !¡K6N¬MVV([P0,k]y)+ ≤ V(P0,k)+ .

°c9T$:n = V([P0,k]y)+

· D ! 1¦¤ KH !Rd′+1 EVD

(z1, t1), · · · , (zn, tn) MNNb DFE 7QE

([P0,k]y)+ ¥ EJ\! Q £Yb¦¤ KH

I! 1, . . . , n

g ∈ P0,k

QMN[g]y(zi) < ti

i ∈ I

·[g]y(zi) ≥ ti

i /∈ I

´ 6¬!PK6NdMNPi ∈ I g(zi) ≤ [g]y(zi)

i /∈ I g(zi) ≥ [g]y(zi)

= K63!(P0,k)

+ MNY E " ¤ºKH I ¸ MV \

i ∈ IbMN

g(zi) > [g]y(zi) OP DE K6

g(zi) > Cy(R)!D[g]y(zi) = Cy(R)

TOP b!·VE "EC ti > Cy(R)

·" \HE'QEP )! ) K6 ![P0,k]y

MV E E3 dDK6 K6 E 7!i

!QE 1, . . . , n J°7)T$TT K6g′

EJ NDD D E [g′]y(zi) ≥ ti > Cy(R)

DPMN ¬K6 C

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³OP

g(zi) ≤ [g]y(zi)

i ∈ I E6QEJvMN

i /∈ IcMN

g(zi) < [g]y(zi) YO¡ DEJKLN

g(zi) < −Cy(R)!D

[g]y(zi) = −Cy(R) QOP#

ti ≤ −Cy(R)·vE6QE3! £EJ\ ¦¤ KH (zi, ti) J°cHT$JDD D EJ ! H KLN

g′!

P0,kMN

[g′]y(zi) < ti ≤ −Cy(R)DMNcK6 O

i /∈ I g(zi) ≥ [g]y(zi) ¨'E!D L K6Nd! P0,k

<MNg(zi) < ti

i ∈ I

g(zi) ≥ ti

i /∈ I

OP V([P0,k]y)+ ≤ V(P0,k)+

ª K6K6""K6 "E:KLEMN"!@E3!E:E5 H!QE µ²¡²¼ Q@EP+

0,k =(z, t) ∈ Rd′ × R : t ≤ g(z) : g ∈ P0,k

⊂(z, t) ∈ Rd′ × R : α.t+ g(z) ≥ 0 : g ∈ P0,k, α ∈ R

.¨@EMN"E KL"O E:DD¡MN

V(P0,k)+ ≤ K0,k + 1.¨==! ! QQE K6# $E 4 )GN2(

√u

24, [P0,k]y + Cy(R), PN,1:2M

tk) ≤ 3

(1728e(2Cy(R))2

u

)2(K0,k+1)

.

°c@ D· E!QE ^´´ ) /T P(T1 > u) ≤9

(1728e(2Cy(R))2

u

)2(K0,k+1)exp

(− Mu

576(2Cy(R))2

)

≤9(12eM)2(K0,k+1) exp(− Mu

2304Cy(R)2

)

u > 576Cy(R)2

M

¨ E73 Vd!3D·!3 E dKE E(T1)

°c@ ·GE(T1) =

∫ ∞

0

P(T1 > u)du

≤v +

∫ ∞

v

P(T1 > u)du

≤v + 9(12eM)2(K0,k+1)

∫ ∞

v

exp(− Mu

2304Cy(R)2

)du

=v + 9(12eM)2(K0,k+1).2304Cy(R)2

Mexp

(− Mv

2304Cy(R)2

)

=φ(v)

v > 576Cy(R)2

M

Q±Y¡KL K)K !E aD φ

PEv =

2304Cy(R)2

Mlog9(12eM)2(K0,k+1).

C

ª¬«OP¥c´¦³° ) %I@O"¢¨"³Oc´ ¨"¯°± °7³3³°7©³ ª¨¡I®I@´ ¸ ° OP¯ ¸ ± O±\µ¨¡³´ «I@°¥bD¡E$/ ¡G

E(T1) ≤ CCy(R)2K0,k

Mlog(M).

¤

zN,R

Oy MV)E£! ·KL FE:V )!DND)!# yN,R

< ¶^ED 6!LKLFE/V '!D D-!#

zN,R ¨HQEJ D E

()ZJNf<h @VegS@ACAL

hE

N−1∑

k=0

1

M

M∑

m=1

q∑

l=1

|zN,Rl,k (PN,mtk


)|2

≤Chβ−1 + CCy(R)2 + ‖fR‖2

∞M

N−1∑

k=0

max0≤l≤q

E(Kl,k(ω)) + C

N−1∑

k=0

infα

E(|yN,Rk (PN

tk) − α.p0,k(P

Ntk

)|2)

+ C

N−1∑

k=0

q∑

l=1

infα

E(|α.pl,k(PN

tk) −

√hzN,Rl,k (PN

tk)|2)

+C

hCy(R)2e

− Mhβ+1

144Cy(R)2

N−1∑

k=0

E

(

N2

(hβ+1

2

3√

2, [P0,k+1]y − yN,Rk+1 , (P

N,mtk+1

)1≤m≤M , (PN,mtk+1

)1≤m≤M))

.

°c@E D /KLEMV¡MN# zN,R

hE

N−1∑

k=0

1

M

M∑

m=1

q∑

l=1

|zN,Rl,k (PN,mtk


)|2

!@KK6"!"MV# yN,R

max0≤k≤N

E1

M

M∑

m=1

|yN,R,Mk (PN,mtk

) − yN,Rk (PN,mtk

)|2,

D¡MNYE QEY!3K6! °d ¸ ³ C

ª¬«OP¥c´¦³° ) %I@O"¢¨"³Oc´ ¨"¯°± °7³3³°7©³ ª¨¡I®I@´ ¸ ° OP¯ ¸ ± O±\µ¨¡³´ «I@°´ vÊ!D"%a¬D¡V!P ·KLN

(AMk )k °7/T

k ≥ 1G

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2

=E‖yN,Rk − yN,R,Mk ‖2k,M

≤(1 + h−1)E

(

‖yN,Rk − yN,R,Mk ‖k,M − ‖yN,Rk − yN,R,Mk ‖k,M)2

+

+ (1 + h)E‖yN,Rk − yN,R,Mk ‖2k,M

≤Chβ +C

hCy(R)2P([AMk−1]

c) + (1 + h)E‖yN,Rk − yN,R,Mk ‖2k,M .

a´´ ) °cDEN/D· ! $EJ !E a´´ ) '= EJN ^´´ QQEJK6N

γ > 0G

hE

N−1∑

k=0

1

M

M∑

m=1

q∑

l=1

|zN,Rl,k (PN,mtk


)|2

≤ChE

N−1∑

k=0

q∑

l=1

T4,k,l,M + T5,k,l,M + Chβ−1 +C

hCy(R)2

N−1∑

k=0

P([AMk ]c)

+ C max0≤k≤N

E‖yN,Rk − yN,R,Mk ‖2k,M + C(1 + γh)

N−1∑

k=0

E(T2,k,M )

+ Cγh

N−1∑

k=0

1

ME

M∑

m=1


) − [αM0,k+1.pm0,k+1]y|2 + C

N−1∑

k=0

ET1,k,M

+ C(1 +1

γh)N−1∑

k=0

ET3,k,M + C(h2 +h

γ)E

N−1∑

k=0

1

M

M∑

m=1

|yN,Rk+1 (PN,mtk+1

) − [αM0,k+1.pm0,k+1]y|2

+ C(h2 +h

γ)E

N−1∑

k=0

1

M

q∑

l=1

M∑

m=1

|zN,Rl,k (PN,mtk


³-E !%D γ = 2C

!%h

LEK6KLNH ))EJ aKL6KLE 1M

∑Mm=1 |yN,Rk+1 (PN,m

tk+1) − [αM0,k+1.p

m0,k+1]y|2

MV< NVN7D3!QE7K6KH!! !L# E 4 5D ¤º!b

E 1M

∑Mm=1 |yN,Rk+1 (PN,m

tk+1) − [αM0,k+1.p

m0,k+1]y|2

!=# E!6! · ¤K6AMk

!EKK6(Ê -MN%^´´ d QQE K6 QE !)FEDD! K6 ¤

P¡FEHK KL Ê /MV K6¡´´ ) E ÊD ! E

C

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O

hE

N−1∑

k=0

q∑

l=1

∫

|zN,Rl,k (·) − zN,R,Ml,k (·)|2µk(dx)

≤Chβ−1 + CCy(R)2 + ‖fR‖2

∞M

N−1∑

k=0

max0≤l≤q

E(Kl,k(ω)) + CN−1∑

k=0

infα

E(|yN,Rk (PN

tk) − α.p0,k(P

Ntk

)|2)

+ CN−1∑

k=0

q∑

l=1

infα

E(|α.pl,k(PN

tk) −

√hzN,Rl,k (PN

tk)|2)

+Cy(R)2Kl,k

Mlog(M)

+C

hCy(R)2e

− Mhβ+1

144Cy(R)2

N−1∑

k=0

E

(

N2

(hβ+1

2

3√

2, [P0,k+1]y − yN,Rk+1 , (P

N,mtk+1

)1≤m≤M , (PN,mtk+1

)1≤m≤M))

.

# #

= = *; #2 #.-'+*/'7 +*/-' :D& # #&'

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β+12

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N2

(hβ+1

2

3√

2, [P0,k+1]y − yN,Rk+1 (·), (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M)

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hβ+1

)2(K0,k+1+1)

≤C exp(CK0,k+1 logCy(R)

hβ+1).

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max0≤k≤N

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2

+ hE

N−1∑

k=0

1

M

M∑

m=1

q∑

l=1

|zN,Rl,k (PN,mtk


)|2

≤CCy(R)2 + ‖fR‖2∞

M

N−1∑

k=0

max0≤l≤q

E(Kl,k(ω)) + CN−1∑

k=0

infα

E|α.p0,k(PNtk

) − yN,Rk (PNtk

)|2

+ CN−1∑

k=0

q∑

l=1

infα

E|α.pl,k(PNtk


tk)|2 + Chβ−1

+C

h2Cy(R)2 exp

(

CmaxkK0,k log(

Cy(R)

hβ+1) − Mhβ+1

144Cy(R)2

)

.

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·M

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M

KLEE!¡# $

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tk KLH H! E KEJL!

yN,Rk (·) ·¬!zN,Rl,k (·) aDL!'D L!¬QE

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!PEE NKL MV b! DED\b D · cMN¬EQEpl,k

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·MNbMN) k O\

0 ≤ k ≤ N − 1

0 ≤ l ≤ q Kl,k = K ¯E3 ! · E

εk(K)FEHK6K6"!3! E KLEJ

infα

E(|yN,Rk (PN

tk) − α.p0,k(P

Ntk

)|2)

+

q∑

l=1

infα

E(|α.pl,k(PN

tk) −

√hzN,Rl,k (PN

tk)|2).

C9E

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max0≤k≤N

E1

M

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m=1

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)|2

+ hE

N−1∑

k=0

1

M

M∑

m=1

q∑

l=1

|zN,Rl,k (PN,mtk


)|2

≤C(1

N)β−1 + CNK

Cy(R)2 + ‖fR‖2∞

M

+ CN−1∑

k=0

εk(K) +CCy(R)2

h2exp

(

CK log(NCy(R)

T) − MT β+1

144Nβ+1Cy(R)2)

.a´´ )

¨ E \MN MN E4K57D ¬ MNK,N

M

! # )#£Ê c! E!5MN

β > 1 MN

maxk εk(K)!¬

0MN

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maxk εk(K)!@)6EK6KLN6V /

06MN ∑N−1

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M aD H!7EJQEEK65Ê@DN- E4K6 d°c · ED! ∑N−1

k=0 εk(K) → 0K6PMNK

D ¬# )DKLK6DEaD K(N)

¥ EJ¬ K6 DE!)E6QEJ)! !DEJD3! D @EL!QEFE5QENK6 MNMN) EEεk(K) ≈ K− 2

d .

´ D7o! 6!D%ECNK− 2

d → 06MV ∑N−1

k=0 εk(K)!=

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M<MN

ε(M) =

CK log(NCy(R)

T)− MTβ+1

144Nβ+1Cy(R)2→ −∞ ·¶)6EJKLK6YV vMN

Cy(R)2N2 exp(ε(M)) →0 E@DE$dDFE ·V !!

M = CCy(R)2Nβ+1K log(NCy(R)

T)

6DENC

E EJ! ¯d!·E MNK6!3LDKLMNR

! c!d# )Ê ¬!3

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M = CNβ+1K log(N)¡D E

CEJ E!

= = `= )+& T< /:< V, #(&UQ] 09b] \S 0fTe` ]aS9N¶[ , JöE =ECD=!L!K6

d′, q ·Lo

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(l, k) ± EJ K6H

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M À K OP FEDK6 E!"# !P!

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tk :K66!KL QE¡!6FE-DKL 5 V )E "!6E£¡MN N ED MNcK: !QEc5NVD! DMN7 :K6 D5MN

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M·HMN'LD

M ≈ Nβ+1+εK

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K

E 5E! "·)MNL 5KLCNK Cy(R)2+‖fR‖2

∞

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hK2d

,

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C G(

1

N)β−1 +N1+

2(β+2+ε)3d C− 2

3d .

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# E D EN Nbc!:K KL!¡DMNY!6G

N = C 2(3d+2)β+2(2+ε) .

C M

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a¡ QEHD£E¡ K64ε = 0

DE-!¡3 C− 17

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M EJ3 K:¡!K

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E D ¤cKH c ENtk

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n t=|wk-pvmxwk

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T5,k,l,M

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m0,k+1]y[α

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m′

0,k+1]y|(PN,mtk

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m0,k+1]y|(PN,m

tk)1≤m≤M

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m′

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) W

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Lφ

aD·

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t

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M

M∑

m=1

|yN,M (tk,Wmtk

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pk(·)

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εN−1 ≤ infα

E|y(tN−1,WtN−1) − α.pN−1(WtN−1

)|2 +‖φ‖2

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M

' k ≤ N − 2

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∞E(Kk(ω)))

Mh.

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144Cy(R)2)E

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N,mtk+1

)1≤m≤M)) +#

+- +-+- #$' +-$ # , + $ ‖·‖k+1,M

#$ ‖·‖k+1,M # $"!# + #$

ª¬«OP¥c´ ³° @° »°7¯ ¸ ´ ¨¡¯ ¸¥b

k = N−1L3! D·KLNd!QEJdD£EJ!3!6 KLO N¥b

k ≤ N − 2V!

βMk/!5G

βMk = arg infβ

1

M

M∑

m=1

|y(tk+1,Wmtk+1

) − β.pk(Wmtk

)|2.

OP !5# E !!βMk

@E/G

EMk

1

M

M∑

m=1

|yN,M(tk,Wmtk

) − y(tk,Wmtk

)|2

≤EMk

1

M

M∑

m=1

|αMk .pmk − y(tk,Wmtk

)|2

=EMk

1

M

M∑

m=1

|αMk .pmk − EMk (βMk ).pmk |2 + EM

k

1

M

M∑

m=1

|EMk (βMk ).pmk − y(tk,W

mtk

)|2

= EFEH!¡ $E MNEMk (βMk )

3FEH -!

infα

1

M

M∑

m=1

|y(tk,Wmtk

) − α.pmk |2.

±Y!KLPKLP!-K6KHP!! PE !E)KLE PMNT1,k,M

¯E£! N!=E QEJ ¬ EMk

1M

∑Mm=1 |αMk .pmk − EM

k (βMk ).pMk |2 !εk+1

¸ MN

EMk

1

M

M∑

m=1

|αMk .pmk − EMk (βMk ).pmk |2 =EM

k

1

M

M∑

m=1

|EMk (αMk ).pmk − EM

k (βMk ).pmk |2

+ EMk

1

M

M∑

m=1

|αMk − EMk (αMk ).pmk |2

' E MN@E @!®DNED· !%FE D· ((pmk )∗

)

1≤m≤MK6LK6®! K6KH®!®! ¬¥ E'DN DK6K6® Eg! ! K6 DN 3!K6KL!®K6KH!)! <%)P 3E\!# EVFEJ !K6¡¤º!FE QEJ!6FE-KLEJD

EMk

(V − EM

k (V )V − EMk (V )∗

) V

D·L!RM !=DV!

([αMk+1.pmk+1]y)1≤m≤M

d¨"c!QE6-D£E6 o /! f

! !@QE!Z=K6K6 D5G

E1

M

M∑

m=1

|αMk .pmk − EMk (βMk ).pmk |2 ≤(1 + h)E

1

M

M∑

m=1

|αMk .pmk − βMk .pmk |2

+ (1 + h−1)E1

M

M∑

m=1

|βMk − EMk (βMk ).pmk |2.

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³±¶E !DNED !FEgD· (

(pmk )∗)

1≤m≤MK6-! KLE 'K6 K6¡!@K6KH"!¡! QE

(1 + h)E1

M

M∑

m=1

|[αMk+1.pmk+1]y − y(tk+1,W

mtk+1

)|2 = (1 + h)εk+1.

¥bd 3D!K6NL dKK6EJK6dMNT3,k,M

ª DTK6! E¤ ¬ E !¡E: ¤OP 5MN /!E6-D£EJL o /! 5=! !E6!

Z! · EJ !5 EJH KL EJ !D! K6-´´ ®E5 )!5D KLN: KLNE PH !!"MV: ®¡EMN# E K6! D "E®! "!D·'E6!QELDE Ed

fH! !'!

Z)EJ) !L KLNEc E MN oE5MN/ aEJD 5KLE NQE ¥ EJ5KLEMN-!KL$J ! KL K6Nb¬Ed! FE3! KLEJ N ! MNDFEHJE5KLED $

#

¨ JE @D"MV# EVDE $EKLN/= D K6N= OP#¬5MN |∆Wml,k| + |∆Wm

l,k| ≤ Cw(R)√h Y³KLEMV"MN:# "!:VD£EJ EJ %

aD· /!R

"E "!¡E:KK6 ^E =MNE: -´´ ¥bPPE P ¡PQEJT\" KL EJN!H P E 3H $E ! a´´ D'MV¬5 NV!EJN)!: D K6H H KL¤ EJ$ <E DE @%E MNE ®EJ K6"E K6 E <EE yN,R,Mk (·) = [αM0,k.p0,k]y(·)

zN,R,Ml,k (·) = [αMl,k.pl,k]z(·)

(αMl,k)0≤l≤q

3! Q¶E¡G

infαl

1

M

M∑

m=1


)∆Wm

l,k

h− αl.p

ml,k|2,

infα0

1

M

M∑

m=1


) + hf(tk, XN,mtk

, yN,R,Mk+1 (PN,mtk+1

), zN,R,Ml,k (PN,mtk

)) − α0.pm0,k|2.a´´ /±¶E®!T D-ECD ^´´ ¤ ´´ E5!D'MN-# E%K6ED

(PN,mtk+1

,∆Wmk )1≤m≤M

QE(PN,m

tk+1,∆Wm

k )1≤m≤M ¨=JE:D ¤º!MN!@ K6N3QE3 3=JE5@QE ¤D E 3FEL @MV<) ! $EJ a´´ M ¤º´´ !QEJD£EP!# E4K6ECD 3E E "G

ª¬«OP¥c´ ³° @° »°7¯ ¸ ´ ¨¡¯ ¸8hf<`cf .£[ ,C[afW A A

O ' k ≤ N − 2

Eh

M

M∑

m=1

|αMl,k.pml,k − zN,Rl,k (PN,mtk

)|2

≤CCy(R)2

ME(Kl,k(ω)) + inf

αE|√hzN,Rl,k (PN

tk) − α.pl,k(P

Ntk

)|2 + Chβ

+ CE1

M

M∑

m=1

|[αM0,k+1.pm0,k+1]y − yN,Rk+1 (PN,m

tk+1)|2

− CE1

M

M∑

m=1

|E[αM0,k+1.pm0,k+1]y − yN,Rk+1 (PN,m

tk+1)|2

+ CE

(

N2

( hβ2

√

18Cw(R)2Kl,k(ω), [P0,k+1]y, (P

N,mtk+1

)1≤m≤M , (PN,mtk+1

)1≤m≤M)e− Mhβ

36Kl,k(ω)Cy(R)2Cw(R)2

)

PN,mtk+1

= Tk(PN,mtk

,∆Wmk,k+1)

+ !(∆Wm

k,k+1)1≤m≤M'$ ! (∆Wm

k,k+1)1≤m≤M+ ! , + $ "+ $E(·) = E(·|(∆Wm

k,k+1)1≤m≤M,0≤k≤N−1)

,'SVegUh ¶S%ACA $ ! $ + $ +- # #, #, +-, & +-$ # ', + $ %$ !#

#$ $ #$ + %, +-% '$+-, & # $ # , #)+- ' , + "&-# $ , %$"+-$tk (PN,m

tk+1)1≤m≤M

, +- (∆Wm

k,k+1)1≤m≤M *!#$ # +# , #'$ %$ "& +-,

+ ! ! +- ! +-$1^´´ J ! ' $ %, ' + # , + $ $ + & + $ , ## ' +-$ , +-, & + # , +# +- $ #

hβ, ##$ # # +-$ , %$ "& +-, !# "+- #$ + #$ +-$ , + )+ $ , $ #$+- +-$ , +-, & + *! # ', + $ %$ !##$ $ #$ ,#$ +-%

³KLEMNMN

zN,Rl,k (PN,mtk

) = E(yN,Rk+1 (PN,mtk+1

)∆Wm

l,k

h)

(∆Wm

l,k)1≤m≤M D ! !QE !

(∆Wml,k)1≤m≤M

· E(·) = E

(· |(∆Wm

k,k+1)1≤m≤M,0≤k≤N−1

) ¬OP# NV!EJN'%D·βMl,k

!

infα

1

M

M∑

m=1

|yN,Rk+1 (PN,mtk+1

)∆Wm

l,k

h− α.pml,k|2

KE

ª¬«OP¥c´ ³° @° »°7¯ ¸ ´ ¨¡¯ ¸±YP! dK6EJ 3DK6KL

T5,k,l,M

D£EαM0,k+1

¬ ! T!QEJN¬!(∆Wm

k,k+1)m·¬E- · EP5 $EJ H!Lª¬ED ¤ ¸ D ¬EJ L!5E-KK6 ^E MV6 ^´´ J DKLK6: K6¡ ¡! K6"K6! KLKH:!6! :!5FE-! 6 ¤ $E ³3¡K6 1

M

∑Mm=1 |αMl,k.pml,k − αMl,k.p

ml,k|2

<°c=DK6QEEN MNQEJ a´´ / a´´ -KLE EJ MN"¬DV6DNαMl,k

αMl,k

N VD $ <ª "MN ¬K6 7 V!N$VD ¬MVP# L d! ! EJ ^EJ Nd K6KLN"! !6T !=/!pl,k

¸ E 6!* E a¡EO MNPMl,k = Id

QO1

M

M∑

m=1

|αMl,k.pml,k − αMl,k.pml,k|2 =(αMl,k − αMl,k).P

Ml,k (α

Ml,k − αMl,k)

=‖αMl,k − αMl,k‖2.

¨ β

¬MN1 < β ≤ 2

·5E!'FE%J EJPMk = P(·|(PN,m

tk)1≤m≤M )

¸ # NVPMl,k = Id

@E

PMk

(

‖αMl,k − αMl,k‖2 ≥ hβ−1

)

=PMk

(Kl,k(ω)∑

i=1

∣∣

1

M

M∑

m=1

pml,k,i[αM0,k+1.pm0,k+1]y

∆Wml,k

h− [αM0,k+1.p

m0,k+1]y

∆Wml,k

h∣∣2 ≥ hβ−1

)

≤Kl,k(ω)∑

i=1

PMk

(∣∣

1

M

M∑

m=1

pml,k,i[αM0,k+1.pm0,k+1]y

∆Wml,k

h− [αM0,k+1.p

m0,k+1]y

∆Wml,k

h∣∣2 ≥ hβ−1

Kl,k(ω)

)

≤Kl,k(ω)∑

i=1

PMk

(

∃ψ ∈ [P0,k+1]y :∣∣

1

M

M∑

m=1

pml,k,iψ(PN,mtk+1

)∆Wm

l,k

h− ψ(PN,m

tk+1)∆Wm

l,k

h∣∣2 ≥ hβ−1

Kl,k(ω)

)

=

Kl,k(ω)∑

i=1

PMk

(

∃ψ ∈ [P0,k+1]y :∣∣

1

M

M∑

m=1

pml,k,iUmψ(PN,mtk+1

)∆Wm

l,k

h− ψ(PN,m

tk+1)∆Wm

l,k

h∣∣ ≥

√

hβ−1

Kl,k(ω)

)

(Um)

!JEFE3E EJ# # ! ^·P! !ENP!P PEJJEJFEJE EJ7 K: EFEJN1

−1E£DPQE 1

2

V´ V! EdPD¦¤ G ![P0,k+1]y

E dψ ∈ [P0,k+1]y

VQ gj ∈ G V PQE

Cy(R) "MN1

2M

M∑

m=1

|ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2 + |ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2 ≤ hβ

18Cw(R)2Kl,k(ω)

ET ®MNCw(R)

/"MN |∆Wml,k| + |∆Wm

l,k| ≤ Cw(R)√h ¯-MV

D£EJ! QEc! G N2

(h

β2√

18Cw(R)2Kl,k(ω), [P0,k+1]y, (P

N,mtk+1

)1≤m≤M , (PN,mtk+1

)1≤m≤M) ¶O

M

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³

ψ ∈ [P0,k+1]y·gj

<EJN# E 4 D ¤º! T¨@E

∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− ψ(PN,m

tk+1)∆Wm

l,k

h∣∣

≤∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− gj(P

N,mtk+1

)∆Wm

l,k

h∣∣

+∣∣

1

M

M∑

m=1

pml,k,iUmgj(PN,mtk+1

)∆Wm

l,k

h− gj(P

N,mtk+1

)∆Wm

l,k

h∣∣

+∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− ψ(PN,m

tk+1)∆Wm

l,k

h∣∣

¨@E

∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− gj(P

N,mtk+1

)∆Wm

l,k

h∣∣2

=1

M

M∑

m=1

|ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2|∆Wm

l,k

h|2

≤2Cw(R)2

2hM

M∑

m=1

ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2 + |ψ(PN,mtk+1

) − gj(PN,mtk+1

)|2

≤ hβ−1

9Kl,k(ω)

EJNdMN 1M

∑Mm=1 |pml,k,i|2 = 1

QEV <ª KLK63FEKK63 E JEJFE K6 ∣

∣ 1M

∑Mm=1 p

ml,k,iUmgj(PN,m

tk+1)

∆Wml,k

h− ψ(PN,m

tk+1)

∆Wml,k

h∣∣Q/ ¡G

PMk

(

∃ψ ∈ [P0,k+1]y :∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− ψ(PN,m

tk+1)∆Wm

l,k

h∣∣ ≥

√

hβ−1

Kl,k(ω)

)

≤PMk

(

∃j :∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− gj(P

N,mtk+1

)∆Wm

l,k

h∣∣ ≥ 1

3

√

hβ−1

Kl,k(ω)

)

.

ª¬«OP¥c´ ³° @° »°7¯ ¸ ´ ¨¡¯ ¸¥bvYDc!vK6¡D! ¡YQEvEJ !

(∆Wmk,k+1,∆W

mk,k+1)1≤m≤M

¨=PMk+1

D "QE4 ´ vV N¡G

PMk+1

(

∃j :∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− gj(P

N,mtk+1

)∆Wm

l,k

h∣∣ ≥ 1

3

√

hβ−1

Kl,k(ω)

)

≤N2

(h

β2

√

18Cw(R)2Kl,k(ω), [P0,k+1]y, (P

N,mtk+1

)1≤m≤M , (PN,mtk+1

)1≤m≤M

)

maxj

PMk+1

(∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− gj(P

N,mtk+1

)∆Wm

l,k

h∣∣ ≥ 1

3

√

hβ−1

Kl,k(ω)

)

¯JXm = pml,k,iUmgj(PN,m

tk+1)

∆Wml,k

h− gj(P

N,mtk+1

)∆Wm

l,k

h E

EMk+1(Xm) = 0

· |Xm| ≤2Cy(R)Cw(R)√

h|pml,k,i|

Q°c%E MNQEN# $EJ !«V !=N¡G

PMk+1

(∣∣

1

M

M∑

m=1


)∆Wm

l,k

h− gj(P

N,mtk+1

)∆Wm

l,k

h∣∣ ≥ 1

3

√

hβ−1

Kl,k(ω)

)

≤2 exp

(

− Mhβ

36Kl,k(ω)Cy(R)2Cw(R)2 1M

∑Mm=1 |pml,k,i|2

)

=2 exp

(

− Mhβ

36Kl,k(ω)Cy(R)2Cw(R)2

)

E"# NV 1M

∑Mm=1 |pml,k,i|2 = 1

PMl,k = Id

¶¨"E-!D6 !-E!:FE K6 Nαl,k

·αl,k

E3 ¬KLKLN! !¡ (!LH!pl,k

¤

´´ QO±\µ¨¡³´ «I@° I®¨"P´ ´¦° ¥d¨"©³u±¶° ¸ °7 ¸ ³

ksyj.pt=l=|

pt/|-'| q| ~ mw |l=x|wk~| ksywk v| ~syql@p1|l YpYw sxpvl=|

¯3ECN@!QEJ3E:E"´EHV max

0≤k≤NE|Ytk − Y N

tk|2 ≤ C

√h

:# E KLE !Y

E5E!= °7 ¸ ³! D=! Q /QE^´ C¤ ´ E")DE )! f

" £E (y, z)

b" )! P)V :EJ!) EJKLEJ %!Y

QEY N v¯¡E P! K6N$ ENP ! ! · !QE ²¡«P µI% Y ²¡«P¥7 QMN¡# /NEHV JE5G

|Yt0 − Y Nt0| ≤ Ch.

ª H E!QEP# P!PE£JE%! ³PI M b ¥7 M M ® PEPD!N=D£EJ!¡ EYKLE¬ N3QE!"V !¡DND

¥c D! E!/NVMN EJ ECE G> v`7f-, Z@ .S%A ACA

$ d = q = 1 , $ ! $ b(t, x) σ(t, x)

$ +-$ , $ $ # #$ , $ $ C∞

b

+-$ , + ' $ $ , + $ ! $ φ(x) ' $ , + ! $ - $ #%$+-, , + +- ,! $ ,

|φ(x)| ≤ C exp(C|x|).

$ $ $ σ '$ #$ #,%, + #$ |σ(t, x)| ≥ σ0 > 0 ∀(t, x)

´´´ <P° ¸ ° »P°d¯ ¸ ´ ¨"¯ ¸¯E E K6¡MV6# °7 ¸ ³ ^´ /"E- ! ¡ E )

(y, z)D! ¡KL QE c! !QE MNK6 !EE¶K6QEJ

XT

°7)QEJ D $£E ¬D! 3 °d ¸ ³ EN6G

−dYt = −(θ(t,Xt)Zt + rYt)dt− ZtdWt,

YT = φ(XT ),

θ(t, x) = b(t,x)−r

σ(t,x)

¬!PDFEJC∞b

¬ VP´´´ PPKK63MV QE D !$JKH

XN 7D KLE3! °c \!X

·b)EVD °7 ¸ ³ !K6\D VQEY NtN

=φ(XNT ),

^´´´ /Y Ntk

=Etk(YNtk+1

) − hθ(tk, XNtk

)ZNtk− rhY N

tk,

^´´´ hZN

tk=Etk(Y

Ntk+1

∆Wk).^´´´

#

(+*& #2 /:<!¯3E ¬!D¡TD3!¡! KLN K6¡ JEJNG()ZJNf<h @VegS@ACA A

, $ +

|Yt0 − Y Nt0| ≤ Ch.

E3D¡D QE = EHFE aD·η : [0, T ] → t0, · · · , tN

! ¡QEJ

η(t) =N−1∑

k=0

tk1[tk,tk+1[(t) + T1T (t).

c a´´´ ¤ ´´´ Q=¡GY Ntk

= Etk

(Y Ntk+1

1 − θ(tk, XNtk

)∆Wk

1 + rh

)

D¡MNY= Ek

!6GY Nt0

= E( 1

(1 + rh)Nφ(XN

T )ξNT)

ξNT =

N−1∏

k=0

(1 − θ(tk, XNtk

)∆Wk).

ª¬«3O¥c´ ³° ´ ° ¸¸ ° ° ª¨"¯ °7³µ¡°7¯ª¬° ³´ "°7³u±Y´¦¯P°cOP´¦³3°± N · ! D

Y Nt0

cD aK6 cMN 6E MVEN EP O JH ¬GYt = E

(φ(XT )ΓtT |Ft

)

dΓts = −Γts(rds+ θ(s,Xs)dWs) ,Γtt = 1.

°c@ JEJND!°7 ¸ <=Y0 = E

(exp(−rT )φ(XT )ξT

)

ξT = exp(−∫ T

0

θ(t,Xt)dWt −1

2

∫ T

0

|θ(t,Xt)|2dt).

¨DQEJ E =!@EEJ MN ¤ !EC6MVY0

$ KLDKLK6®# EJD ! QE£ EDQE /FEQEJ % MN ¤ Q

! QQEdQdP

∣∣FT

= ξT OP |Yt0 − Y N

t0| KLE QEFEHKLK6"!" ¬K6¡G

∣∣E(exp(−rT )φ(XT )ξT

)− E

(exp(−rT )φ(XN

T )ξT)∣∣

+∣∣E(exp(−rT )φ(XN

T )ξT)− E

(exp(−rT )φ(XN

T )ξNT)∣∣

+∣∣E(exp(−rT )φ(XN

T )ξNT)− E

( 1

(1 + rh)Nφ(XN

T )ξNT)∣∣.

¯3E ¬3D QEJD@!¡D KL$ (hf[ .£[B@eS',CSNhegS

ª ¡KL"¡%^ED K6=KEJMNQE3MNexp(−rT ) − 1

(1 + rh)N= exp(−rT ) − exp(−N log(1 + rh))

= exp(−rT )(1 − exp[rT − T

h(rh− r2h2

2+ o(h2))]

)

=Ch+ o(h).

±Y! @EK6 D N!3 EJNKL !FE $ +* (+*#&' 2 #*/7 # (+-'!+*¯3E! E! ¯E M YQED

Dk,p !=DEDv!I%E E£V

E

´´´ <P° ¸ ° »P°d¯ ¸ ´ ¨"¯ ¸07SegegS@A ACA

$ ! $# # '$ -#$# $ d (Xt)t≥0

$ +- ! !# #$

C∞ # #$ (XNt )t≥0

$ ! + , #

Xt = x+

∫ t

0

b(s,Xs)ds+

q∑

j=1

∫ t

0

σj(s,Xs)dWjs ,

XNt = x+

∫ t

0

b(η(s), XNη(s))ds+

q∑

j=1

∫ t

0

σj(η(s), XNη(s))dW

js .

, 1 ≤ k ≤ d

Xk,t −XNk,t =

q∑

i,j=0

cX,0i,j,k(t)

∫ t

0

cX,1i,j,k(s)∫ s

η(s)

cX,2i,j,k(u)dWiudW j

s

+ *!, + ! $ #$ $ dW 0t = dt

! + +# (cX,i1i,j,k (t))t≥0 : 0 ≤ i, j ≤q, 1 ≤ k ≤ d, 0 ≤ i1 ≤ 2 + +-+-$ supN,t ‖cX,i1i,j,k (t)‖k′,p <∞ ' k′, p ≥ 1

¯3E $EJKLN =!¡FE aKH ! EJ-QEQE JE5 §¬¯ M G8hf<`cf .£[ ,C[afW A ACA

F ∈ (D∞)m

#, + + "! +-,%, + %$γF

%$ # # , + + γ−1

F ∈ ∩p≥1Lp φ ∈ C∞

b (Rm)G ∈ D∞ , (det γF )−1 ∈ D∞

' ', %$ -"! α %, '$ +- +, +-,)+ % Hα(F,G) ∈ D∞ #,%, E(G(∂αxφ)(F )

)= E

(φ(F )Hα(F,G)

).

, p > 1 , %$ -"! α %, '$ ! $ +-$ C(p, α) #$ # $+ #,

n1, n2 %$ -"!

k, d, d′, b, b′ #$ +-$ "& +-,#$

pα #,

‖Hα(F,G)‖p ≤ C(p, α)(‖γ−1

F ‖n1k ‖F‖n2

d,b‖G‖d′,b′).

±Y7 K6d JEJN KLQE !QE ¥c M CMVN KL7 aK67 D ! °7 ¸ £EJVE5 $EJKLN¡G()ZJNf<h @VegS@ACA A

F H '$ +- %$'& +-,

Z'$ +- %$'& +-, ! $ %$ #,%,

Z0 = 0 , , + , $ EH(Z) , + $

Xt = Ht +

∫ t

0

Xs−dZs

$ $ +-

EH(Z)t = E(Z)tH0 +

∫ t

0+

E(Z)−1s d(Hs − [H,Z]s)

E(Z)t = exp(Zt − 12[Z,Z]t)

ª¬«3O¥c´ ³° ´ ° ¸¸ ° ° ª¨"¯ °7³µ¡°7¯ª¬° ³´ "°7³u±Y´¦¯P°cOP´¦³3°°c)QQ3E£=!=K6KL EN¡G07SegegS@A ACA

F q, p > 1 $ +

supt

‖Xt‖q,p + supt

‖ξt‖q,p + supt,N

‖XNt ‖q,p + sup

k,N‖ξNtk‖q,p <∞

+ *!(ξNt )t

$ +-

ξNt = 1 −∫ t

0

ξNη(s)θ(η(s), XNη(s))dWs.

±Y: EJ"P K6PK6"! D ÊJD K6N! ÊJ MNb, σ

θL ! ! L $ d¥b5-!5KL c E /N =!

6D ( XN

ξN

) DK6KL6LD KLE®! °c)!# °7 ¸ ( X

ξ

) ·:E MN)E! D·KLN'!- EJDFEMN' ! !%!-D L!QE

Dq,p ¬I%EJ DV6D N3!¡ °7 ¸ ( X

ξ

) ¡3QE(!:! 3 !:DE!ξ ¡

QE D ! ¡TD· = EN¬FEH !¡!θ \3! E!QK6NMN

supk≤N,N ‖ξNtk‖p <∞ p ≥ 1

7@E:! Q /!ξNt

QE51 ≤ i ≤ N

GξNti = ξNti−1

(1 − θ(ti−1, XNti−1

)∆Wi−1).^´´´

OP =Np ≥ 1

G

E|ξNti |2p =E

(

|ξNti−1|2pEti−1

2p∑

k=0

Ck2p(−1)2p−kθ(ti−1, X

Nti−1

)k∆W ki−1

)

=E

(

|ξNti−1|2pEti−1

p∑

k=0

C2k2p (−1)2(p−k)θ(ti−1, X

Nti−1

)2k∆W 2ki−1

)

≤(1 + Cph)E(|ξNti−1|2p)

h

LEK6KLNT· / E¬E)!" $EJ PE) !"!θ Q¨=! ! 3E ÊD KLNMN

supk≤N,N ‖ξNtk‖p <∞ °c dD! u

MNtk−1 < u ≤ tk

d°co EJN6MVaD ' $ -!EDDK6:$ D QEMNξNti ∈ D∞ °c!T NFE'a´´´ H

i ≥ k + 1G

DuξNti

= DuξNti−1

−Du[ξNti−1

θ(ti−1, XNti−1

)]∆Wi−1.

M

ª¬«3O¥c´ ³° ´ ° ¸¸ ° ° ª¨"¯ °7³µ¡°7¯ª¬° ³´ "°7³u±Y´¦¯P°cOP´¦³3°P QFEHKL ||| · |||2

E¡G|||φ|||2 =

√

E|φ(XT )|2 +√

E|φ(XNT )|2 +

(E

∫ 1

0

|φ(XN,λT )|2dλ

) 12

XN,λt = λXt + (1 − λ)XN

t

Y¯3EJP¡! E! EJ¡ DMNφ !DFE

C∞ !5 DK6QED C∞

0

-!5D! /! KLMN∣∣E(exp(−rT )φ(XT )ξT

)− E

(exp(−rT )φ(XN

T )ξT)∣∣ ≤ C|||φ|||2hECD

φ!¡DFEJ

C∞0

Q°c= ·DE!φ

KLE @! ! /D ! EN" ! aDφm

!EC∞

0

MV!5φ

cFEKL ||| · |||2 Nª 3V3TD· =3 D¡ DKLK6

E(exp(−rT )φ(XT )ξT

)− E

(exp(−rT )φ(XN

T )ξT)

=

∫ 1

0

E(exp(−rT )ξT∂xφ(XN,λ

T )(XT −XNT ))dλ.

°c@ EJN "KLK6´´´ Q=NE(exp(−rT )ξT∂xφ(XN,λ

T )(XT −XNT ))

=1∑

i,j=0


T )cX,0i,j (T )

∫ T

0

cX,1i,j (s)(

∫ s

η(s)

cX,2i,j (u)dW iu)dW

js

).

¯E E ¬ PK6"D!QEJN!i = j = 1

°c= EJN¬E aKH P!"!QEJ ¯E M 3 KL¡! Q/ ¡G


T )cX,01,1 (T )

∫ T

0

cX,11,1 (s)(

∫ s

η(s)

cX,21,1 (u)dWu)dWs

)

=

∫ T

0

E(Dsexp(−rT )ξT∂xφ(XN,λ

T )cX,01,1 (T )cX,11,1 (s)(

∫ s

η(s)

cX,21,1 (u)dWu))ds

=

∫ T

0

∫ s

η(s)

E

(

Du

[Dsexp(−rT )ξT∂xφ(XN,λ

T )cX,01,1 (T )cX,11,1 (s)]cX,21,1 (u)

)

duds.

±Y@KLE(Du[Dsexp(−rT )ξT∂xφ(XN,λ

T )cX,01,1 (T )cX,11,1 (s)]cX,21,1 (u)) '% D %FE'aK66! K6K66!LK6

E(∂αxφ(XN,λ

T )G) MN HKE ¡ EFE )´´´ c# $EJ !3ª¬ED N¤ ¸ D ¬E 3E

C‖φ(XN,λT )‖2

³KLEMNcMN ¶FE ´´´ D£EJ(XN,λ

t )t )VD ! ´º aK6 K6\ MN

²¡«P¥7 Q´ DFEH!K6d = 1

3DDE"·3 V ¡!3! 3DKLK6"!QEJ µI% # OP =NQEKLNMNE(exp(−rT )ξT∂xφ(XN,λ

T )cX,01,1 (T )

∫ T

0

cX,11,1 (s)(

∫ s

η(s)

cX,21,1 (u)dWu)dWs

)≤ C‖φ(XN,λ

T )‖2h.

C

´´´ <P° ¸ ° »P°d¯ ¸ ´ ¨"¯ ¸±Y3EK6¡E !¡KK6" QQE K6DD !

∣∣E(exp(−rT )φ(XT )ξT

)− E

(exp(−rT )φ(XN

T )ξT)∣∣ ≤ C|||φ|||2h.#-SN¶[B@VegS,CSNhegS

±¶¡! K6"KL∣∣E(exp(−rT )φ(XN

T )ξT)− E

(exp(−rT )φ(XN

T )ξNT)∣∣

K6 # E KLEJ )! ! ¬!³3E!ξT

QE ¬EE ¬EJ EJ d! ED1

KEJJEH!'!)JEJ) $EJ ξNT

ª K6K6L) LK6¡K6\3MNφ

3!QEC∞

0

!5D! -!¡KLMN∣∣E(exp(−rT )φ(XN

T )ξT)− E

(exp(−rT )φ(XN

T )ξNT)∣∣ ≤ Ch|||φ|||2.¯"E KLPFEL! D

ξT − ξNT!' E!!® KL)´´´ v³KLEMNP! E!=MV

dξt = −ξtθ(t,Xt)dWt3MNξNT

EHEKL QE !¡ MNQEJ /! ´º ξNt = 1 −

∫ t

0

ξNη(s)θ(η(s), XNη(s))dWs.

¨=!D¡ KLE:! D¡DK6K6ξt − ξNt =

∫ t

0

ξNη(s)θ(η(s), XNη(s))dWs −

∫ t

0

ξsθ(s,Xs)dWs

= −∫ t

0

(ξs − ξNs )θ(s,Xs)dWs +

∫ t

0

[ξNη(s)θ(η(s), XNη(s)) − ξNs θ(s,Xs)]dWs

=E(α)t

∫ t

0

E(α)−1s εNs [dWs + θ(s,Xs)ds]

αt = −

∫ t

0θ(s,Xs)dWs

a vb KLc´´´ ¬YFE EJ E(·) TεNs = ξNη(s)θ(η(s), X

Nη(s))−

ξNs θ(s,Xs) ¨=PEJ D ¡KL"!

εNs3FE aK6:G

εNs =ξNη(s)θ(η(s), XNη(s)) − ξNs θ(s,Xs)

=ξNη(s)θ(η(s), XNη(s)) − ξNη(s)

[1 − θ(η(s), XN

η(s))(Ws −Wη(s))]θ(s,Xs)

=ξNη(s)[θ(η(s), XNη(s)) − θ(s,Xs)] + ξNη(s)θ(η(s), X

Nη(s))θ(s,Xs)(Ws −Wη(s))

=ξNη(s)[θ(η(s), XNη(s)) − θ(η(s), Xη(s))] + ξNη(s)[θ(η(s), Xη(s)) − θ(s,Xs)]+

ξNη(s)θ(η(s), XNη(s))θ(s,Xs)(Ws −Wη(s)).¨@JELEJE MV3E6KK6¡D MNMV) K6K6<DKH QE K6! Q)aK:"!!QE ¡·! N $EJ /QEQE !6D! =MN

ξNtk ∈C

ª¬«3O¥c´ ³° ´ ° ¸¸ ° ° ª¨"¯ °7³µ¡°7¯ª¬° ³´ "°7³u±Y´¦¯P°cOP´¦³3°

D∞ MNsupN,k ‖ξNtk‖k′,p < ∞

k′, p > 1 bª" )ED·K6)5 E!g±YK6K6´´´ Q´ v! D QEK6N

∣∣E(exp(−rT )φ(XN

T )ξT)− E

(exp(−rT )φ(XN

T )ξNT)∣∣ ≤Ch‖φ(XN

T )‖2

≤Ch|||φ|||2.

¤

C

´´´ <P° ¸ ° »P°d¯ ¸ ´ ¨"¯ ¸

C

ksyj.pt=l=|

i vymxl@pt@kr | j mxql v| kst@pvmwkkpT l=|w1t=pv|Yv| 6t=m~ksx6t@p q| l t=l=mxyl%syq|l ~kpv|

#

¨=D !¡E:KK6"!VQEK6 MV¡X

Xi,t = Xi,0 +

∫ t

0

bi(s,Xs)ds+

∫ t

0

σi(s,Xs)dWs, 1 ≤ i ≤ d

3=)aD¡# NV 5G> v`7f-, Z@ .S%A ACA

F # $ , $ ! $ b

σ $ ! $

$ - # #$ +, + *! # + +-, $ '$ #$ $ $ $ '$ ! $ - $ '$ #,%, "!# '

σσ∗ σσ∗ ≥ ε0Id+ *!

ε0 > 0 °c' ·V¬E !3E£JE dE£D'VD¬! ´º )aK6 K6d MN EJ MNd E! I v ¯ ¬ $EKLNMNP JP´ P Q ED·QE QD QE# °d ¸ ³ a´ /¬PEJ! KE D D P=(Y, Z,K)

MNY Q5GYt = g(T,XT ) +

∫ T

t

f(s, Ss, Ys, Zs)ds−∫ T

t

ZsdWs +KT −Kt,

Yt ≥ g(t,Xt), t ≥ 0,

∫ T

0

(Yt − g(t,Xt))dKt = 0.^´´´ /

¯\b a ! °d²¡²¡¥ + M E N¶¶\QED !E\MN\D D b(Y, Z,K)

bEJ MN'TH!: EJH! D/:! D ) E K6

´´´ <P° ¸ ° »P°d¯ ¸ ´ ¨"¯ ¸# E3O N± N EJ QQEJD 3 FE" JE ¨L! 7D a !3JEJ

(Yt)t·

(Zt)t !FE¡ 5'EJD

S!E 3 a ª a $T K6T) MN:FELE!a P !ED

g(t,Xt) ª EDK6 E!D! EK6 DE 7 QE\ KL!QE3 "D£E! =EK6 D£EJ@EEJ

g(t, x) = (K − ex)+ ¯E c!QE VD E6D QE MV <dd KL!3KLV!Qc dE K6 3!QEE´3´´ DK6¡!¡E:DNE N"!FED $ ° E ! MV¬ D!5E K57 bD MN ! D!N !3!bV !)D D5P D MNH!QEPFELE´ T%JE P "FE6KLV!QD£E !-D6E K6H :DK6-!FEJD $ c± E!QEJ EJ! K6:E K6 N ECD) EJ !H¥c D£E!< K6 E <±YNVP ED

gNP JEJN

> v`7f-, Z@ .S%A ACAg(·, ·) , # #$ $ %$ -"! 1

2

#$ '$ #$ !'#$ , + +- +, + +-, ³KLEMN"MN:HK6 :! EJKL DE % ! KL

1vD:NV - Q · E3! MN ED

(K − ex)+ NYx ≥ log(K)

¨ ! Q gR

!5KEJ:EQE !φN,R

fR

!QE¡E-QEJ :´´ ¶±Y¡Cy(R)

·Cz(R)

)! QDK6K65 D !K6K6 b©3gDE Dc KL EJ *!=D 7!6FE- ´´ K63!3¬JE D KLN$ ± EJ K6 N3aKL E!:G→→→ ¨= E

k = NQE

yN,R,MN (·) = gR(tN , ·)

→→→ Oy% EP!)! D EJ tk(αMl,k)1≤l≤q

! QbDK6K6®K6 K6 3!%J¤ K6"!¡KL!¬D£E ¡Ginfαl

1

M

M∑

m=1

|yN,R,Mk+1 (XN,mtk+1

)∆Wm

l,k

h− αl.p

ml,k|2

ECDXN,mtk+1

KH !¬FEKKL ^E HMVPN,mtk+1

!QE EQE ´´ !PE b!XN,mtk

·c ¤D K6\ E E

∆Wmk

QEXN,mtk+1

= XN,mtk

+b(tk, XN,mtk

)h+σ(tk, XN,mtk

)∆Wmk

¨! Q ¡E

zN,R,Ml,k (·) EzN,R,Ml,k (·) = [αMl,k.pl,k]z(·)

¥cαM0,k

¡! Q DKLK6K6 K6 ¬!=K6"!K6 !DE G

infα0

1

M

M∑

m=1

|yN,R,Mk+1 (XN,mtk+1

) + hf(tk, XN,mtk

, yN,R,Mk+1 (XN,mtk+1

), zN,R,Ml,k (XN,mtk

)) − α0.pm0,k|2.

¨=! Q 3EyN,R,Mk (·) QEJ

yN,R,Mk (·) = max([αM0,k.p0,k]y(·), gR(tk, ·)

) C

ª¬«O¥c´ ³3° @OP±\µ¨"³´ «I%°¥d¨"©P³±\° ¸ °7 ¸ ³3³

→→→ ¨= MN !6# Et0

¨b b KE Y7KK6dE K6cEb E E aK KL7KLEMNMN FED· VQE´´ OP V# E K63MNQEK6¬K:FE "!)DT! ·¤D c!QEcFEQE ¬´´ 3!"DD<7MN !¡D QEJMV3 E*aDE KEJ yN,R,Mk! )E@ED

gR(tk, ·) ±YKK6"¦V! E K6¡E6! ¦!6 ·EQEJ V !QE §¬¥7¥7 § KLE E)!HD MV:! :!'MVEN ¤

QDEJ ¡:K6¡EDH!5D£E D !I@E E£ ¡:D! bEJD!E D$ :DE¡! ! "! !EN!Z

QE"NV E 5E MN5E D·= 'D# ¯356 E K6N:! E5 V6! E K5H5 °7 ¸ ³ & D 3!QEFEHQE "VKL MN"KEJE£DEE V"K6 MV ¯PE K6 FE MN ! DK6K6 N(yN,R,M (XN), zN,R,M (XN))

(Y N , ZN )

! D E:5KL7! ^´´´ /bMV3! D KEJNQEJN$

PHKLE)EJQE ! ^´´ ¤ ´´ ®! Q H! D E(Y N , ZN )

! ^´´´ /!¡KLE ·E!5G→→→ ¨=

Y NtN

= g(tN , XNtN

)

→→→ O = ENtkQ/! Q

ZNtk

QE

hZNtk

= Etk(YNtk+1

∆Wk),^´´´

/! Y Ntk

YMN

Y Ntk

= Etk(YNtk+1

) + hEtkf(tk, XNtk, Y N

tk+1, ZN

tk),

^´´´ 3)Q@=

Y Ntk

= max(g(tk, X

Ntk

), Y Ntk

) ¨ ! ¡ E K6": VDE

(Y N,R, ZN,R) K6E £EJN

gf

EgR

·fR

O D£E%!@E =!®I%E !

XN E Y N,Rtk

= yN,Rk (XNtk

)Y N,Rtk

=

yN,Rk (XNtk

)ZN,Rtk

= zN,Rk (XNtk

)

PHK K6)MN:E @´´ T" EJ :H PDKLK6=VDE E °7 ¸ ³ & D

C9E

ª¬«O¥c´ ³3° @OP±\µ¨"³´ «I%°¥d¨"©P³±\° ¸ °7 ¸ ³3³,'SVegUh ¶S%ACACA

$ ! +- +-$ + *! , $ ! $"+ , + ! $ #

& #$ ! "&- + √hh

14 # , + #$ ! , +'!#,

g , %$ +, #$ & #$ # +-,

¯J¡! E MND·PE K6"E=D£EJ3! EKL D£E =# ED" QE¡!5DEC1,2 DKLK6:QEJ¡KL 5!QE 5DE¡! QEC

(K − ex)+

¶¨"Y¡KL¤ MN¬¬E NV¬ ED g

¬ EK6D DD ! 3!QE I v # + ,2 #Y-+* "\% #" +*#+* *#& B *; #:<¨'^E !QED·¡QE # NV JE> v`7f-, Z@ .S%A ACA

O $ g !#, +

C1 #$ ' '$ #$ !' #$

x

g(t,Xt)g(t,XN

t )+ $ , #, $

g(t,Xt) =g(0, X0) +

∫ t

0

Usds+

∫ t

0

VsdWs + At,

g(t,XNt ) =g(0, XN

0 ) +

∫ t

0

UNs ds+

∫ t

0

V Ns dWs + ANt ,

A AN $ ! $ %$ !# + $ %$ "&-)+, #, , dAt

dANt

$#%$'& ', # +- +# dt + ! , ! $ - $ %$ "&- +%, +-$ supt E(|Ut|p) +

supt E(|UNt |p) + supt E(|Vt|p) + supt E(|V N

t |p) <∞ ' p ≥ 2

³KLEMN'MV¬QEK6 ' %D£E'! /EKL D£E !K6 1 # NVJ¤¡´´´ : Q ¡D£E

g(t, x) = ψ(t, exp(x))E£D

ψ(t, ·) D <OP / aK:! ´º 6 EN ¬ K6%E QE !QE ² ¸ G

g(t,Xt) = ψ(t, exp(Xt)) =g(0, X0) +

∫ t

0

Usds+

∫ t

0

VsdWs + At

g(t,XNt ) = ψ(t, exp(XN

t )) =g(0, XN0 ) +

∫ t

0

UNs ds+

∫ t

0

V Ns dWs + ANt

ECDtk ≤ t < tk+1

Ut =∂tψ(t, exp(Xt)) + ∂xψ−(t, exp(Xt)) exp(Xt)b(t,Xt) +

1

2σ2(t,Xt),

UNt =∂tψ(t, exp(XN

t )) + ∂xψ−(t, exp(XN

t )) exp(XNt )b(tk, XN

tk) +

1

2σ2(tk, X

Ntk

),

Vt =∂xψ−(t, exp(Xt)) exp(Xt)σ(t,Xt),

V Nt =∂xψ

−(t, exp(XNt )) exp(XN

t )σ(tk, XNtk

).

´´´ <P° ¸ ° »P°d¯ ¸ ´ ¨"¯ ¸ª K6K6

∂xψ− ' %!E =D£EL!'EK6 D£E ! KL

1d LD! ! $E !¡# NV´´´ : Q $

±YMN ® EKL ¡3P! P!E I v #<% EJT MNE5D! g

!DEC1,2 VNN KLN(!6:! "G

EH6K6K66O b±¶E% $E 5!g

H 6)K6)DN ) VDdDEK

KN,0 N¨$D vK6dDMN 5 KL¬!QEc# NV´´´ <± MN /!5 EJ MNE:KLEMN¡O 5MN¶!

0 ≤ dKt ≤ f(t,Xt, g(t,Xt), Vt) + Ut−dt = ktdt,^´´´

0 ≤ dKN,0t ≤ f(t,XN

t , g(t,XNt ), V N

t ) + UNt −dt = kN,0t dt,

P%P KK6DN PPVD"DEPMN)!QEPD£EP g

!¡DEC1,2 <°c=QE D $

supt,N E|kN,0t |p <∞ ·supt E|kt|p <∞

p ≥ 2

E3 K6¡O QE3¡DE$Q=EE:KLN3 " EJ

E sup0≤t≤T

|Y N,1t − Y N,0

t |2 +

∫ T

0

|ZN,1t − ZN,0

t |2dt ≤ C√h

"¡!V EEJE- K6"O a Ô EH! Q -!(Y N,0, ZN,0)

Yª D ¡ Ê QEP EC)VFE6V :! HQEP) K6K6O 7CDK6KLNK6D¡ EJ¡G ±¶E/6 MVLK6 EJ !/D :! I v # Y¯¡E-!¡!¡KK6"!QE@ N $E4 ! =DE5D QE) ¨:

∆Yt = Y N,0t −Y N,1

t

∆Zt = ZN,0

t −ZN,1t

HKEMN¬MN∆Yt

EPE!VQEK6MN JE¡]tk, tk+1[

−d∆Yt = [f(t,XNt , Y

N,0t , ZN,0

t ) − f(t,XNt , Y

N,1t , ZN,1

t )]dt− (ZN,0t − ZN,1

t )dWt + dKN,0t .

OP %EJ MNQEE aKHP! ´º 6= Nd|∆Yt|2 = − 2∆Yt[f(t,XN

t , YN,0t , ZN,0

t ) − f(t,XNt , Y

N,1t , ZN,1

t )]dt+ 2∆Yt∆ZtdWt

− 2∆YtdKN,0t + |∆Zt|2dt.

ª (!5 NC > 0

YMN

2|∆Ys||f(s,XNs , Y

N,0s , ZN,0

s ) − f(s,XNs , Y

N,1s , ZN,1

s )| ≤ C|∆Ys|2 +1

2|∆Zs|2

^´´´

DMV dH EJN\MNf

c±Y D d $E 7! d*Ô O#®NP E MNQEJNP aKH! ´º !Λs|∆Ys|2

[t, tk+1]

ECDΛs = exp(Cs)

ª¬«O¥c´ ³3° @OP±\µ¨"³´ «I%°¥d¨"©P³±\° ¸ °7 ¸ ³3³^

C3FEHD EJN!¡# MNQE/ D ! ¡G

Λt|∆Yt|2 +

∫ tk+1

t

Λs|∆Zs|2ds

=Λtk+1|∆Ytk+1

|2 + 2

∫ tk+1

t

Λs∆Ys[f(s,XNs , Y

N,0s , ZN,0

s ) − f(s,XNs , Y

N,1s , ZN,1

s )]ds

− 2

∫ tk+1

t

Λs∆Ys∆ZsdWs + 2

∫ tk+1

t

Λs∆YsdKN,0s − C

∫ tk+1

t

Λs|∆Ys|2ds @ E a´´´ G

Λt|∆Yt|2 +1

2

∫ tk+1

t

Λs|∆Zs|2ds

≤Λtk+1|∆Ytk+1

|2 − 2

∫ tk+1

t

Λs∆Ys∆ZsdWs + 2

∫ tk+1

t

Λs∆YsdKN,0s .

´ \P !LDN 3!3KL!®KLKH!)! %VH!)DD QE=KLK6!µ"¬E !D Q¥bDFE@E-G∆YsdK

N,0s =[g(s,XN

s ) − Y N,1s ]dKN,0

s

=[g(s,XNs ) − EsY N,1

tk+1+

∫ tk+1

s

f(r,XNr , Y

N,1r , ZN,1

r )dr]dKN,0s

≤Es[g(s,XNs ) − g(tk+1, X

Ntk+1

) −∫ tk+1

s

f(r,XNr , Y

N,1r , ZN,1

r )dr]dKN,0s .

ª" % KLK6N=FE! K6EJ D QEJ 3°7 ·%=DN /FE! Dg(s,XN

s ) − g(tk+1, XNtk+1

) =E@E MVN/ aKH ®! ´º oD MN¡ /K6 EJ)D

Ch!QEb# MNQED ¤º!$ JEJbdD£Ebb!dK K6TD·¡:KE EJ ® N EJN YO#T E*^´´´ ¡ V5´´´ E MV " K6KLPO H!= MV v 3¬JEE # NV´´´ Q/¡G

E

∫ tk

t

Λr∆YrdKN,0r

≤CE

∫ tk

t

Λr|r − tk| + |XNr −XN

tk| +∫ tk

r

|f(s,XNs , Y

N,1s , ZN,1

s )|dskN,0r dr

≤Ch 32 .

¨=NQQEJKLN¡GE[Λt|∆Yt|2 +

1

2

∫ tk+1

t

Λs|∆Zs|2ds] ≤Λtk+1[E|∆Ytk+1

|2 + Ch32 ]

≤Λt(1 + Ch)[E|∆Ytk+1|2 + Ch

32 ].

sl-t=pv| wkr l@p q|

kst=l%pr | j.syl't@pv|

-kt@st@ wkr l@p q|

ksyj.pt=l=|

syjkj | q| ~ksj.pt=l@| jkl ~ q|w1t@

E3D·¡ <! 3EVD FEH -!¡# °7 ¸ ³xG

Yt = Φ(X) +

∫ T

t


t

ZsdWs

^´ /

W'KLK6!%! KL

qX

'FE o!%# MNQEJ! VD QEJ MN@! KL d

GXi,t = Xi,0 +

∫ t

0

bi(s,Xs)ds+

∫ t

0

σi(s,Xs)dWs, 1 ≤ i ≤ d.

Φ *aD· :!55FE'ED:!6FE-!T

SYDDd"¡DK6! bQE¤º!! \D QE ¬QE ´ J¥b ! EEJc!:DEEJD QEJV¤!!N$:EVD ¬FED! KL QE

Φ(X)QE

ΦN(PNtN

)J

ΦN 7 aD! ·KL 3(PN

tk)0≤k≤N

¬PD QEP!"I%E L!P!K6d′ ≥ d

!N d

·¤K67DKLENd ! QEJdDd!(XN

tk)0≤k≤N

N°7L! EJdK6V'E N KLN ! EJJEFE d! EJ¬ !PKE V E!VQEK6MNK6 D !¡FEHD! 'K6E ¨@E5 ·! ¡!EQE ´3´´ !@EJ K6¬ ·EJ!3EVD FEH ¤ ®!^´ / ¶ª¡! EJ K6PN¡E ¡"!¡ K6!:K6 !DE MN ® EJ K63L! )!D · EJ =@K6! NJEJ

[0, T ]a

N +1!QEJ !! D ELN J

(tk = kh)0≤k≤N

h = TN

°c' ·V! EVD "FE'(Ytk , Ztk)

!-D QEMN5 E¡!5! D EJ tk

E"!¡D· P!PQE! aD· T±¶ELQEJ) PEVD Ytk

J p0,k(P

Ntk

) = p0,k5q

QE5 H5EVD 5D QED6!q

DK6EJN5!Ztk

N5 (pl,k(P

Ntk

) = pl,k)1≤l≤q <±Y¡DE! QEv!E:E

pl,k J

Kl,k

´ 7³° ¸ ©±\Oc ¸ ¯©I%°7³´ "©P° ¸³3E P) MV:K6P!HKL !PDE "N"E ¡!"KHEJ !I@ ¤ ª¬E%!FE@D QE !/I@E C

(PNtk

)0≤k≤N ±Y'KHL!'KHFE !-I@N·¤ª¬EJ + P KHFE N

(PN,mtk

)0≤k≤N,1≤m≤M v±Y"E!3KEJ D¬! $@!¡D K6!K6 !D£EJ EVD "!

pl,kE

M KHFE ¬N3 Kl,k(ω)

O\EN !d b! HE K6\ N !QEbbQE b´v ´´ b! DQ"D -! aD·!¡QE

ksyj.pt=l=|

qmpQn q| 7mwk~t@pvmwk q| .sx'|

E5FE-E:VK6 MN ¶¡!T N¡ !*aD "!6QE \¥ EDQ!DE dEC7$T E"!D 5!Dd! c Vd!aD· 5!QEJFEHKK6"QE

¥ $ pl,k

l

k ¥!!"$#% &'()*+'-,.(!/0D ⊂ Rd′ *'1(2

P0

* 3 D =

∏d′

i=1 ]P0,i −R,P0,i +R]4125 6().()7+ ((5,8 '#9:(

δ

; D = ∪i1,··· ,id′Di1,··· ,id′

Di1,··· ,id′ =]P0,1−R+ i1δ, P0,1−R+(i1 +1)δ]×· · ·×]P0,d′ −

R + id′δ, P0,d′ −R + (id′ + 1)δ] ¯<75

pl,k(·)'09" .= &'(0'>('?@' !A'(

B#CD,N@ '#FEpl,k(·) =

(1Di1,··· ,i

d′(·))

i1,··· ,id′ ªG(()H#C%&'((I(HJ2K5

L< $@M'G·N()N(GM'-,41L,8 '#OG @P@'>Q ((P V¥RP80 NP M <I#II @0P@'>Q'(()

1, x1, · · · , xd′

p0,k

(1

pl,kl ≥ 1

;

p0,k(x) =(1Di1,··· ,i

d′(x), x11Di1,··· ,i

d′(x), · · · , xd′1Di1,··· ,i

d′(x))

i1,··· ,id′,

pl,k(x) =(1Di1,··· ,i

d′(x))

i1,··· ,id′, 1 ≤ l ≤ q.

ªGQ'S,% ()'L &'T()OQ#Q(H()J%K ?> H1

0L N()0

C0 80UC"B#HC @@'! p0,k

pl,k

1 ≤ l ≤ q

´ 7³° ¸ ©± ; ¸ ¯©%°7³´ "©P° ¸

´ 'G!'5'%%#2 &'T()7'()'?(!"$ #%

0 ≤ l ≤ q vªGT(() &$L # "&'T()<'> ()'<N(OB 6().()9(

¹ µ¡±¶ N(P '1()M1(PPB( 1(MPN N¥ ·¤

'01($1 MM P 0 (M<#I20

&'( '>('$NB

20()'( 1(-I

PN ()(PN,M+i)1≤i≤20

0N(

(PN,m)1≤m≤M V¥ $NP() !H3()N(

tk (PN,M+itk

)1≤i≤20

P () ¤().()9 ¹

(Ck,i)1≤i≤20

Ck,i = x : |x − PN,M+i

tk| < infj 6=i |x − PN,M+j

tk| ° )

R pl,k(·) =

(1Ck,i

(·))

i

ªG(()G#G &'T() (R() V¥R(()- L* G41 <#J%K ' 1(01( '-,41'M J¤¹ 8 0 @'OP T()( $ ¥ M 4@M J%K5 (F'HH'-,410' # 8 " '6().()0

1, x1, · · · , xd′ p0,k

(1

pl,kl ≥ 1

*'S,41Ck,i

;

p0,k(x) =(1Ck,i

(x), x11Ck,i(x), · · · , xd′1Ck,i

(x))

i,

pl,k(x) =(1Ck,i

(x))

i, 1 ≤ l ≤ q.

¯<O()O'(() # ()' > O(1, 0)

?"?0 '() 41 J2K5

#

¯<*'p0,k

'0<"F# @Gd′

"#MT( )&!dy

(pl,k

l ≥ 1

*'0 "B#HC 8 0C$C)&!dz

"! # $ %#

&<O£L 0 ()(')'() 2+* (),H41().-,0/.D6(-N(tk0B" ().

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M(I

h;=87 (A@8 C41H'(-#(), B')'C4ED3;F'G'C4IHJ;F'G'C4LK0T(M'G'C4ON34&<C<'' L/.3+*.(,F N()QO? ()-'G'<C' %OB" ().#.D'01(I#P0<328<4Q '

d = q = 1T(F '

S"!J@@41Q @ 8RM"'CSUTWV8'-,

XGY RZV\[?H?]_^ E

dStSt

= µdt+ σdWt.

` F'(Q()</.Iacb>VJd/6((ST −K)+

;@'I#.37'> #4FeGI f;5 E

f(t, x, y, z) = −θz − ry + (y − z

σ)−(R− r)

C'θ = µ−r

σ

; OH4@f4` ) T()H 3@8041C

ST(IC/.D ()!1(CE

µ σ r R T S0 Kg 4 g N g 4OD g 4 g K g 4 gUh g 4EN i gfg i gUg

e6j((kjml)n.(koFprqkn+sUtvu5sUw)wGj<lGxsUt\yzpfou|pfl6sU~9j3Uj<tJy\j<o\wZyJjm~/.LsUxJq)n9sftj<8x\w)o\tqkjy\j~/.Ipfw)*Uj5tqpfovqkpfo32y#.Ln9tqk<wG75q

Rj5q(x\wG75q)jvpfoqCpfo32

rpCUj5u

r < RXGY R6j5wkNr]_^4e6j5qGqkjn+xj5wG=j5u5qkn+sUty\j8pfw)uC\xFpfw

' 4cd a V `MV& acdB' a V

wkp?x\xsUwGqcp?o32 @3xs?qk5l)j<l n+y\|pf~+j<l y\j Y RV\[fH?]\u5w)<jZo\t\jZt\sUt TA~9n+t\<pfw)n q)6y\pft\l ~9j6y\w)n Uj<wf;o\n\l)n+t\sUt?p?o\y\wkp?n+qBlGn98x\~9j58j<tq

f(t, x, y, z) = −θz − ry.&MsUo\lMpCfsUt\lBu JsUn9lGnu5j5qMj23j5x\~+j u<pfwB~+j>y\wGn Uj<w

fj5lGq6t\sUt3T ~9n+t\|p?n+w)jBpfn+l tJsUo\l6xsfoUsUt\l6t\<pft\8sUn+t\l u<sUtJtFpfn+q)w)jM~_p"?pf~+j<o\w

Y0

4 a tj j5q<;3~+jMy\w)n Uj<wfl)j wG<5u<w)n q

f(t, x, y, z) = − µ−R

σz −Ry + (y − z

σ)+(R− r)

≥− µ−R

σz −Ry.

e6j>yJj<w)tJn9j5w6qkj5w)8jmqCpftqZ~9j y\w)n Uj<wu5sUw)wGj<lGxsUt\yFpftqM,-o\t\jmyJ@ tFpf8no\jmyJj>xsfwGqkj=j<o\n+~9~+j u5~_pflGl)no\jpCfj<uo\t o\tJno\jqCpfo 2 y#.Ln9tqk<wG75q

R;sUt l)pfn+q(j<t oJqkn+~9n+l)pftq(~+j8qkJ<sUw<8jvy\jvu<sU8xFp?wkpfn+l)sUt xsUo\w

~9j5lvacb>VJd XGY a>[r]_^o\jY0

l)j5wkp lGo\x<wGn9j5o\wpfoxJw)n 2 u<pf~9u5o\~9zxFpfw~9p=sUwG o\~9jzy\j R6~9pfuCSTV3uC\sU~+j<l6pCUj5u qkpfo32y:.In+tqk5w)7q

R4pfn9l sUtlkpfn q <*p?~9j5j5tq y\pft\l ~9jMu<pfl6yJj~9p wG<x\~+n9u<p?qkn+sUtyFpft\l ~+j

8s y<~+jMR6~_p?u SUTWV u JsU~9j5lo\jxsfo\w u<sUo3wGn9w ~ .IsfxJqkn+sUt#;U~+jUj5t\y\j5o\w6y\j w)p(q)sUoGsUoJw)l j<8x\wGo\tq)j<w , ~_p pft oJjf;3u?.Lj<lGqWT ,rT y\n9wGj!o\jf; l)n sUtvtJsfqkj

(Y BSt , ZBS

t )~+j<l l)sf~9oJq)n9sUtJl yJj ~/.Ia bmVJd pfl)lGs u<n+<jpfoy\w)n Uj<w

−µ−Rσz − Ry

; pf~+sUw)lY BSt ≤ ZBS

t

σ

; ZBSt

σ

w)j<xJw)<lGj<tqCpftq~9j>sUtqCpftqBn9tUj<lWqkn j5t p?u5q)n9sUt#4 n9t\lGn/;\sUts q)n9j5tq n+j<t ~ .I5*pf~+n q) y\j<l yJw)n Uj5w)l

fj5q −µ−R

σz − Ry

~+j ~9sUtJ*zy\j~_pl)sf~9oJq)n9sUt(Y BS

t , ZBSt )

jqy\sUtJu ~ .I5*pf~+n q)

Y0 = Y BS0

4Y BS

0

lGj u<pf~9u5o\~9jmj23x\~9n+u<n qkj<8j5tqZj5qrpfoJq7.156

4Q t qkj<lWqkj n+u<nB~_p pfl)j#"%$4 Q tt\s?qkj

K~9jt\sU wGjy\ju<j5~9~9oJ~9j<ly\j#csfw)sUt\sfnMo3qkn9~+n+l)5j<l54& sUo\w

u<sfxFp?w)j5w ~9j5l p?~9*UsUwGn qkJ8j<l p£Uj<usUo lkpftJl lGn9 o\~_prqkn sUt(y#.Ln9t\u5w)5j5tq)l w)s10Bt\n9j5t\l pfo323n+~9n9pfn w)j5l<;rt\sfo\ly\5u<n9yJsUt\lBy\j>= p?n9w)jPrpfw)n+j<wl)n9 o\~ qCp?t\<8j5tq

N;M

j5qK4 a txFp?wGqkn+u<o\~+n+j<w5;JtJsUo\lZ= pfn+l)sUtJl¬?pfwGn9j5w

Ny\j q)j<~9~+j-l)sfwGqkj' oJjN = [N0(

√2)j−1]

sfj = 1 . . .

yJ<l)n+*Ut\j ~+j t\sf w)j y\jHrpf~9j5o\w)lmy\jN

o\j~/.LsUt qkj5lGqkj X

[·] y\<lGn9*Ut\j n9u<n ~_pxFpfwWqkn+j j5tqkn(<wGj1^4a t\l)oJn+qkj?;t\sUo\l>= p?n9l)sft\l?pfwGn9j5wK

j5qM

y\j q)j<~+~9jl)sfwGqkjo\j

K = [K0(√

2)(j−1)αK ]j5qM = [M0(

√2)(j−1)αM ]

p£Uj<uN0 = 2 = M0 = K0

4 sUo\wu \po\j=?p?~9j<oJwy\j

(αK , αM)t\sUoJl8w)j<xsUwWqksUtJl8~9jzxJw)n 2 8sP@Uj5t y\sUtJt\xFpfw~/.Ipf~+*UsUwGn q)\8j

X u<pf~9u5o\~+Bj<t~_p?t)|pftq6N g =sUn+lc~ .Epf~+*Usfw)n qk\j1^;f~/.L<u<pfwGqWT qA@ x jBy\jMu<jMx\w)n 2 ;~_p>u<sUoUj<wGq)o\w)j 8sP@Uj5t\t\jMj<tt0jqM~/.L<u<pfwGqWT qA@ x j>y\j ~_p u<sfoUj<wWqko\w)j?4*csUn9u5n#~+j<lBwG<lGo\~+qkp?qklBs qkj5t o\l

j

Prix

105

5

6

7

4.5

5.5

6.5

7.5

Prix avec browniens auxiliairesPrix sans browniens auxiliairesPrix Black-Scholes

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

+-,/.

e ' Bd a"HJ4 ma e Q V m&VRd Q &M'Wac&V%!m' `' 'WdMa V

j

Couve

rture

105

0.2

0.3

0.4

0.5

0.6

Couverture avec browniens supplémentairesCouverture sans browniens supplémentairesCouverture Black-Scholes

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Pr

ix

105

1

2

3

4

5

6

7Avec browniens auxiliairesSans browniens auxiliaires

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Co

uvertu

re

105

1

0.5

Avec browniens auxiliairesSans browniens auxiliaires

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 ! "$#&%$'()* +*,-.#!( !/%-#&0213#!*"$4*(5*%$6"$+7#!*

989#!#!#&"$*:$*(6!*% $;*()<=".%-*">?@A()B()?(5#C".ADE/1F*.% 9"$#G<H">I%I89()<!?I%4 !*(5*%JK"$#L<H"::%-".:I89()<>#L$#&M%ONQPR#!$#L<":**#!$#2STU/:%$%$5@$#%-/SU<"$4*$*%-V%$W%-OJX1Y/! ".#&%-'()Z$$#!( !/%$#213#!"$4*(5*%$6"$+7#!*989#!#!#&"$*N

[\\


iPN h

# %

! "#!$&%(')+*-,.)0/1/1)324/5'6,7"98:8/;$=<> "#!$?%@8A$&2/B*"DC"&)EGF

& sfo\lMpIHUsUt\lBs qkj<toyFp?t\lM~9p x\pfwGq)n9j>'G' ~9j>w)5l)o\~ qCp?qBlGo\nJHrpftq<4

max0≤k≤N

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2

+ hE

N−1∑

k=0

1

M

M∑

m=1

|zN,R,Mk (PN,mtk

) − zN,Rk (PN,mtk

)|2

≤CCy(R)2 + ‖fR‖2∞

M

N−1∑

k=0

max0≤l≤q

E(Kl,k(ω)) + CN−1∑

k=0

infα

E|α.p0,k(PNtk

) − yN,Rk (PNtk

)|2

+ CN−1∑

k=0

q∑

l=1

infα

E|α.pl,k(PNtk


tk)|2 + Chβ−1

+CCy(R)2

hexp(− Mhβ+1

144Cy(R)2)N−1∑

k=0

E

(

N2

(hβ+1

2

3√

2, [P0,k+1]y − yN,Rk+1 (·), (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M))

.

X ' 4LK\4 iP^dMpfx\xj<~+sUt\l X HUsfn9wMxFpfwGq)n9j ')'W^ oJj (

yN,R(PN), zN,R(PN)) j<lGqm~_p8lGsU~9oJq)n9sUty:.Io\tJj y\n+l)u<wG5q)n9lkprqkn9sft

j<tqkj5x\lcyJj X '-4+if4+iP^ j5q y#.Lo\t\jM~+s u|pf~+n+lkprqkn+sft y\qkj<wGn+t\<jZxFpfwcoJtKHUj5u5qkj5o\wR4 e6jq)q)jM~9s u<pf~9n lkp?qkn sUt

t\sUoJlZxj<wGjqZy\jmsUtqkwGj<w o\jyN,R(PN)

j5lGq sUwGt\mxFpfwoJt\j>u<sft\lGqkpftq)jCy(R)

o\n#yJ<xj<tJyy\j~_p ~+s u|pf~+n+lkprqkn+sft#4d pfxJxj<~9sft\l j<tMLFt oJj~/.Ipfx\xJw)s123n+p?qkn sUtw)5l)o\~ qCpftq6yJj ~/.Ipf~9*fsUwGn q)\8jByJj ~_p xFpfwGq)n9j'G'6j5lGqMt\sfqk5j (

yN,R,M (PN), zN,R,M (PN)) 4

& sfo\lMpf~9~+sUt\lyJ<y\o\n+w)j y\owG<l)oJ~+qCprqMu5n T y\j<lGl)o\lM~9p x\wGsUxsUlGn+qkn+sUtlGo\nJH?p?tq)j

" !#"%$&'()$+*,$"-/. $0$&1&-,23#-,"546$&78$&9;:<=7>-,"δ

-<M

?-<@"BA%'-δ = hγ

4'C7γ > 1

2

-<M = h−(β+1+γd′) $&1&-,2

β > 1 DCEC&*%F-<'HG

maxk

E1

M

M∑

m=1

|yN,R,Mk (PN,mtk

) − yN,Rk (PN,mtk

)|2 + hE

N−1∑

k=0

1

M

M∑

m=1

|zN,R,Mk (PN,mtk

) − zN,Rk (PN,mtk

)|2

≤o(1) + C(R)N−1∑

k=0

P(PNtk/∈ D)

CIo(1)

?-<J;1&-<7>"0

A%6$&Jh

K-<JL1&-<7>"0 M

N(OQPRTSUPWV a tpfx\x\~+noFpftq X ' 4LK\4 iP^pIHUj<u o\txFpfwkp? qkw)jβ′ qkj5~ o\j 1 < β ′ < β

sUtzs q)n9j<tq

max0≤k≤N

E1

M

M∑

m=1

|yN,Rk (PN,mtk


)|2

+ hE

N−1∑

k=0

1

M

M∑

m=1

|zN,R,Mk (PN,mtk

) − zN,Rk (PN,mtk

)|2

≤CCy(R)2 + ‖fR‖2∞

M

N−1∑

k=0

max0≤l≤q

E(Kl,k(ω)) + C

N−1∑

k=0

infα

E|α.p0,k(PNtk

) − yN,Rk (PNtk

)|2

+ C

N−1∑

k=0

q∑

l=1

infα

E|α.pl,k(PNtk


tk)|2 + Chβ

′−1

+CCy(R)2

hexp(− Mhβ

′+1

144Cy(R)2)N−1∑

k=0

E

(

N2

(hβ′+1

2

3√

2, [P0,k+1]y − yN,Rk+1 (·), (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M))

.

sUoJq y#.Ip sUwGy#; t\sfo\l l)pIHUsUtJl' oJjyN,Rk (·) jq (

√hzN,Rl,k (·))1≤l≤q

l)sUtq y\j<l-=sft\u5q)n9sUt\l-`n9xJl)uC\n+qYXo\t\n =sUw)8<8j5tqZj5t

kjqN4* n9t\lGn/;JxsUo\wB~9p pfl)j. $;\t\sUoJlMs qkj<tJsUt\l

E(Rp0,k

(Y N,Rtk

)2)

≤E(|Y N,Rtk

|21Dc(PNtk

))

+∑

i1,··· ,id′E(1Di1,··· ,i

d′(PN

tk)|yN,Rk (PN

tk) − yN,Rk (xi1,··· ,id′ )|2

)

≤ Cδ2 + Cy(R)2P(PNtk/∈ D),

sUxi1,··· ,id′

j<lGqo\t x sfn9tqpfw n+qkw)pfn+w)jyJjDi1,··· ,id′

j5qsU tJsUo\lpIHUsUtJloJqkn+~9n l)~_pxJw)sUx\wGn9qkzy\j`n9x\lGuC\n+qYX yJj

yN,Rk

lGo\wD4 a txsUlkp?tq , x\w)5l)j<tq

η = δh−12 → 0

xFpfwZ@ xsfqk<l)j?;sUts qkn+j<tq

infα

E|α.p0,k(PNtk

) − yN,Rk (PNtk

)|2 ≤ Chη2 + Cy(R)2P(PNtk/∈ D).

iPN&Z

e ' Zd aK\4% mdB' Z' Q &V '#` &ca bma V md BdMa V

bj5l u<pf~+u<o\~+lZl)n98n+~9p?n+wGj<ll)sUtq=H?pf~9p ~+j<lZxsUoJwE(Rpl,k

(√hZN,R

l,tk)2) jqMsUtj5ty\5y\o\n q

maxk

E1

M

M∑

m=1

|yN,R,Mk (PN,mtk

) − yN,Rk (PN,mtk

)|2 + hE

N−1∑

k=0

1

M

M∑

m=1

|zN,R,Mk (PN,mtk

) − zN,Rk (PN,mtk

)|2

≤C(hβ′−1 + η2) + CCy(R)2

N−1∑

k=0

P(PNtk/∈ D) + C

Cy(R)2 + ‖fR‖2∞

M

N−1∑

k=0

max0≤l≤q

E(Kl,k(ω))

+CCy(R)2

hexp(− Mhβ

′+1

144Cy(R)2)N−1∑

k=0

E

(

N2

(hβ′+1

2

3√

2, [P0,k+1]y − yN,Rk+1 (·), (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M))

.

Kl,k(ω)j5lGq p?~9sUwGl pGsUwG6xFp?w ~9j6t\sU w)jy:.I@ x j5w)u5o j<l y#.Ipfw)7qkj

δu<sftqkj5t oyFp?t\l ~+jyJsUvp?n9t\j

D4

% Q 'u|pfw l)j<oJ~9l u<sU8xJq)j<tq6~9j5l @ x j5w)u5o j<lyFpft\l ~9j5l o\j<~+l l)sUtq6x\w)5l)j5tqkj5ly\j5l-lGn9 o\~9p?q)n+sft\l

(PN,mtk

)1≤m≤M;u<sU88jsUt ~/.Ipzy\ ),n+t\y\n( oJ8yFp?t\l(~9pzxFpfwGq)n9j'G'C4 n+t\l)n ;sUt p

Kl,k(ω) ≤ (2Rδ

)d′ l)sUn q

Kl,k(ω) ≤ C(R)δ−d′ H3o ~9j(uC\sUn 2y\j

Rj5qδj :j5u5q)o\<l>x\wG<u5<y\j58j<tq|4

'A~ H n9j5tq

maxk

E1

M

M∑

m=1

|yN,R,Mk (PN,mtk

) − yN,Rk (PN,mtk

)|2 + hE

N−1∑

k=0

1

M

M∑

m=1

|zN,R,Mk (PN,mtk

) − zN,Rk (PN,mtk

)|2

≤C(hβ′−1 + η2) + CCy(R)2

N−1∑

k=0

P(PNtk/∈ D) +

C(R)δ−d′

Mh

+CCy(R)2

hexp(− Mhβ

′+1

144Cy(R)2)N−1∑

k=0

E

(

N2

(hβ′+1

2

3√

2, [P0,k+1]y − yN,Rk+1 (·), (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M))

.

Q toJqkn+~9n+lGj ,x\wG<l)j5tq ~9j>u<sUtqkwU~9j>y\ot\sU w)j>y\j u5sUoMHUj5wGq)o\w)j N2

j<t=sUtJu5qkn+sUtzy\jδj5qhyJsUt\t\

xFpfw X 'G' 4IHJ4LK^ X HUsfn9w xFpfwWqkn9j('G'W^4#e6sU8j(sUtpvuC\sUn+l)nM ≈ h−(β+1+γd′) pIHUj β ′ sfts qkn+j<tqLFtFp?~9j<8j5tq oJj(~+j<lBy\j5o32zqkj<wGj5lMn+xJ~9n oFpftq

MyFpft\lM~ .In+t\ o\p?qkn+sUtzu5n T y\j<lGl)o\lBqkj5t\y\j<tq HUj5w)l

04

¤

& sfo\l y\<lGn9w)sft\l?HU<wGnJLFj5w ~_pH?pf~+n9y\n q)y\ju<jq)q)jx\wGsUxsUl)n qkn+sUt#;u?.Ij5lGqWT ,rT y\n9wGj HUsUn9w(l)n6j<t = pfn+lkpftq?HrprTw)n+j<w

N, δj5qM

y\j qkj5~9~+j l)sUwWqkj oJj-~+j pGsfwkpftqMyJj(~_px\wGsUxsUlGn+qkn+sUtzq)j<t\yJj&HUj5w)l0; ~/.Ipf~+*Usfw)n qk\8j

u<sft-HUj5w)*Uj n+j<t#4& sfw)pf~9j58j<tq|;\t\sUo\l y\j H3wGn9sUtJl <*p?~9j<8j<tqB= pfn9wGj q)j<t\yJw)j(~9j xFp?wkpf 5q)w)j

RHUj5w)l~ .In+tMLFt\nxsfo\wp?lWT

l)oJw)j<wo\j ∑k P(PN

tk/∈ D)

qkj5t\yAHUj<wGl0j5q oJj

yN,R0 (PNt0

)qkj<tJy HUj5w)l

Y0

4#& sfl>x\w)j5n+j<wGlqkj5lGqklt oJ5w)n( oJj<l>t\sUoJl>sUtqkwGj<tqo\j~+j xFpfw)pf 5qkwGj

R;:l<.Ln9~ j5lGq uC\sUn+l)n l)o8l)pf8j<tq *fwkpft\yy5l ~+j

y\ oJq|;t#.Ln9tFoJjMxFpfl y\jMvp?t\n5w)jBlGn9*UtJnJLFu<p?qkn HUjBlGo\w ~_p u5sUt HUj<w)*fj<t\u5jf4n+t\l)n ;tJsUo\l y\<u5n9y\sUtJl y#.LoJq)n T~9n+l)j5wZoJty\sfvp?n9t\jmy\j>qCp?n9~+~+j L32Jj?4

iPNf

' 4cd a V `MV& acdB' a V )= 24/ 2:%) $98!$&% ) 2/1$&8 )=%:%) " ) ':0/1)=% !) *

') "&2><& sfo\lcu<sft\l)n+y\<wGsUt\lc~_pm87<8jZa bmVJdj5qc~9j5l 75j5l H?pf~+j<o\wGlcy\jZxFpfwkp? qkw)j5lo\jMxsfo\w ~9jZuCFp?x\n+q)w)jHjqMwkp?x\xj<~+sUt\l o\j ~9j>x\wGn 2y\j wG5=<wGj<t\u5j(j5lGq

7.1564

"!#%$&'(*),+

& sfo\l j j<u5q)o\sUt\l qksfoJq6y#.Ip sfw)yvy\j5l qkj<lWqklp HUj<u~9p p?l)j . $43e6sU8jBsUtv~/.Ip yJ ),(y\n q|;sftvy\<lGn9wGjqkj5lGq)j<wB~_p?H?pfwGn_prqkn9sftlGn9 o\~ qCp?t\<jmy\j

M, δj5qN4Factj :jq|;F~+j>y\sfvpfn+t\j

Dj5lGq=LJ23 oJt\j>=sUn9lxsUo\w

qksfoJqkjm,[40, 180]

j5q~9p(u5sUt\lGqkpftq)jCy(R)

j<lWq0LJ23>,100

4\e j<l H?pf~+j<o\wGlx\pfwkpfn+l)lGj<tql)o8lkp?8j<tq~_p?w)*Uj5l8xsfo\w8t\jxFpfl n_p?n9l)j5w~+j<l8w)5l)o\~ qCprqkl<4 a tj :jq|;co\t u|pf~+u<o\~Zy\jxJw)s p n9~+n qkyJjl)sfwGqkn+jzy\jy\sUpfn+t\jt\sUo\ln9t\yJno\j o\j~_p l)sUwWqkn9j y\j

[40, 180]X l)o\w~9j5lp?qkoJw)n+q)<ly#.IsfxJqkn+sUt\l o\jt\sUo\l

u<sft\l)n+y\<wGsUt\lC^ j<lWq6j 23q)w)7<8j5j5tq wkpfwGjMj5q L32Jj5wCy(R)

,100 = S0 = K

xFpfwkp?n+q qkw<l wkpfn+l)sUt\t\p ~9j?4& sfo\lB= pfn9lGsUt\l Hrpfw)n+j<w

Ny\j q)j<~+~9j>lGsUwGqkj oJj

N = [N0(√

2)j−1]sU

j = 1 . . .y\<lGn9*UtJj ~9j tJsU w)j

y\j0Hrpf~9j5o\w)l y\jN oJjB~ .IsUt-qkj5lGq)j X

[·] y\5l)n+*Ut\jZn+u<n3~_pxFp?wGqkn+j6j<tqkn5w)j|^4UactJl)o\n qkjf;ft\sUo\l = p?n9l)sft\l;Hrpfw)n+j<wδj5qM

yJj q)j<~+~9jml)sUwWqkjo\jδ = δ0√

2(j−1)αδ

j5qM = [M0

√2

(j−1)αM]4

& sfo\lx\wGj<t\sft\lN0 = 2 = M0

jqδ0 = 50

4 &MsUo\l L32Jsft\lo\t\j&H?p?~9j<oJw y\jαδ

xsUoJw~9po\j5~9~9j t\sfo\lqkj5lGq)sUt\ly\n :5w)j5tq)j<l H?pf~9j5o\w)l y\j

αM4 sUoJw6u \po\j H?pf~+j<o\w y\j

αδtJsUo\l6w)j5xsUwGq)sUt\l6~9jMx\w)n 28sP@Uj5t

y\sUtJt\8xFpfw ~/.Ipf~+*UsUwGn q)\8j X u<pf~9u5o\~9 j<t ~9pft)<pftq(N g =sUn9l>~ .Epf~+*Usfw)n qk\j1^;:~ .I5u|pfwWqGT qA@ xj8yJj8u<j8x\w)n 2 ;~_p u5sUoMHUj5wGqkoJw)jz8sP@Uj<tJt\jj<t

t0j5q ~/.L<u|p?wGqGT/q @ xjy\j~9p u5sUoMHUj5wGq)o\w)j?4 sUo\w yJsUt\t\j5w o\t\jn9y\5jy\j

~/.LsUwGy\w)j y\j*Uw)pft\y\j5o\w6y\j<lx\pfwkpf 5q)w)j5l<;3y\n+l)sft\l6xFpfw6j 23j<8x\~9j o\jmxsfo\wj = 10

;3u<j5~_p-lGn9*UtJnJLFj! oJjN = 64

j5qZlGo\nJHrpftq~9j5l H?p?~9j5o\w)lyJjαδ

j5qαM

sUty\sUt\tJj>y\pft\l~9j qkp ~+j|pfovu<n T y\j<lGl)sUoJlZ~+j<l H?pf~+j<o\wGly\ou<sfo\x\~9j

(δ,M)

αδ \ αMi iU4EN D D34EN H

g 4 h X h 4EDUN3; h K^ -. 0/2143652.7/98 X h 4ODfN3;ED g K Z^ X h 4EDUN3; iUiPN&ZUNU^ X h 4EDUN3; h NfNfH h ^i X iU4EN h ; h K^ X iU4EN h ;EH h DU^ - 612.436/2:<;9=98 X iU4EN h ; iUiPN&ZUNU^ X iU4EN h ; h NfNfH h ^iU4EN X g 4ED&ZJ; h K^ X g 4ED&Z3;EH h DU^ X g 4OD ZJ;ED g K Z^ ->:T6/2=3(17=2198 X g 4ED&ZJ; h NfNfH h ^D X g 4 g N3; h K^ X g 4 g N ;EH h DU^ X g 4 g N3;ED g K Z^ X g 4 g N3; iUiPN&ZUNU^ ->:T?:71430.2121257.98

a t*fwkpfl5; sUtvp(n9tJy\no\M~+j<l H?pf~9j5o\w)l u<sUwGw)j<lGxsUt\yFp?tqklpfo328l)j<oJn9~+l yJju5sUt-Hfj<w)*fj<t\u5jf4\a tj :jq|; xsUo\wuCFpo\jH?pf~+j<o\wy\j

αδsUt l5.Ep?xj<w)u5j!H wkp l)o\w8~9j5l8*Uw)pfx\\n(o\j<l8u5n T p?x\w 5l<;o#.In+~Zj23n9lWqkjo\t\jHrpf~9j5o\w

l)j5o\n9~ xsUoJwαM

j<tvy\j5l)l)sfo\l6y\j ~_p/ oJj<~9~+jZ~/.Ipf~9*fsUwGn q)\8jZl)j5 ~9jMtJj xFpfl u<sUt HUj<wG*Uj<w j5q6pfoy\j<lGl)o\l6y\j~_p/ oJj<~+~9j n9~ u<sft-HUj5w)*Uj?4 pfw j 2Jj5xJ~9j(xsUo\w

αδ = 0.6~+j(lGj<o\n+~ oJj ~ .IsUts lGj<w HUj t o\85w)no\j5j5tq

j<lWqαM = 1.5

y\sft\u sUtz=sUtJu<j ~9j>u5sUo\x\~+j(6.25, 362)

4csUn+u<n:~9j<lZw)5l)o\~ qCp?q)lBs q)j<to\lMxsUo\wB~9p Hrpfw)n9p?qkn+sUtl)n9 o\~ qCp?t\<jmy\j5lMxFpfw)pf 5qkwGj<l

N, δjqM

i hUg

αδ = 0.6j5qαM

H?pfwGn9j` p x\w)sUxsfl)n+q)n9sft ' 4 i t\sUo\lvn+xsUlGj oJj

δq)j<t\yJjHUj5w)l

0xJ~9o\lKH n+q)j#o\j

h1246e6j5u<n n98xsUl)j

αδ >12

4 e.Lj<lGq Go\lGq)j~9ju|p?l-pIHUj5uαδ = 0.6

4` j8l)j5o\n9~cxsUoJwM

y\j8~_pzxJw)sUxsUlGn+qkn+sUt' 4 iU; n9u5npIHUj5u

d′ = 1X j5qMj5txJw)j<t\pftqM~_p?H?p?~9j<oJwB8n9t\n+8pf~ jmxsUo\w

β^;\y\sUtJt\j

αM > 2.64

j

Prix

105 156

7

8

6.5

7.5

1 1.5 2 2.5 Prix BS

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,1

t 1 PN,1

t 1 PN,2

t 1 PN,2

t 1 PN,3

t 1 PN,3

t 1

j

Std Pr

ix

105 15

1

2

3

4

5

6

7 1 1.52 2.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105 15

0.1

0.2

0.3

0.4

0.5

0.6

1 1.5 2 2.5 Couverture BS

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1


j

Std Co

uvertu

re

105 15

1

0.5

1 1.52 2.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 αδ = 1

M αM > 3 ! #"%$&(')*

j

Prix

105

6

7

8

9

6.5

7.5

8.5

1 1.5 2 2.5 3 3.5 Prix BS

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Pr

ix

105

1

2

3

4

5

6

7

8

91 1.52 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

+


j

Couver

ture

105

0.2

0.3

0.4

0.5

0.6

0.7

0.81 1.5 2 2.5 3 3.5 Couverture BS

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Co

uvertu

re

105

1

0.5

1.5

1 1.52 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 αδ = 1.5

M αM > 3.5

*

j

Prix

105

6

7

8

5.5

6.5

7.5

2 2.5 3 3.5 Prix BS

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1


j

Std Pr

ix

105

1

2

3

4

5

6

7

8

9

2 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105

0.3

0.4

0.5

0.6

0.7

0.35

0.45

0.55

0.65


P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Co

uvertu

re

105

1

0.5

1.5

2 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

# % ' ' %$& ' ' ( $& # ) % αδ % αM > 1 ' % $ # '%) ' % # % ) αM # *

% % # )0 # ' * ( %

αδ = 1 )' $& αM = 1, 1.5 $ ) αM = 2

') αM ≥ 2.5

* ' & % $& & % * # ) &' " % # " %* % ' #& $ % ! ' # "%*

# $ # %' #% ' ) # ) '%) #%%$ '&')(' " % $ # * +&#$&$& %*-, !.

αδ = 1 '%) #%

0/

t\jzlGj< ~+jxFpfl u<sUt HUj<wG*Uj<wu<~9pfn9wGj<8j<tq HUj5w)l~_p u5sUoMHUj<wWqko\wGjRZV ; pf~+sUw)l'o\jzxsUo\wαδ = 1.5

;uf.Lj<lWq~+jMu|p?l<43dBj<pfw o\sft\l&o\j ~9j5l q)\<sfw <8j5l 5qCp ~9n+lcyFpft\l ~+j<l xFpfwWqkn+j<l ' jq6'G' t\j u<sft\u<j5w)t\j5tq)pfpfn+l y\j<l>w)5l)o\~ qCp?q)lmy\j u5sUt-HUj5w)*Uj5t\u<jl)o\wmy\j<l>u<sfoMHUj<wWqko\wGj<l xsUt\uqko\j5~9~+j<lmvp?n9lMqksUo-GsUo\w)lmlGo\wy\j5ln+tq)<*Uw)pf~9j5l y\j ~_p u5sUoMHUj5wGqkoJw)jmlGo\w6~ .In+tq)j<w Hrpf~9~ j

[0, T ]43` .Ls lGj<w H?p?qkn+sUtvy\j u<sUt HUj<w)*fj<t\u5j>xsUoJw

y\j5l6u5sUoMHUj5wGqkoJw)j<l6xsUt\uqko\j5~9~+j<l l<.Lj23x\~+no\j u<j5wGqkpfn9tJj<8j<tq xFpfwhZN,R

tk= Etk(Y

N,Rtk+1

∆Wk)sUsUt

u<sfx\wGj<t\y o:.Io\tJj sUt\t\jzpfx\xJw)s|2Jn+vprqkn+sft y\jY N,R n9tJy\o\n+q-o\t\j sUt\tJjp?x\x\w)s|23n9p?qkn+sft y\j

ZN,R 4Z\5sUw)n( oJj<8j<tqMp HUj<u

αδ = 0.6;3sUty\j!H w)pfn+qu5sU8j5t\u<j5wZ,-u<sft-HUj5w)*Uj5w0HUj<wGlZ~9j3H3w)pfn x\w)n 2vxsUoJw

MlGo8lkpf88j<tq-*Uw)pft\y#4&M|pft\8sUn+t\l<;~_pH n+q)j<l)lGjvy\ju5sUt-HUj5w)*Uj5t\u<jvj<lGq(qkj5~9~+j<8j<tq ~9j5tq)j o\j

~/.LsUtztJj xj5oJqM~/.Ls l)j5w HUj5w p HUjy\j>qkj5lGqklBwkpfn+l)sUt\t\p ~9j?4Q t HUsfn+q y\sUt\u o\j~/.Ipf~+*UsUwGn qk\8j l)ju5sU8x sfwGqkjy\jpft\n5w)ju5sU\5w)j<tqkjp HUj<uu<j o\j~/.LsUt xj5oJqy\5y\o\n9wGjy\j5lj5lGqkn+vprqkn+sUt\l yJj~_pxFpfwWqkn9j'G'-j5q87<8jzxJ~9oJq fqn+j<o32 o\jxJw)!H o#4 a tj :jq|;c~+j<ll)j5o\n9~+l

αMj<t =sUt\uqkn9sft y\j

αδt\<u5j<l)lkp?n9wGj<lxsUo\w s l)j5w HUj<wo\tJjzu5sUt HUj<w)*fj<t\u5jy\j~/.Ipf~9*fsUwGn q)\8j

l)sftqmx\~9o\l pflo\j u<j5o32 x\w) H o\l>xFpfw~9px\w)sfx sfl)n qkn9sft '-4+i j5qmsUt u<sU8x\wGj<t\y vpf~ u<sU88j<tq~+j<ll)j5o\n9~+lBpfx\xFpfw)pfn+l)lkpftqBt oJ5w)no\j5j5tqBxj<oMHfj<tqM= pfn9wGj>u5sUt HUj<w)*fj<w X x\pfwBj 23j<8x\~9j|^~+jmqkj<wGj

CCy(R)2

hexp(− Mhβ+1

144Cy(R)2)N−1∑

k=0

E

(

N2

(hβ+1

2

3√

2, [P0,k+1]y − yN,Rk+1 (·), (PN,m

tk+1)1≤m≤M , (P

N,mtk+1

)1≤m≤M))

.

Q w<; n9~ pfx\x\pfwkp?n+qo\j ~ .Isftxsfo\w)wkp?n+qmj :j5u5q)o\j<w ,u<j5wGqCp?n9t\lmj<tJy\w)sUn qklmyJj 8j<n9~+~+j5o\w)j5lvp-GsUw)p?qkn+sft\l54 pfwZj23j5x\~+jf;\~9sUwGl o\j>yFpft\lB~9p x\pfwGqkn+j>'G'C;\sUt:HUj5oJqMvp-GsUw)j5w

P([AMk ]c)X y#.LsUzx\w)sIH3n+j<tqB~9jmq)j<w)8j

u<n TAy\j5l)lGo\lC^ ; sUts qkn+j<tq

P

( 1M

∑Mm=1 Um|gj(PN,m

tk+1)|2 − |gj(PN,m

tk+1)|2

√1M

∑Mm=1 |gj(PN,m

tk+1)|2 + |gj(PN,m

tk+1)|2

>1

3h

β+12

∣∣∣∣(PN,m

tk+1, PN,m

tk+1)1≤m≤M

)

≤2 exp

(

−Mhβ+1 1

M

∑Mm=1 |gj(PN,m

tk+1)|2 + |gj(PN,m

tk+1)|2

18 1M

∑Mm=1

(|gj(PN,m

tk+1)|2 − |gj(PN,m

tk+1)|2)2

)

.

sUo\wBpfw)w)pft\*Uj5wB~+j>pGsfwkpftqZy\j ~/.Ln9t\oFp?qkn+sUtu5n T y\j<lGl)o\l<;FsUt5u<w)n qMpf~9sUwGl

1

M

M∑

m=1

(|gj(PN,m

tk+1)|2 − |gj(PN,m

tk+1)|2)2

=1

M

M∑

m=1

(|gj(PN,m

tk+1)| + |gj(PN,m

tk+1)|)2(|gj(PN,m

tk+1)| − |gj(PN,m

tk+1)|)2

≤8Cy(R)2

M

M∑

m=1

(|gj(PN,m

tk+1)|2 + |gj(PN,m

tk+1)|2).

X ' 4LK\4EDU^

'A~ pfx\xFp?wkpfn qq)w)sUx =sUwWq>yJj vp-GsUwGj<w (|gj(PN,mtk+1

)| − |gj(PN,mtk+1

)|)2 xFpfw

4Cy(R)2 4#a tj j5q<;#5qkpftqy\sUtJt\ oJj

PN,mtk+1

= Tk(PN,mtk

,∆Wmk,k+1)

jqPN,mtk+1

= Tk(PN,mtk

,∆Wmk,k+1)

sU sft wkp?x\xj<~+~9ji h N

o\jTk

y\<lGn9*UtJj~9pqkw)pft\lGn+qkn+sUt y\j~_puCFpfn+t\jy\j pfwGSfs HPN ;cn+~@ p y\jz=sUwWqkj<l8uCFpftJu<j<l o\j

gj(PN,mtk+1

)j5qgj(P

N,mtk+1

)lGsUn9j5tqMxJw)s uC\j<l54 n9tJl)n/;Jl)n

PN = XN j5q!o\j ~_p u<~9pfl)lGj G X o\nw)j5u<sUo H3wGj[P0,k+1]y − yN,Rk+1

^j<lWq oJt\j u5~_pflGl)j y\j>=sUt\uqkn9sft\lB`n9x\lGu Jn+q>X>sUtzs qkn+j<tq

|gj(PN,mtk+1

) − gj(PN,mtk+1

)| ≤ C|∆Wmk − ∆Wm

k |

j5q-sUt HfsUn+q oJj~ .IsUt *pf*ft\jo\t = pfu5q)j<o\wh4 a t *U<tJ<wkp?~/;u5sU8j8~9pu5~_pflGl)j

[P0,k+1]yt#.Lj<lWq xFp?l

o\t\j8u<~9pfl)lGj8y\j=sUt\uqkn9sft\l `n9x\lGu \n q>X[P0,k+1]y − yN,Rk+1

t:.EpxFp?l y\j8wkpfn+l)sUt y:.I7qkw)j8o\t\j8u<~9pfl)lGj8y\j=sUt\uqkn+sUt\lZ`n9x\lGu \n q>Xmj5qBy\sft\u G t\sftxJ~9o\l54 pfn+lZy\pft\lZt\sfqkwGj>u|pflZxFpfwWqkn9u5o\~9n+j5w6y\j>~_p pfl)j . $;Jn9~t#.Lj<lWqBxFp?ly\n 8u5n9~9jMy\jmlGj>u5sUt-Hrpfn9t\u5w)jo\jm~_p(p-GsUwkp?q)n+sUt X '-4IKJ4ODU^ j5lGqqksfoJqy\jm87<8j q)w)sUxv=sUwGq)jX HUsfn9w>5*pf~+j<8j<tq ~_pl)sUo\l TAlGj<u5q)n9sft K\4ED34IHzsf sUt l<.Ipfxj5w )5sUn+qo\j~9j<lm=sUt\uqkn9sft\l>j<lGq)n98<j5l t\jl)sUtqxFpflB~+sUn9ty#.L75qkwGj(`n9xJl)uC\n+qYX1^4

"!#%$&'(*),+

& sfo\l x\wGj<t\sft\l u<jq)qkjv=sfn9lαδ = 0.6

jq ~9p p?l)j . $ -%<3?: 8P4 csUn9u5n ~9j5l-w)<lGo\~+qkp?qkl-s qkj5t o\l xsUoJwy\n :5w)j<tqkj5l=H?p?~9j5o\w)lBy\j

αM4

j

Prix

105 15

6

7

8

5.5

6.5

7.5

8.5

1 1.5 2 2.5 Prix BS

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Pr

ix

105 15

1

2

3

4

5

6

7

8 1 1.52 2.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105 15

0.4

0.5

0.6

0.7

0.8


P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Co

uvertu

re

105 15

1

0.5

1 1.52 2.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

"!#$%&'&)(&*,+&&&-.*,/&'0&&%12*%31*$45(&

αδ = 0.66+)+78&1:9;<>=0*,&*%@?A11*84B<C0D*

yN,RE1*,&)

F170GHI<>&JK&+*,=1,*L MHCM)GB17*7NOJP,..K=&+)&<>QR1*=&C

0JK(*+&>17*αM

NTSUQLV4B&<>1WXO,&<'0-=&.0*KαM > 2

NTYZ*U5(URL ;![\

αδ = 0.63X?],&*,(^01_U(*+&*_RL&(&*.*,&

t0N2`aG))L<>=&00*,=Fb?c11*,54B<C^.$d

yN,R"&, ^.K931*=&)L-&(*+&K'R

&(&*.*,N

egfihjflk m%noqp2rtsUuTvw xqyUp7zB|xqyUvEsu~ni

?A+*$.M<>U1_&&<>&)1*$4C9F@?c0,.t0<Cq=&+<C_31*4C9L.

0,.).tk17*

0 ≤ k ≤ NN2O,J).=*&,.)J%.*&*RC=\*<C\.0

1*$42WH&1* )jj = N

L\.031*,$4C&,=+)09%L\.15GNSj08jUF,U!8(&0&_1*.<C/\.*,&

N = 20WK0 = 20

&&q*,M = 5000 \L17*QP,&ZR&&*J>@?c+*,$M<>5VW2J.*&QJJK1*4"=&L1*

b?c0+*.M<>g\1*RQ*<'0gJ )|B M0& 5(J B43X?]=&*,R_B430,.

tj17*j = 4, 9, 14, 20 3*<3*,H?A,J^.*,&0&<>&)8W)@?c***_,*0g1*4>&,+*.17*_&_(&*_

S=&0(=& 17Q0<'R^.0j<'7=&&&\.Q&N

)


S

Prix

80 90 100 110 12085 95 105 115

10

20

5

15

25

Prix algorithme Prix Black-Scholes

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,1

t 1 PN,1

t 1 PN,2

t 1 PN,2

t 1 PN,3

t 1 PN,3

t 1

. + !"t20 = T = 0.5

S

Prix

90 100 110 120 130

10

20

30

40

5

15

25

35


P0

t 0 t 1 t 2 t 3 t 4 t 5PN,1

t 1 PN,1

t 1 PN,2

t 1 PN,2

t 1 PN,3

t 1 PN,3

t 1

#$%&(' N*),+-\.Q1*,$4t14 = 0.35

.


S

Prix

80 90 100 110 12085 95 105 115 125

10

20

30

5

15

25


P0

t 0 t 1 t 2 t 3 t 4 t 5PN,1

t 1 PN,1

t 1 PN,2

t 1 PN,2

t 1 PN,3

t 1 PN,3

t 1

. !"t9 = 0.22

S

Prix

90 100 11095 105 115

10

20

5

15


P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,1

t 1 PN,1

t 1 PN,2

t 1 PN,2

t 1 PN,3

t 1 PN,3

t 1

# $%&(' N ' +!-Q1*4&t4 = 0.1

' 4cd a V `MV& acdB' a V )= 24/ 2:%)( / )& sfo\l pf~9~+sUtJl qkj<lWqkj5w ,x\w)5l)j<tq ~_p:H?pfwGn_p?q)n9sft l)n+ o\~+qkpft\5j8y\j<l-xFpfwkp? qkw)j5l

N;δj5qM

yFpft\l-~+ju|p?l yJj8~9p pfl)j . $ j5q xsfo\w ~ .Epf~+*Usfw)n qk\j x\wG<lGj<tq)yFp?t\l ~_pxFpfwGq)n9j ')'G'C;uCFpfxJn+qkwGjD ;xsfo\w ~+j<lacb>VJd w) F<uC\n+j<l<;:sf sUt x\wGj<t\y ~9j(pr23n9 o\,vuCFpo\j x\pfly\j(qkj<8x\lj5tq)w)j ~+j(w)5l)o\~ qCp?qy#.Lo\tx\wGs ~5j y\jw)<*fw)j<lGl)n+sUt jq(~ .Is lGqCp?u<~9j?4bmpft\l u<j8u<pfl<;u<sU88jn+t\y\n( oJyFpftJl ~_pxFp?wGqkn+j ')'G'C; ~+j<lqkJ<sUw<8j<l 'G'C4ED3;?'G'C4EHJ;?'G'C4LK>jq ')'C4EN l)sUtq j5t\u<sfw)j H?pf~9p ~+j<l yFp?t\l u5ju|p?l j5q y\sUtJu~9p x\w)sfxsUl)n qkn+sUt(' 4 i<*Upf~9j5j5tq<4#&MsUo\l>t\sUo\l p?qGqkj5t\y\sUt\l>y\sUtJu8,s lGj<w HUj<w ~9j 87<8j lGj<o\n+~ y\j u5sUt-HUj5w)*Uj5t\u<j=sft\u5q)n9sUty\j

αδxsUo\w

αM4

& sfo\lBu \sfn9l)n+l)lGsUt\ld = q = 1

4 ` .Iacb>VJdMd oJj t\sUo\l HUsUo\~+sUt\lBpfx\x\wGs u Jj<wBj<lGqM~_p l)o\n H?pftqkj

dStSt

= µdt+ σdWt,

dYt = (−rYt − θZt)dt− ZtdWt + dKt,

YT = (K − ST )+, Yt ≥ (K − St)+, 0 ≤ t ≤ 1,∫ T

0

Yt − (K − St)+dKt = 0

pIHfj<uθ = µ−r

σ

4#V n ~/.LsUt u<sU8xFpfwGj p HUjj 2Jj58x\~9j-u5sUw)wGj<l)xsft\y,f(t, x, y, z) = −ry − θz

jqg(t, x) = (K − ex)+

4e6j5u<n u<sfw)w)j5l)xsUtJy y\sft\u pfou|pflmy#.Lo\tx\o3qpf8<wGn9u<pfn+ty\jlGqkwGn9Sfj

Kj5qu5sUoMHUj5wGqZlkp?t\lu5sUtq)wkpfn+tqkj yJjmxsUwWqkj=j<o\n+~9~+j X y#.LsUo\ty\wGnJHUj<w

f~+n9t\<pfn+w)j

j<t(y, z)

^4`j<lBxFpfwkp? qkwGj<lByJj ~9p y3@3t\pfn(o\j y\j

SjqMy\j ~/.LsUxJqkn+sUtlGsUtqB~9j<lZl)o\n H?pftqkl

µ σ r T S0 Kg 4 g N g 4IK g 4 g N i i gfg i gfge6j5qMj23j5x\~+j j5lGqMq)n9wG>y\j Y R& g Kf] sf~9j5lMpfoJq)j<o\wGlMyJsUt\t\j5tq oJtxJw)n 2y\j w)5=5w)j5t\u<j

Y0 = 13.554

& sfo\loJqkn+~9n+lGsUt\l ~9p-87<8jxJw)s u<5y\o\wGj>xsUo\w qkj5lGqkj5wZ~9p&Hrpfw)n9p?qkn+sUtl)n+ oJ~+qCp?t\5jyJj<lZx\pfwkp? qkw)j5l o\jxsUo\wZ~9j>u<pflMyJj ~ .IsfxJqkn+sUtzp HUjqCpfo 2zy#.Ln9tqk<wG75q<4 αδ = 0.6

j

Prix

105

14

15

16

17

18

1 1.5 2 2.5 3 Prix de référence

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1


j

Std Pr

ix

105

10

5

1 1.52 2.53

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105

-0.5

-0.4

-0.3

-0.55

-0.45

-0.35

-0.25 1 1.52 2.53

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Co

uvertu

re

105

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 1 1.52 2.53

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 αδ = 1.

j

Prix

105

14

15

16

17

13.5

14.5

15.5

16.5

17.5

1.5 2 2.5 3 3.5 Prix de référence

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1


j

Std Pr

ix

105

10

5

15

1.52 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105

-0.5

-0.4

-0.45

-0.35

1.52 2.53 3.5

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Co

uvertu

re

105

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 1.52 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 αδ = 1.5

* ' " % ) µ )% % 0.08

* ) %' % ' 0% ( % # * ) % )' % "$&(' % ' ' "*

+


j

Std Co

uvertu

re

105

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 1.52 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

1 2 3 4 5 6 7 8 9 10

-0.6

-0.5

-0.4

-0.3

-0.65

-0.55

-0.45

-0.35

-0.251.52 2.53 3.5

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 αδ = 0.6

)' *µ = 0.08

*

j

Prix

105

13

14

15

16

17


P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105

-0.6

-0.5

-0.4

-0.55

-0.45

-0.35

1.52 2.53

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

Q ts l)j5w HUj <*pf~+j<8j<tqv~/.Lj23n+lGqkj5t\u<j yJj l)j5o\n9~+lvyJj u<sft-HUj5w)*Uj5t\u<jxsUo\wαM

j<t=sUtJu5qkn+sUty\j ~_pH?p?~9j<oJw y\j

αδ4 e6j<l l)j5o\n9~+l lGsUtq u5sU8xFpfwkp ~9j5l , u5j<o32s l)j5w HU5lxsfo\w ~+jBu|pfl acb>VJd"t\sUtw) F<uC\n+jf4

bjvx\~+o\l<;j5t u<sU8xFpfw)pftq(y\j5l?Hrpf~9j5o\w)l(y\jαδ

n9yJj<tqkno\j5l-xsUoJw-~+ju<pfl-w) F<uC\n j5q tJsUt3TAwG F5uC\n/;sUtl<.Ipfxj5w )<sfn+q!o\j ~_p u5sUt-HUj5w)*fj<t\u5j-yFpftJlM~9j>u|pflBw) F<uC\nj5lGq x\~+o\lB~+j<tq)j j5q!o\j ~/.LsUtp qkj<tJyFpft\u5j,8pfx\x\wGs u \j5w ~+j x\wGn 2pf85w)n9u<pfn+tzxFp?w pfo TAy\j5l)l)oJl<4 Q tu<sUt\lWqCp?q)j(5*pf~+j<8j<tqMj5t s l)j<w H?pftq~9j5lMu<pflαδ = 1.5

pIHUj5u~_p pflGj . $ jqαδ = 0.6

p HUj<u~9p p?l)j . $ -%<36: 8 oJj~/.Ipf~9*fsUwGn q)\8ju5sUw)wGn9*Uj~9j y\w)n = qMxsUo\w wGj5q)w)sUoMHUj5w ~9j xJw)n 2zw)n+l o\jT t\j<o3qkw)j?4#e6j5ly\j<o 2 j 2Jj58x\~9j5l8sUtqkwGj<tqM<*pf~+j<8j<tqo\j~9sfw)l o\j(~ .Ep?x\x\w)s|23n9p?q)n+sfty\j

Yj5lGqmlGo8lkpf88j<tq sUt\t\j?; ~ .Ep?x\x\w)s|23n9p?q)n+sftl)oJw~9p8u5sUoMHUj5wGqkoJw)j

j<lWq sUt\tJjf4Pact LFt#;Pj5t(u<sfxFp?wkpftq ~9j5l u|p?lαδ = 0.6

xsUoJw ~9j5l pflGj<l . $"jq . $ -%<3?:48sUt u5sUt\lGqkp?qkjo#.LsUt*p?*Ut\j j|pfo\u5sUo\x ,pGsfoJqkj5wBy\j5lMxsU~ @ t U8j<lZ~9s u<pfo32y\j y\j5*Uw)

1xsUo\wMpfxJx\w)s uC\j5w

Y4

iP[?K

% 4

xJw <lp HUsUn+w HU<wGnJLF ~9p H?p?~9n+y\n qk>y\j5l w)5l)o\~ qCp?q)lMy\j u<sUt HUj<wG*Uj<tJu<j-y\sUt\t\5l yFpft\lM~_p xFpfwWqkn9j>')'C; t\sfo\lxsUoMHUsft\lMu<sfxFp?w)j5wB~+j<lZy\j<o32pf~9*fsUw)n q)\8j5l\u<j5~9o\n#y\j ~_p xFpfwGq)n9jm'6jqMu<j5~9o\n#y\j ~_p xFpfwGq)n9j>'G'C4 ` .Lo\tj<lWq pfl)l)oJw yJj<l n qk5wkp?q)n9sUtJl(y\j n9u<pfw)y , u Fp/o\jn9t\lWqCpftq y\jy\n+l)u<wG5q)n9lkprqkn9sft

tkjq uC\j<wGu Jj,

pfx\xJw)s uC\j<wZj<tz87<8jq)j<8x\lYtk

jqZtk

4\`j>l)j5u<sUtJyt:.Ep xFpfl j<lGsUn9ty#.In qk5wkp?q)n9sUtJl6y\jcn+u|p?w)yu|pfw~9jl)uC\<pj<lWq j 23x\~9n+u<n qkjf;jq(qkwGsUoMHUjxFpfw y\j5l-wG<*UwGj<l)lGn9sUtJl-lG<xFpfwG<j5l

YjqZ4& sfo\l-pf~9~+sUtJl#HUsUn9w

u<sf8j<tqBl)j u<sfxsUwWqkj5tqB~9j5lBy\j5o32pf~+*UsUwGn qk\j<ll)o\wBo\tj 2Jj5xJ~9j>qA@ xjf4& sfo\lt\sfo\lx\~9p)5sUt\l yFp?t\l~+j(u<pflyJj-~/.LsUx3qkn9sftj<o\wGsUx<j5t\t\j pIHUj5u-y\n( <wGj<tqkn+j<~ y\j qCpfo32y:.In+tq)<w)7q|;pIHfj<uzo\t\j pflGj+. $4 `jw)<lGo\~+qkp?qy\o qk\5sUw 5j'C4EHt\jxj<w)8j5q qkJ<sUwGno\j5j5tqxFpfl y\j= pfn+w)jqkj5t\y\wGj j5t75jqkj5x\l

N;MHUj5w)lM~ .In+tMLFtJn:j5q

δHUj<wGl

04 n9t\lGn#yFpft\lBo\tzx\w)j5n+j<w6qkj5x\l5;FtJsUo\l

pf~+~9sft\l LJ23j5wNjqδ; o\t y\sfvpfn+t\j

D = [40, 180]j5qM= pfn9wGj>qkj5t\y\w)j

MHUj<w)l ~/.Ln9tML\t\n/4\& sUo\lp?~9~+sUt\l

pf~+sUw)lu<sU8xFpfwGj<wZ~+j<lBu5sUt HUj<w)*fj<t\u5j<lyJj<lMy\j5o32p?~9*UsUwGn qkJ8j<l54 N = 5

;δ = 20

M

Prix

51. 10 52. 10 53. 10 54. 10 55. 106

7

8

6.5

7.5

Algo explicite Algo itérations de PicardPrix BS

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1


M

Std Pr

ix

51. 10 52. 10 53. 10 54. 10 55. 10

0.1

0.2

0.3

0.4

0.5

Algo explicite Algo avec itérations de Picard

P0

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 N = 10 δ = 10

M

Prix

51. 10 52. 10 53. 10 54. 10 55. 106

7

8

6.5

7.5

Algo explicite Algo avec itérations de PicardPrix BS

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

M

Std Pr

ix

51. 10 52. 10 53. 10 54. 10 55. 10

0.1

0.2

0.3

0.4

0.5


P0

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 N = 20 δ = 5

e! ' BdMa N34 e Q md 'WV Q & bma V%m` Q dB' a V

M

Prix

51. 10 52. 10 53. 10 54. 10 55. 106

7

8

6.5

7.5


P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

M

Std Pr

ix

51. 10 52. 10 53. 10 54. 10 55. 10

0.1

0.2

0.3

0.4

0.5


P0

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 N = 50 δ = 1

M

Prix

51. 10 52. 10 53. 10 54. 10 55. 106

7

8

6.5

7.5


P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

M

Std Pr

ix

51. 10 52. 10 53. 10 54. 10 55. 10

0.1

0.2

0.3

0.4

0.5


P0

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

Q txj<o3q6w)j5vp?w o\j5w oJj ~9j5l pf~+*UsUwGn qk\j<l l)jMu<sU8xsUwGq)j<tq j5l)l)j5tq)n9j5~9~9j5j5tq y\jM~9p 87<8jZ= p)5sUtv,u<j5u<nFx\w<l oJjM~ .Ep?~9*UsfwGn qkJ8jpIHUj5uMn qk5wkp?q)n9sUtJl y\j cn+u|pfwGy qkj5t\y,>y\sUt\tJj<wco\tJjBx\~9o\l xjqkn+q)j0H?pfwGn_pftJu<jo\j~ .Ep?~9*Usfw)n qkJ8jj 23x\~9n+u<n qkjf4 Q t xj5oJq xj<t\lGj<w o\j~9pu|pfo\lGjj<lWq ~_pn+t\n+8n lkp?q)n+sft lGn9 o\~+qkpftJ<jj<t

(Y, Z)~9pzxFpfwWqkn9jj5t

Z∆Wxj5oJq GsUo\j5w(~+j8wU~9jy\j H?pfw)n9p ~+jy\j8u<sUtqkw f~9jf4 Q t xj5oJq-j5tMLFt

w)j5vp?w o\j<w o\j~+j n_p?n9l j<tqkw)j~9jzx\wGn 2 y\sft\t\x\pfw8~ .Epf~+*Usfw)n qkJ8jj5q~+jzx\w)n 2 yJjwG5=5w)j<tJu<j j5lGq~95* 5w)j<8j<tqBx\~9o\l= pfn ~9jyFpft\lB~9j>u<pflBy\j ~/.Ipf~9*fsUwGn q)\8jmj 23x\~9n+u<n+q)jf4& sfo\l p?~9~9sft\l(, x\w)5l)j5tq j<l)l)p1@Uj5w y\jv= pfn+w)j H?pfw)n+j<w(l)n+ o\~+qkpft\58j<tq(~+j<l-x\pfwkpf 5q)w)j5l-yFpft\l-~/.Ipf~9*fs?Tw)n qk\8j p HUj<un qk<w)p?qkn+sUt\lcy\j n+u|pfwGy#; xsUoJw HUsfn9w l)n~9j<l wG<lGo\~+qkp?qkl l)sftq6n9y\j5tqkn(o\j<l6,(u<j5o32s l)j5w HU5lxsUo\wZ~/.Ipf~9*fsUwGn q)\8jmj 23x\~9n+u<n qkjf4 αδ = 0.6

jqαM

H?p?w)n+jf4

j

Prix

105 15

2

3

4

5

6

7

1 1.5 2 2.5 Prix BS

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Pr

ix

105 15

10

20

5

15

25 1 1.52 2.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105 15

-0.1

0.1

0.2

0.3

0.4

0.5

0.6


P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1


j

Std Co

uvertu

re

105 15

1

0.5

1 1.52 2.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 αδ = 1

j

Prix

1051

2

3

4

5

6

7

8

9

1 1.5 2 2.5 3 3.5 Prix BS

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Pr

ix

105

10

20

5

15

1 1.52 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105

0.1

0.2

0.3

0.4

0.5

0.6

1 1.5 2 2.5 3 3.5 Couverture BSP0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1


j

Std Co

uvertu

re

105

1

2

0.5

1.5

1 1.52 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 αδ = 1.5

j

Prix

105

1

2

3

4

5

6

7

2 2.5 3 3.5 Prix BS

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Pr

ix

105

10

20

5

15

2 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105

0.1

0.2

0.3

0.4

0.5

0.6


P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1


j

Std Co

uvertu

re

105

1

0.5

2 2.53 3.5

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 # '$ #0% ' '% # % ) ' "$& ! ' #* )' %

#' αM ' # αδ % $ # )

0 "$& $ &') $ $& * $ %$ % $& "% * #"%$& ) ' # ' $ $& ! #"%$&(% %' *


iZD

& sfo\l>pIHUsft\lmy\ ),qkj5lGqk ~_pKHrpfw)n9p?qkn+sUtl)n+ o\~+qkpft\5j(y\j<lmxFpfw)pf qkwGj<lN;δ; jq

MyFpft\lm~+j-u|p?lmy\j

~_p p?l)j3. $ xsUo\wBo\t\j(acb>VJd w) F<uC\n+jf4& sUoJl pf~9~+sUt\lyFpft\lMu5j(uCFpfxJn+qkwGj>qkw)pfn+q)j<wZx\~+o\lMj5ty\5qkpfn9~~_pH?p?~9sUwGn9lkprqkn+sft jq-u<sUo HUj<wWqko\w)jvy#.LsUx3qkn9sft\l(pf8<w)n+u|pfn+t\j5l<4 Q oJqkwGj~/.Ipf~+*Usfw)n qk\jx\wG<l)j5tq)yFp?t\l(~_pxFpfwWqkn+j 'G'G'C;ruCFpfx\n qkwGj6D3;PsU(~/.LsUt x\w)j5t\y ,Mu Fp/o\j6n9tJlGqCp?tq yJj y\n9lGu<wG5qkn+lkp?q)n9sUt

tk;P~+j pr23n9 o\ j5tq)w)j

~/.Ls lWqCpfu5~9j6jq ~_pl)sf~9oJq)n9sUt-y\o x\w)s ~5j6y\jsfn9t\yJw)j<l u<pfw)wG<l<;fsUt xj<oJq n9pf*Un t\j<w 5*pf~9j58j<tq yJj<o32pfoJq)w)j5lMqk\s yJj<lMxsfo\wMpfx\x\wGs u Jj<wM~9p l)sU~+oJqkn+sUty#.IoJt\j(a bmVJd 4 & sUo\lMpf~9~+sUt\lZ,x\wG<lGj<tq y\5qkpfn9~+~+j<wu<j5lMy\j<o 2pfoJq)w)j5lM85q)\s y\j<l

W = 6

tJj 85q)\s y\j yJj xJw)j<o HUj(y\j ~ .Ij 2Jn+lGq)j<t\u5j jqMy\j ~/.Lo\t\n9u5n+q)>,~ .Eacb>V3d wG F5u \n+j

Yt = g(T,XT ) +

∫ T

t

f(s,Xs, Ys, Zs)ds+KT −Kt −∫ T

t

ZsdWs,

Yt ≥ g(t,Xt),

∫ T

0

(Yt − g(t,Xt))dKt = 0X ' 4 h 4 iP^

u<sft\l)n+lGqkj , y\!LFtJn9wBoJt\j lGo\n+q)j y#.Iacb>VJd tJsUtw) F<uC\n+j<l

Y nt = g(T,XT ) +

∫ T

t

f(s,Xs, Yns , Z

ns )ds+ n

∫ T

t

(Y ns − g(s,Xs))

−ds−∫ T

t

Zns dWsX ' 4 h 4EDU^

y\j l)sU~+oJqkn+sUt(Y n, Zn)

4 Q ty\!LFtJn+q>pf~9sUwGlKnt = n

∫ t

0(Y n

s − g(s,Xs))−ds

;:jq>sUt8sUtq)w)j X HUsUn9wY a + U[r]_^o\j~9sUwGl o\j n qkj<tJy4HUj5w)l~/.Ln9t LFt\n/; (Y n, Zn, Kn)

qkj<tJy4HUj5w)l(Y, Z,K)

l)sU~+oJqkn+sUty\jX '-4 h 4+iP^ 4` .In+y\<jj<lWqpf~9sUwGl8xsUo\w

nLJ23f; y#.Ipfx\x\wGs u \j5w~9p l)sU~+oJqkn+sUt y\j ~ .IoFp?qkn+sUt X ' 4 h 4EDU^8, ~ .Ep?n9y\jzy\j

~/.Ipf~+*Usfw)n qk\8jzj 2Jxsfl) yFpftJlv~_p xFp?wGqkn+j')'C4 a tj j5q|; X ' 4 h 4EDU^j<lWqo\t\j acb>VJd pIHUj5u o\ty\w)n HUj<w`n9x\lGuC\n+qYXf4 dBj<pfw oJsUt\l o\j H3o~9j<l8w)<lGo\~+qkp?qkly\j Y a + [P] ; Yn Y

jq oJj ~ .Isft x j5oJq

l<.Ip?qGqkj<tJy\w)jp HUj<uu<j5qGqkjqk\s y\j,q)w)sUo HUj<w y\j5l sUw)tJj<l n9tJ=5w)n9j5o\w)j5l X y\jx\~+o\l j5t x\~9o\l>x\wG<u<n+l)j5loFpft\y

n→ ∞ ^6y\ox\w)n 2zpf8<wGn9u<pfn+t#4

W 6

tJj p?oJqkwGj(n9yJ<j u<sUt\lGn9lWqkj-,o3qkn9~+n+l)j5wB~9j5l wG<l)oJ~+qCprqkl yJj Y RZe a g Dr]/4`j<lpfoJqkj5o\w)l u<sft\l)n+y<wGj<tqm~9ju|p?lMsf~/.Ls lWqCpfu5~9j

g(t,Xt)j<lWq o\t\j l)j5n+vp?wGqkn t\*p?~+ju|,fy\~9,f*yJj ~9p =sfw)8j

g(t,Xt) = g(0, X0) +

∫ t

0

Usds+

∫ t

0

VsdWs + At

pIHfj<uU

j5qV

y\j<l x\wGs u<j<lGl)o\l xJw)!H n+l)n ~9j<lGHf<w)n L pftq ~9p u<sUtJy\n+q)n9sUt y:.In+tqk5*Uwkp n9~+n qkE∫ T

0|Vt|2 + |Ut|2dt < ∞ jq

Ao\t xJw)s u<j5l)l)oJlvu|,fy\~9,f*\; pfyFp?xJqk?;cu<wGsUn9lGlkpftq|; j5qKHf<w)n L pftq~9p

u<sft\y\n+q)n9sft y:.In+tq)<*Uw)p n9~+n qkE|AT | < ∞ 4 bmpft\l u<ju|p?l<; sUt lkpfn q o\jzuC\j5w)uC\j<wo\t\jl)sf~9oJq)n9sUt ,

X '-4 h 4+iP^6w)j H n9j<tq ,uC\j5w)uC\j<wMo\tzqkw)n+x\~9jq(Y, Z, α)

HU<wGnJL pftqM~ .Eacb>V3d

Yt =g(T,XT ) +

∫ T

t

f(s,Xs, Ys, Zs) + αs1Ys=g(s,Xs)[f(s,Xs, g(s,Xs), Vs) + Us]−ds

−∫ T

t

ZsdWs,

Yt ≥g(t,Xt)X ' 4 h 4IH^

pIHfj<uE∫ T

0|Yt|2 + |Zt|2 + |αt|2dt

4a t u<sU8xFp?wkpftqp HUj= pfn+q o\j-~ .Isftl)pfn+qy\ ),J;Jj<tzoJq)n9~+n+lkpftq6~_p-wGj<pfw o\j-4IHJ;o\j>~9jx\w)s u5j<lGl)o\lMu5w)sUn+l)l)pftq

Ky\j>~9p-l)sU~+oJqkn+sUtyJj>~/.Iacb>VJd

X '-4 h 4+iP^6l)j5wkpy\j ~_p-=sUw)8j

dKt = αt1Yt=g(t,Xt)[f(t,Xt, g(t,Xt), Vt) + Ut]−dt,

pIHfj<u0 ≤ αt ≤ 1

4r` .In+t\u<sUtJt o\j xsUoJw y\qkj<wGn+t\j<wKy\j H3n+j<tq yJsUt\u ~9j x\wGs u<j5l)l)oJl

α4 sUo\w 8sUtq)w)j5w

~/.Lj23n+lGqkj5t\u<jjq ~ .IoJt\n9u5n+q) yJj~/.L oFprqkn9sftu5n T y\j<lGl)o\l5; ~+j<l pfo3qkj<oJw)l oJq)n9~9n l)j<tq>o\t\jwG<*UoJ~_pfwGn9lkprqkn+sft y\jX '-4 h 4EH^ 4Fa txFp?wGqkn+u<oJ~9n+j<w5;JoJt\j =sfn9lu JsUn9lGnoJtxFpfwkp? qkw)jmy\j w)5*Uo\~9pfw)n+lkp?q)n+sUt

n;\n9~+ln9tqkw)s y\oJn9l)j5tq

o\t\j l)oJn+qkj>y\j>=sUt\uqkn9sft\lφn ∈ C∞ qkj5~9~9j5l o\j

0 ≤ φn ≤ 1j5q

φn(x) =1l)n |x| ≤ 1

2n,

φn(x) =0l)n |x| ≥ 2

2n.

xFpfwWqkn9w y\j>u<jq)qkj>lGo\n+q)j>yJjm=sUt\uqkn+sUt\l<;3~9j5lBp?oJqkj5o\w)lZy\ LFt\n+l)l)j5tqMoJt\j>l)o\n qkjmy#.Iacb>VJd xFpfw)pfqkwG<jxFpfw

n

iZfK

e c' BdMa h 4 Q Z' Q &mV%cdM'We ' &a V

Y nt =g(T,XT ) +

∫ T

t

f(s,Xs, Yns , Z

ns ) + φn(Y

ns − g(s,Xs))[f(s,Xs, g(s,Xs), Vs) + Us]

−ds

−∫ T

t

Zns dWs.

X ' 4 h 4LK^

a t u5sU8xFpfwkp?tq p HUjqkj5w)8jαs1Ys=g(s,Xs)

4`j<lcpfoJqkj5o\w)l sftqkwGj<tq p?~9sUwGl o\jB~_pl)sU~+oJqkn+sUt y\jZu<jq)qkjMacb>V3du5sUt-HUj5w)*UjB~+sUw)l oJj

n→ ∞ HUj5w)lc~_pl)sf~9oJq)n9sUty\j X '-4 h 4EH^ 4Fa tsUoJqkwGjf;\sUtp

Yn Y4F`c.Iacb>VJd wG<*Uo\~9pfwGn9l)5j X ' 4 h 4LK^6j<lWq oJt\j acb>VJd

,-y\wGnJHUj5w`n9x\lGu Jn+q>X?4 Q txj<o3qZy\sUtJum~ .EpfxJx\w)s uC\j<w6,-~/.Ipfn+y\jMy\jm~ .Ep?~9*Usfw)n qkJ8jMy\<u5w)n qyFpftJl~_p xFpfwGq)n9j'G'6j5qMj<lGx<w)j5w s q)j<t\n+wBy\j5l sfw)t\j5l lGo\x<wGn9j5o\w)j5l X y\j x\~9o\lBj5tx\~+o\lBx\w)5u<n+l)j<lM~9sUwGl o\j

n → ∞ ^y\ox\wGn 2zpf8<wGn9u<pfn+t#4

)= $K) !)=% :< =

e6j<l y\j<o 2 85q)\s y\j<lZT6x<t\pf~9n+lkp?q)n9sftjqwG<*Uo\~9pfw)n+lkp?q)n+sUt T u5sUt\y\o\n+l)j5tqm,pfx\x\wGs u Jj<w t\sft xFpflMo\t\jacb>VJd pfn+lMo\t\j-lGo\n+q)j(y#.Eacb>V3dx\pfwkpf85q)w)5j(l)sUn qx\pfw ~9j u<s j u5n9j5tqmy\j-x5tFpf~+n9l)p?qkn+sUtl)sUn q xFpfw~9j8u5s j 8u5n9j<tq y\jwG<*Uo\~9pfw)n+lkp?q)n+sft#4 bjx\~9oJl<; ~_p8qk\s y\j8y\jvx5tFpf~+n9l)p?qkn+sUt p?x\x\w)s uC\j

YxFpfw j<t

y\j5l)l)sfo\l j5qM~9p 85q)\s y\j yJj wG<*UoJ~_pfwGn9lkprqkn+sftzpfxJx\w)s uC\jYxFpfwMp?o3TAyJj<l)lGo\l<4

& sfo\lmu<sUtJl)n9yJ<w)sft\lmxsUo\wmu5j-qkj<lWq>~9j-u<pflmy\ox\oJq>pf8<w)n+u|p?n9ty\j ~_plGj<u5q)n9sUt K\4IHJ;:pIHUj5u ~+j<l75j5lH?p?~9j<oJw)lBy\j xFpfw)pf 5qkwGj<l54bmpftJl(oJt x\w)j5n+j<wmqkj58x\l<;t\sUo\l(p?~9~+sUt\l LJ23j<w ~9j5l xFpfwkp? qkw)j5l y\j8w)5*Uo\~9pfw)n+lkp?q)n+sUtjq(y\j8x5tFpf~+n Tlkprqkn9sft#; x\o\n9l-= p?n9w)j Hrpfw)n+j<w-l)n9 o\~ qCp?t\<8j5tq

N; ~+jt\sf w)jy\jv=sUt\u5q)n9sUtJl y\j pfl)jj5q

M4 &MsUo\l

j<lGx<wGsUt\l-p?n9t\lGn;HU<wGnJLFj5w o\t j<t\u<pfy\w)j58j<tq y\jY0

x\pfw ~+j<l>y\j<o 2 85qkJs y\j<l j5q#HU<w)n LFj<wo\j~/.Lj<t3Tu|p?y\w)j5j5tq-y\j H n9j<tq-x\~9oJl(x\wG<u5n9l-pfo =o\w(jq ,j5l)o\wGj o\j~9j5l(u5s j 8u<n+j<tqkl-y\jx<t\pf~9n+lkp?q)n9sft jqw)5*Uo\~9pfw)n+lkp?q)n+sUtpfo\*U8j<tqkj5tq|4

& !#(

sUo\wy\n( <wGj<tqkj<l H?pf~+j<oJw)lmy\jn; t\sUo\lmoJq)n9~+n+l)sUt\l ~9p pfl)j . $ -%<3?:48 p HUj<u

αδ = 0.6jqαM = 2

4& sfo\lmpIHUsUtJlmj5tj :jq H3o yFpftJlm~+j<l u Fp?x\n+q)w)j5lx\w)5u<5y\j<tqkl oJj u<j-uC\sUn 2y\j-xFpfwkp? qkw)j(pfl)lGo\wkpfn q~_p u5sUt-HUj5w)*Uj5t\u5j-y\o n9pfn9lj5q y\j ~_p H?pfwGn_pftJu<jf4 &MsUo\lM8sUtq)w)sUtJlB~9j5lBw)<lGo\~+qkp?qklBy\j u5sUt-HUj5w)*Uj5t\u<j(y\ox\wGn 2 ;\y\j ~_p u<sUoMHUj5wGq)o\w)j j5qMyJj ~+j<o\wZ5u|pfwWqGT/q @ xj xsfo\w

nH?p?~_pftq

0.5, 1, 5, 104

iZN

j

Prix

105 15

20

15

25

0.5 1 5 10 Prix de référence

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105 15

-0.7

-0.6

-0.5

-0.4

-0.65

-0.55

-0.45

-0.35

0.51 5 10

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Pr

ix

105 15

10

20

5

15

0.51 5 10

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Co

uvertu

re

105 15

1

0.5

0.51 5 10

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 "$(5".K* .#%

nS=!><"$# 8 I% @?*"W%-".**"$ *%S * *"$ +*@

[

pft\n5w)j y\sUtq sft p= pfn+q H?pfwGn9j5wN;δj5qM4 Q t lkpfn q oJj~ .Isfty\sUn+q>t\sUw)pf~+j<8j5tq pfx\x\wGs u Jj<w

~9jvx\wGn 2 y\jwG5=5w)j<tJu<jzxFpfw-j<t y\j5l)l)sfo\l<4 e.Lj<lWq n+j<t u<j o\j~ .IsUt s l)j5w HUjzxsUoJwn = 0.5

sUo14

pfwu<sUtqkwGjf;:xsUo\wn = 5

sfon = 10

;:~+j<ly\j<wGt\n5w)j5l H?p?~9j<oJw)lmy\jjq)j<lGq)<j5l x\wG<l)j5tq)j<tq qksfoGsUo\wGl

o\t x\w)n 2~95* <wGj<8j<tq lGo\x<wGn9j<oJw(pfo x\wGn 2y\jw)=<wGj<t\u5jf4 Q t xj5oJq(xj5t\l)j5w o\j8~/.Ipf~+*Usfw)n qk\8j-t#.EpxFpfl j<t\u5sUw)ju5sUt-HUj5w)*UxsUo\w u5j<l9H?pf~9j5o\w)l jq o\jl)nc~/.LsUtqkj<lWqCpfn q y\j<l9H?pf~+j<o\wGl(yJj

jlGo\x<wGn9j<oJw)j<l5;

sUtw)j<x\pfl)lGj<wkp?n+q j5ty\j<lGl)sUo\l y\ox\w)n 2 y\j wG5=5w)j<tJu<jf4e.Lj<lWq <*pf~+j<8j<tq>u5sU\5w)j<tq y#.Ls lGj<w HUj<w o\t\ju<sft-HUj5w)*Uj5t\u<j-8sUn9tJlBw)pfx\n+y\j yFpftJl ~+j u|pflMy\j

n*fwkpft\yu|pfwB~9j u5s j 8u<n+j<tqy\j(`n9xJl)uC\n+qYX y\oy\w)n HUj<w

n9tqkj5w H n9j5tq yFp?t\l ~+j<l>j<lGq)n9p?qkn+sft\lmy#.Ij5w)wGj<o\wGl o\j~/.LsUt pzs qkj5t o#4 ~9o\l>u<j5~9o\n TAu<n j5lGq(*fwkpft\y:;x\~+o\l~9j5lBu5sUt\lGqkpftqkj<lMyFp?t\lM~+j<l sUw)t\j5lMy#.Lj<wGw)j5o\wMl)sftqMn+xsUwWqCpftqkj5l<4

& !##

sUo\wZpfx\x\~+no\j5wu<jq)qkj>qk\s yJjf;Jt\sUo\lZy\j HUsUt\lBl)x<u5nJL\j<wB~9p-=sft\u5q)n9sUtφn4F&MsUo\lZ8s y\nJL\sUt\l6~9<*T

w)j5j5tq ~_p>yJ!LFt\n qkn+sUt oJn j<tj<lWq y\sUt\t\5j y\pft\l Y RZe a g Dr] ;qCpftq y\sUt\t\!o\jMt\sUo\l tJjMx\w)j5t\sUt\lxFpfl

φn C∞

φn(x) =1l)n |x| ≤ 1

2n,

φn(x) =0l)n |x| ≥ 2

2n

j5qφn

w)j5~9n_p?tqZ~+n9tJ|pfn+wGj<8j<tq0,

1l)o\wB~9j5lBn+tqkj5w H?p?~9~+j<l

[− 2n,− 1

n]j5q

[ 1n, 2n]4

a t sfoJqkwGjf; ~/.Ln9y\5j y\j ~9p 85q)\s y\j y\<u5w)n+q)j yFp?t\l Y RZe a g Dr]mj5lGqy\j y\sUtJt\j<wy\j5lpGsfwkpftqkly\j

Y4 'A~n+8xsUwGqkjy\sUt\uy\j*pfwkp?tq)n9w oJjz~_p y\n+l)u5w)5q)n9l)p?qkn+sUt y\j X '-4 h 4IK^&Hrp n9j5t =sfo\w)t\n+w-y\j<l

pfx\xJw)s123n+vprqkn+sUtJlxFpfwBpfo3T y\j<lGl)o\l54 sUo\wBu5j5qGqkj w)pfn9lGsUt#;JsUtH?py\n+l)u<wG5q)n9l)j5w X ' 4 h 4IK^yJj ~_p vp?t\n5w)jl)oJnJH?p?tqkj

Y Ntk

=Etk

(

Y Ntk+1

+ hf(tk, XNtk, Y N

tk+1, ZN

tk)

+ hφn(YNtk+1

− g(tk+1, XNtk+1

))[f(tk, XNtk, g(tk+1, X

Ntk+1

), V Ntk

)

+g(tk+1, X

Ntk+1

) − g(tk, XNtk

)

h]−)

,

hZNtk

=Etk(YNtk+1

∆Wk),

hV Ntk

=Etk(g(tk+1, SNtk+1

)∆Wk).

` .In+y\<j j<lWq o#.Lj<t oJqkn+~9n+l)pftqmo\t p?tFpf~+sU*Uo\j y\oqk\5sUw 5j y\j8u5sU8xFpfw)pfn9lGsUtxsUo\w ~+j<l acb>V3d y\n+lWTu<w5q)j<l<;UsUt xj<o3q = pfu5n9~9j58j<tq 8sUtqkw)j5w o#.Isft8pqksUo3qkj<l ~9j<l u Fp?t\u<j5l oJjZ~/.Ipfx\x\wGs123n9prqkn sUt-y\ju<jq)qkjy\n+l)u<wG5q)n9lkprqkn9sfty\sUt\t\j y\j5lMx\w)n 2l)o\x5w)n+j<o\wGl ,~ .Is lGqCp?u<~9jmj5qMyJsUt\u l)o\x5w)n+j<o\wM,

Y4

e6n T y\j<lGl)sUoJl<; tJsUo\lBx\w)5l)j<tqksUtJl ~9j5lBwG<l)oJ~+qCprqklBxsfo\wBy\n :5w)j<tqkj5l=Hrpf~9j5o\w)lBy\jn4

iZ[

j

Std Co

uvertu

re

105 15

1

0.5

0.51 5 10

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Pr

ix

105 15

10

5

1 2 10 100

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105 15

-0.6

-0.5

-0.4

-0.3

-0.55

-0.45

-0.35

1 2 10 100

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Std Co

uvertu

re

105 15

0.1

0.2

0.3

0.4

0.5

0.6

0.71 2 10 100

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1.% %- 13 <<"$K '6#!* <".#&8,(54*"$#O# <U" .$*N 134".%)*%$"$

!<".#&8 ".: ". ) 1F! ".#&%$'() *%J!<".#&8 ()4".#!O/#! .)"$.$*"$"$5"$.n*"$#&%ON

E"V%-". $#n*.%%-".< "-/

n = 100 S@ "$4* !"$#-/%$#;213#!M%-*".K#!*%<UV*%! <"$#&8?*"$[

~9jo\j<~~ .Epf~+*Usfw)n qk\jmu<sft-HUj5w)*Uj j5lGq n+tJ=<wGn9j5o\wZpfox\wGn 2zpf8<wGn9u<pfn+t#4

)= $K) !)=% #"#/;!" C*!)=

& !#(

& sfo\lzp HUsUt\ly\ ), s lGj<w HU~9j5lx\wGsUx\w)n+5q)<ly\j u<sUt HUj<wG*Uj<tJu<jy\j~9p 85qkJs y\j y\j x<tFpf~+n9l)p?qkn+sUt, x<t\pf~9n qkLJ23j X

n = 0.5, 1, 5, 10^4 tJj n9y\5j tFp?q)o\w)j5~9~9jj<lGqy#.Ep?x\x\~9n(o\j<w~/.Ipf~9*fsUwGn q)\8jp HUju|pflBy\j ~_p pfl)j. $ -%<3?: 8P4- sUo\wq)j<lWqkj<w u<j5qGqkj H?pfw)n9p?qkn+sUt l)n+ oJ~+qCp?t\5jf;1sUt-xsUlGj

n = n0(√

2)(j−1)αnxsUoJw

n0 = 0.5j5qMsftzqkj5lGqkj ~_p u<sUt HUj<wG*Uj<t\u5j(y\j ~/.Ipf~9*fsUwGn q)\8jmxsfo\wBy\n :5w)j5tqkj5l=Hrpf~9j5o\w)lByJjαn4

j

Prix

105 15

10

20

30

40

50

60

70

80

90 0.1 0.3 0.5 0.7 1 1.5 1.8 2 2.5 Prix de référence

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

αn !"

αn = 2"

2.5

" αn = 1.8

! "# #" $%" &'( ) * , + , -, .0/1. ""2& 34 " + " 5762." 7% 8 9%"

αn"! " " " : ;"<" : ; )

=> ? @%" αn = 1.8, 2, 2.5

j

Prix

105 15

12

13

14

15

16

17

18

0.1 0.3 0.5 0.7 1 1.5 1.8 2 2.5 Prix de référence

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 K(*

αn = 0.111*0*1l0J&*g(*+F*10&<>&)(&*,UL(&*g0=\

*0&*,J1*4<C=*080 &HX&gM=*= =JHQ%<C=\.MHJ17=&0.011*H MQ1*B&,g01*4<>=&*08<C,.J(^*j=(0.*,117 *.1<>&

.

HUj<wGl ~+j xJw)n 2pf8<wGn9u<pfn+t#4:`j<l#H?p?~9j5o\w)lαn ≥ 0.7

lGj< ~+j<tqmqkw)sUx =sUwGqkj5l>u|pfw>sUtl)jwGj5q)w)sUoMHUj8p?o3Ty\j5l)l)oJlcy\o x\wGn 2-pf8<wGn9u|p?n9t u<j o\n3j<lWq xj<o u5sU\5w)j<tqcpIHUj5u~9j<l w)<lGo\~+qkp?qkl prq)qkj5t\y\o\l y\j~9p 85qkJs y\jf4dBj<lGq)j<tq(~9j5l#H?p?~9j<oJw)l

αn = 0.3sUo

0.5 oJncy\sft\t\j5tq(~+j<l>w)<lGo\~+qkp?qkl>~9j<l>x\~+o\l u5sU\<wGj<tqkl<4`j<l w)T

l)oJ~+qCprqklBlGo\wB~_p u5sUoMHUj5wGqkoJw)j(u5sUtMLFwGj5tqMu5j<lMs lGj<w Hrp?qkn+sUt\l54 Q tzq)w)sUoMHUj y\sUtJu o\tJj \j<oJw)n9lWqkn( oJj y\jn ≈ h−1

2

xsUo\wpflGl)o\wGj<wo\t\ju<sUt HUj<wG*Uj<tJu<jy\jz~9p 85qkJs y\jy\jx<tFp?~9n9l)p?qkn+sUt j5t y\n+j5t\l)n+sUt14

b-.IpfoJq)w)j<lZqkj5lGqklBj<ty\n+j5t\l)n+sUtxJ~9o\lZ<~+j!HU5j lGj<w)pfn9j5tqBt\<u5j<lGlkpfn9wGj<lMxsUo\w0HUsUn+wl)nu5j5q)q)j \j<o\wGn9lWqkno\jy\5xj<t\yy\j ~_p y\n98j<t\lGn9sUt:4

& !##

bj ~_p 87<8j pft\n(<wGjo\j xsUo\wB~_p 85qkJs y\j yJj x 5tFpf~+n9l)p?qkn+sUt#;JsUtz= pfn q H?pfwGn9j5wB~9j u5s j 8u5n9j<tq y\jw)5*Uo\~9pfw)n+lkp?q)n+sUt

nj<t87<8jZq)j<8x\l oJj

N, δjqM

yFpftJl ~9jZu|pfl y\jB~_p p?l)j . $ -%<3?: 8P4 Q txJw)j<tJyqksfoGsUo\wGl

n = n0(√

2)(j−1)αnxsfo\w

n0 = 0.5j5qZsUtqkj5lGqkj>~9p-u5sUt-HUj5w)*Uj5t\u<j y\jm~/.Ipf~9*fsUwGn q)\8j xsUoJw

y\n :5w)j<tqkj5l=H?p?~9j5o\w)lBy\jαn4

j

Prix

105 15

14

15

16

17

13.5

14.5

15.5

16.5

17.5 0.1 0.5 1 2 Prix de référence

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

αn " + . # " ; = %

". "αn

;2: .") " 4 .! *( ") " = ". :!3 " "8) ." .) ; , =): ."; . "; 6 " "

n 5 =( " " " &" '(

&.; .) ; . 1

% .; . %; n

. $ ); " 2 " "!: " " ( = ) ." .) ; &-

Y =@

" #( ; " ; ) ." $n

: )" n

; )?." "3 2 = " " = @ /1. % %&." ! "

n ; ! " )"" =)5 " ) "

" 2 " / " =φn

" ) ." ) " % " ;

+

)=%: " ?)*!" 0 $ ')') $&%? " "#/1 >Q txj5oJqc<*pf~+j<8j<tq o3qkn9~+n+l)j5w ~/.Ipf~+*Usfw)n q)\8j xsUo\w y\5q)j<wGn+t\j<w oJt qkj<8x\l y#.Ep?w)w)7qcsUx3qkn9pf~ lGo\nJHrpftq~_p 85qkJs y\j y\j `sUt\*flGqCpT V3u 0pfwWq>X?4 d pfxJxj<~9sft\lB~+j>x\w)n+t\u<n+xj>y\j u<jq)qkj qk\s yJjQ tn9tJn+qkn9p?~+n l)j>p HUj<u

yN,MN (·) = g(tN , ·)jqMxsUo\wZq)sUoJq

msftxsUl)j

τm = tN4

Q t u<pf~9u5o\~9j xFpfwoJt\jw)<*fw)j<lGl)n+sUt ~9n+t\<pfn+w)j

yN,Mk (·) x\o\n9lxsUoJwq)sUoJq m ;lGn yN,Mk (XN,mtk

) ≤g(tk, X

N,mtk

)pf~+sUw)l6sUtzpfu5q)oFpf~+n9l)j~+jqkj5xJlZy#.Ipfw)wG75qZsUxJqkn+pf~xsUo\w~_p(qkw)pGj5u5q)sUn9wGj

mj5tzx sflkpftq

τm = tk4

tJj =sUn+lZxFp?w HUj5t o ,~ .In+t\lGqkpftq

t0;FsUtzu|pf~+u<o\~+j>~9j>xJw)n 2pf8<w)n+u|pfn+tu<sU8j

1

M

M∑

m=1

exp(−rτm)g(τm, XN,mτm ).

& sfo\lZxsUoMHUsUt\lBpfx\x\~+no\j5wu<jq pf~+*UsUwGn qk\jj5ty\qkj<wGn+tFpftqB, u \po\j n9t\lWqCpftqyN,Mk (·) xFpfwo\t\jw)5*Uw)j5l)lGn9sUtl)oJwM~9p pfl)j. $4csUn+u<n#~+j<lZw)<lGo\~+qkp?qkl

j

Prix

10512

13

14

15

16

17

18

19


P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

j

Couver

ture

105

-0.6

-0.5

-0.4

-0.55

-0.45

1 1.52 2.53

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1%& b?c0+*$M<>q +, ^7|BVMj*._&O*,=&0 ^.B,OM=&*,&).

5(&L0&*,=&,0..=9-*(=g1*00* + B

αMX?]&1,Q+*.1**.112*,9

αδ,&*,(J1*/<CJ)(&*+&

& 1*.0&0*j17*R'&(&*.* 1*\4B&<>10Q17*αM = 1

1.5

VN+ ?A+*,$.M<>(*+L0&(&*,gJ1*,$4Q*=\=&*,&F17*

αM,&L+*.XN

` .Ep?~9*UsUwGn qkJ8jvy\sft\t\jo\t\ju<sfoMHUj<wWqko\w)j X xsUo\w

αMp?l)l)j Xz*Uwkp?t\y ^(y\o 75jvsUw)yJw)j o\ju<j5~9~9j

s q)j<to\j xFp?wB~9j5lMp?oJqkwGj<lBqk\s y\j5l<4

i1D

$ % B

$ ' *!) " $ /;!) *B24/ *!) "#/ @')(* , 0*!) /;

Q tH?p q)wkpfn qkj5w~9jmu|p?lBy\o8s y5~9j= pfuqksUwGn9j5~:xsUo\wZ~ .I HUsU~9o3qkn9sfty\j<lZ=sUwW0pfwGy\lZl)o\wB5~9j5u5q)w)n9u5n+q)mxsUo\w8sUtq)w)j<w o\j5~9l8l)sftq~+j<lwG<l)oJ~+qCprqkl oJj~9pqk\s y\jzy\sft\t\jyFpftJl8u<ju|p?l<4 a t xFpfwWqkn9u5o\~+n9j5w<; sUty\5l)n9wGj Hrpf~9sfw)n9lGj<wo\t\j sUxJq)n9sUtu|pf~+~#j<o\wGsUx<j5t y\j p?q)o\w)n qk

T = 10GsUoJw)l<4 `j l)sUoJlWT()p?u<j<tq j<lWq o\t

u<sftqkw)p?qZ=sUwG06pfw)yzy\j y3@3t\pf8no\j

dF (t, T )

F (t, T )= σe−a(T−t)dWt.

`j<l Hrpf~9j5o\w)ly\j<lvx\pfwkpf 5q)w)j5ll)sUtqa = 0.19

;σ = 0.23

jq~+jqkpfo32 y#.Ln9tqk5w)75qr = 0.05

j<tH?p?~9j<oJwp?t\t oJj<~9~+jf4 ` p H?pf~9j5o\wMy\o lWqkw)n+Sfj(j5lGq

K = 100jq~9p Hrpf~9j5o\wMn9tJn+qkn9p?~+j>y\o lGsUo\lWT )pfu5j<tq H?p?oJq

F (0, T ) = 1004

pf~+sUw)n+l)j5w(~ .LsUxJqkn+sUt w)j H n9j5tq ,qkwGsUoMHUj<w-~9ju5sUo\x\~+jvy\jxJw)s u<j5l)lGo\l(Y, Z)

lkp?q)n9lG= p?n9lkp?tq(~/.L oFprqkn9sftw)qkwGsU*Uwkp?y\j

Yt = (F (T, T ) −K)+ +

∫ T

t

f(s, Ys, Zs)ds−∫ T

t

ZsdWs

sUf(t, y, z) = −(y − z

σea(T−t))r

4 Q t p?x\x\~+no\j~ .Ep?~9*Usfw)n qkJ8j8yJ<u5w)n+q-yFp?t\l-~_pxFpfwWqkn9j8'G' p HUj<u~9j5l pfl)j5l . $ j5q . $ -%<3?:48 jq t\sfo\l-w)j<xsfwGqksft\l-~9j5l-wG<lGo\~+qkp?qkl(s q)j<t oJl xsUo\w(o\t\jvu5sUoMHUj5wGqko\wGjGsUo\wGtFpf~+n5w)j

(N = 10)jqZo\tJjmqkpfn9~+~ j y:.I@3xj5w)u5o j<l

δ = 2x\o\n9l6xsUo\w6o\tJj>u5sUoMHUj5wGqkoJw)j D =sUn+l6x\pfw

GsUo\w(N = 20)

j5qMo\tJj qkpfn9~+~+jy#.I@ xj<w)u5o j<lδ = 1

4

M

Prix

41. 10 43. 10 45. 10

11

12

13

14

15

11.5

12.5

13.5

14.5

N=10,delta=2,HC(1,0)N=10,delta=2,HC N=20,delta=1,HC N=20,delta=1,HC(1,0)

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1

M

Std Pr

ix

41. 10 43. 10 45. 10

0.1

0.2

0.3

0.4

0.5

N=10,delta=2,HC(1,0)N=10,delta=2,HC N=20,delta=1,HC N=20,delta=1,HC(1,0)

P0

t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 1*,$4;>*=\=&*&C17*LC1XWX& 1*LYSW &%

14.72N *,(C*

j&*7JU_&g<>=&0*=0*,H?cB.00R%HC(1,0)

Jb?]=&&*,iGH17&J1U<>17* )12*

(N, δ) = (20, 1)'17*

(10, 2)P&&*;5(-0&U*=$

R'1*.0J, N

!"$#%$&('*)+,$.-!/0&-1243 65748-)+:9-

;=9".&.= :V4B&<>102=&*,& B46X?c0=&*\>KR"1*,.01*=&=&&F%J&=&**%9(& &C&F0&@N;&17&)8WF M+&%$.0.*<>0

Φ(S) = (ST −K1)+ − 2(ST −K2)

+ N QVM$4Q1*.<>/*&&

µ σ r R T S0 K1 K2< N <>= < N*) < N < < N < < N*) = ?<@< A= B<A=

C=&,+g1*BSi(r)

1*,$4> B M0&q=(=8(&0,.*,Ki

_.B4KX?c0=&*\rN

BT.LQ1*,=&0*Y0

1*L&<',0=80*L1*$4_R B M&W

'

BS1(R) − 2BS2(R) "!BS1(r) − 2BS2(r) $#BS1(r) − 2BS2(R) % '& BS1(R) − 2BS2(r) ( '#")

*+-,/.102.1354$*6,+-*7,/8:9<;"=5><?A@CB:@EDF;"?AGH4I=/4$.135;"*KJML' NOPQ&RTSVUXWY;"*7GZ@<+-,["9F=\.1]H@<=^6+:@C02@<3,/=\;".23G:=\@X_?A.2>F=\@F3O01.2`"*-@F3O3\;"*R,aB-@F3CbZ;"=\*:@F3E.2*-cd9<=5.1@F+:=\@F3OGZ;"+:=

Y0

4$01;"=53Q^e+-@f0V4gB:@F=\*:.2><[email protected]`"*:@acd;"+:=5*-.1,O+:*:@bZ;"=\*:@O35+:GZ9<=5.2@<+:=5@Ih*i=5@<GZ;"=\,/@jD<.k_lB:@F3\35;"+:3Q024m?A;on"@<*:*:@jB:@F3QG:=\.kp;$b-,/@<*e+:3QGH4$=0rq'4$01`";"=\.k,58:?A@C3\+:=

50024$D<@F?A@F*6,53@X,

@F*R,/=\@G:4$=5@<*6,/8->F35@<3s0rqt9FDF4$=\,M_l,nGZ@u@F?AG:.1=5.1^e+-@$WGZ;"+:=fB-.1vw9F=5@<*6,53A,neGZ@<3sB-@ucd;"*:D<,5.2;"*-3AB:@7bH4$3\@@X,B:.kvx9<=5@F*R,/@<3[I4I02@<+:=53B:@

M

N = 5 N = 20 N = 50

M D = [60, 140] D = [60, 200] D = [40, 200]

δ = 5 δ = 1 δ = 0.5

%y$z ( )R!-J)- "U ( % J) &"!"U ( '#"&HJd|: % !"U! %o & ( J)- %"% U ( % |YJ) % #RU ( t| z J)- !I|6U

)$| z & J)-')R!"U ( ')")HJ) ) ( U ( ') z J)- %o Uz-% & & % J)-') ( U '&"#HJ) ) U '&"&HJ)-') U

(R I# z &"):J)-') % U '&R!:J) ) % U '&"#HJ)-') % U

~ e % 9F3\+:01,/4I,/3;"b-,5@F*e+-34y["@<DE0V4jbH4$3\@Cif

bH4$3\@O b:4$35@Q bH4$35@Q bH4$3\@OmFHkM 16

=59<`".2;$*:3B:@Q;"=5;$*:;".64

=5@F`-:;"=<10

=5@<`::;$=F10

=5@<`:-;"=FN = 5 N = 20 N = 20 N = 20

%o$z ( I( J)- ( )RU |: !$)HJ % % U ( ') z J)- !"U ( $( J)- $( U! %o ( )"!:J)- %y( U ( ( #HJ)- % )RU '& % J)- %"% U ( ) ( J)-') z U

)$| z &I|YJ)-')"#RU ( ')R!:J)- )I|6U '&"):J)-')"#RU &R-J)-')$|6Uz-% & & J)-') ( U '&"#HJ)- ) U z #:J)-') ( U &R!-J)-') U

(R $# z &$)HJ)-') U '&$|YJ)- ) % U z #:J)-') U &R!-J)-') % U

~ E 9<35+:0k,4I,53;"b-,5@F*6+:34y[$@FDE01@F3b:4$35@<3@X,sFHkxo

+02@F3Q=\9F3\+:01,/4I,/3QB:@Ai@<,QfWYN,M

t0

35@F?jb:01@D<;"*6["@<=5`"@<=O["@<=532.95

@<,5,5@[T4$02@<+:=Q*:@GZ@F+-,GH4$3<,5=5@G-=59FB-.1,/@GH4I=+:*:@D<;"?jb:.2*H4I.23\;"*A02.2*-94$.1=\@B-@G:=5.pf0V4ID "_MeD 8:;$02@<3¡eBH4$*:3DF@<,@Xp@F?AG:02@IWe0V4*:;"*e_02.1*:9F4$=5.,/9QB:@

f4A+:*=59<@F0.1?fG:4$D<,35+:=

Y0

Zq'.1*6,/@<=5G:=\9<,/4I,/.1;"*]H*H4I*:DF.1>F=5@E@F3\,^6+:@C02@Q["@F*:B-@F+:=B:@a0rqt;"G-,5.2;"*7B:;$.1,4I01,/@<=5*H4T,/.k["@F?A@F*6,@<?fG:=\+:*6,5@F=@<,G:=\<,/@<=B:@a0rq'4$=\`"@F*R,GZ;"+:==\9FG:01.2^6+:@F=02@CGH4n";$vB:@O0rqt;"G-,5.2;$*

% &R!

lj 7 MP

N = 5 N = 20 N = 50 N = 50

M dy = 1Wdz = 0 dy = 2

Wdz = 1 dy = 4

Wdz = 2 dy = 9

Wdz = 9

%o$z z :J)- ( &RU ( ) % J)- |6U ( ') J)- " U ( t|R&HJ % !""U! %o zR J)- )RU &$|HJ)- %o U '&R-J)-')"&RU ( ') J) % U

)I| z z J)-')R"U & J)-')R"U '& J)-')$|6U '&R:J)-') ( Uz-% & z J)-')R!"U & J)-')$|6U '& J)-') U '&"#HJ)-') % U

(" $# z $&HJ)-') ( U & % J)-') U '& % J)-') % U '&R!:J)-') % U

~ (C 9F35+-01,4T,/3;$b-,/@<*e+:34["@FDO0V4jbH4$35@gA

*gD<;"?AGH4$=/4I*6,02@F3B:.kvx9<=5@F*R,/3D8:;".pfB:@cd;"*:DX,/.2;$*:3B:@bH4$35@IW;"*sGZ@F+,=\@F?f4$=5^6+:@<=^6+:@024CD<;"02;$*:*:@N = 5

B:@C Jd4$b-02@F4$+ UG:=\9F3\@F*6,5@B:@<3=59<35+:0k,4I,5335.1?f.1024I.1=\@F34y["@<Dj+:* *:;$?b-=5@C9F^6+:.k[I4$01@F*6,B:@cd;"*:DX,/.1;"*:3OB:@AbH4$3\@f^6+:@f024D<;"02;$*:*:@

N = 5B:@f Jd4$b:02@F4$+ e % UX4$*:3E02@4$b:01@4I+ W

02@<3B:@F+epmB:@F=\*:.2><=5@F3D<;"02;$*:*:@F3?A;"*6,5=5@<*6,^6+qt+-,5.202.135@<=+:*:@b:4$35@B:@GZ;"01ne*$?f@<302;eDF4$+pAGZ@F+,4$?A9X_02.1;"=\@F=C35.2`$*:.1]:D4I,5.1["@<?f@<*6,j024 G-=59FD<.23\.2;"*x ;$+:34y[e.2;$*:3jB:9 ;"b:35@<=\["9D<@F0V4 B:4$*:30V43\;"+:3M_l35@<D<,/.1;"*|: GZ;"+:=02@<3bH4$3\@F3i @<,sFHky .1*H4$01@F?A@<*6,W6[e+02@4Ib:02@F4$+u ( W-024abH4$3\@OB-@QGZ;"0kn*"?f@<3`"01;"bH4$+epGH4$=\[e.1@F*R,a9<`R4$01@F?A@F*R,74T,5,/@<.2*:B-=5@A024g[I4$01@F+:=C4I,\,/@<*:B:+:@IW02;"=\35^6+:@A*:;"+:3a4$+-`"?A@F*6,5;"*:3E02@*:;"?jb-=5@fB:@mGZ;$01ne*"?A@F3EBH4$*:3a024ubH4$35@I\;"+,/;"*:3C^6+:@sGZ;$+:=C0V4uB:@<=5*:.1>F=5@fDF;"01;"*:*:@AB:+ 4$b:01@4$+ ( W02@<3gb:4$35@<3 0q'.1*:3\,/4$*6,

tk*:@i3\;"*6,G:4$3 J4["@FD

d = q = 1Up0,k = 1, x, · · · , xdy

Jd=5@F3\Gp1,k = 1, · · · , xdz

Ug?f4$.131,

x−minm SN,mtk

maxm SN,mtk

−minm SN,mtk

, · · · , ( x−minm SN,mtk

maxm SN,mtk

−minm SN,mtk

)dyJ@X,70V4 ?f<?f@

D8:;"35@jGZ;"+:=p1,k

UX@X,5,/@j*:;"=\?m4I02.1354I,/.k;"*7@F3M,Q*:9FD<@F353/4I.2=5@j02;"=\35^6+:@a02@a*-;"?jb:=5@aB-@jGZ;"01ne*"?A@<3B:@X_[e.2@<*6,`$=/4$*:BxWHGZ;$+:=9<[e.k,/@F=Bq 4[";".1=B:@F3?m4I,5=5.1DF@F3B:@O=59<`"=5@<3535.1;"*u?m4I0D<;"*:B:.k,/.1;"*:*:9<@F3F

! "#$ &%('*)+

h* DF;$*:35.1B:>F=\@,5;"+,\;"+:=\3d = q = 1

@X,A02@g?A;eB:><02@sB:@u0V4$D"_MD8:;$02@F3GZ;"+-=S

4iDF;"*:B-.1,/.1;"*,/@<=5?A.2*H4I01@B:@A0q Q [I4$+-,uG:=\9F3\@F*6,

Φ(S) = ( 1T

∫ T

0Stdt − K)+

W 35;$.1,O02@DF4$3EBqt+:*:@A;"G-,/.1;"*4$3\.V4I,5.2^6+:@$W-@<,*-;"+:3G:=5@<*:;"*:301@F3GH4$=54$?f>X,/=\@F33\+:.1[T4$*6,53E¡

µ σ r T S0 K)-')"# )- )- % % % )") % )");"+:=4$G:G:=\;eD 8-@F=DF@X,5,5@ED<;"*:B:.k,/.2;$*s,/@<=5?A.2*H4$01@GH4I,58_B:@<GZ@F*:B-@F*6,FWYDF;"?A?A@Q3\+:`"`"9<=59OBH4$*-3024jGH4$=\,5.2@WH*:;"+:3G:=\@F*:;"*-3

d′ = 2@<,35.2?j+:01;"*:3

PNtk

= (SNtk ,1

k+1

∑ki=0 S

Nti

)∗Jd[";".1=ELte) % SVUXY@<3=\9F3\+:01,/4I,/3

G:=\9F35@<*6,59F3QBH4$*:302@E,4Ib:02@F4$+it|s35;$*6,DF;"8-9F=5@<*6,53QD4I=02@CG:=\.kp7B:;$*:*:9jG:4$=0q 4$01`";$=5.,/8:?@E*qt@F3\,QGH4I3,/=\>F39F01;".2`$*:9B:+uG:=5.pgB:@O=59Xcd9F=5@<*:DF@

7.04B:;"*:*-9EBH4I*:3EL'e) % S

;"?A?A@?A@F*R,/.1;"*:*:9BH4$*:3Lt) % SrWy0rqt+-,5.201.k3/4I,5.1;"*QB:@ 1N+1

∑Ni=0 S

Nti

GZ;"+:=4$G:G-=5;eD8:@F= 1T

∫ T

0Stdt

@<3\,02;$.2*CBqt<,5=5@;"G-,5.2?f4$0ThQ*jGZ@F+, ["9F=\.1]H@<= D<@^6+:.35@GH4$3\35@3\.-;$*D8H4$*-`"@

PNGZ;"+:=?A.2@<+pa4$G:G:=\;eD 8-@F=

1T

∫ T

0Stdt

;$?f?A@ G:=5;"GZ;$359 BH4$*-3 L'e) % SW *:;"+:3 4$G-G:=5;eD8:;"*:3 1T

∫ T

0Stdt

G:4$=

% &"#

N = 5 N = 20 N = 50

M δ = 5 δ = 1 δ = 0.5

D = [60, 200]2 D = [60, 200]2 D = [60, 200]2

%y$z #- ("( J)'| % U |:t|6:J ( z "U ( t| z J %y( ') z U! %o #-'#R!:J) % U #- $z J)I#RU !'# ( J ( "U

)$| z #- z )HJ) )"&"U #- $#HJ) |6U #-t| z J)-'|"&RUz-% & #- z"( J) )$|RU #-'&R!:J) )$#RU #- z #HJ)- %y U

(R I# z #- z"( J) ) U #-'& z J) ) ( U #-'&"&HJ)- )I|6U

~ '| 9<35+:0k,4I,/3GZ;"+:=0rqt;"G-,5.2;"*g4$35.24I,/.1ê+-@Q@F*u+-,/.102.1354$*R,0V4jbH4$3\@Cim

1N

∑N−1i=0 SNti (1 + µh

2+ σ

2∆Wti)

WFDF@^6+:."DF;"*:B-+:.1,PNtk

=(SNtk ,

1k

∑k−1i=0 S

Nti

(1 + µh2

+ σ2∆Wti)

)∗GZ;"+:=

k ≥ 16 @F3=59<35+:0k,4I,53Jd[";$.2= ,4$b:01@4I+m !"U 35;"*R,b:.1@F*A?f@<.20101@<+:=534["@FDD<@D 8:;$.kpGZ;"+:=

PN @

G:01+:3FWy*:;"+:3;"b:3\@F=M[";"*:30V4DF;"8:9<=5@<*:DF@B-@0q 4I02`";$=5.,/8-?A@^6+:.6G:=5@<*:Ba@<*a.1*:G:+-,B:@<335.1?j+:0V4T,/.k;"*:3 B:@SN35;$+:3024CG:=\;"bH4$b-.202.,/98-.23\,5;"=5.1^6+:@jJ

µ 6= rU@X,DF;"=\=5.2`$@02@B-=5.1c ,GZ;"+-=B:;"*:*-@F=02@G:=\.kpf=\.23\ê+-@X_*-@F+-,5=5@$

z (" %o$z ! %o )$| z z-% & (R $# z=5.p # )-'&") |:t|R& #-'# z # % ! #- z"z # &"& e ) DF4$=\,M_l,nGZ@ |- ) z z ) %"% |:'#$| % %$% )- % ) )R ) ) ~ e! 9F35+-01,4T,/3GZ;$+:=0rqt;"G-,5.2;$*4$35.24I,/.1^6+:@Q@F*u+-,/.102.k3/4$*R,+:*:@O?f@<.20101@<+:=5@Q4$G:G-=5;yp.1?m4T,/.1;"*sB:@1T

∫ T

0Stdt

@<,024b:4$35@E JN = 20

Wδ = 1

WD = [60, 200]2

U

! $*)+!) $ '*)+ )*) * )+#

4G:02+-GH4$=\,B:@<3@Xp@<?fG-02@F3^6+:@*:;"+:34y[";"*:3,5=/4$.k,/9<3\+:3\ê+xq G:=59<35@F*R, 35;"*R,@Xvx@<D<,/+-9F3@F*aB:.1?f@<*:35.1;"*1

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∏3i=1 S

it)

13 )+

@F3[I4$02@<+:=53B-@F3GH4$=54$?A><,/=\@F33\;"*R,

µ = 0.05Wr = 0.05

WT = 1

WY02@<3DF;$?fGZ;"354$*R,/@F3B:@S0

[I4I02@<*6,

100@X,K = 100

H4?m4T,/=5.1DF@σ

[I4$+,σ =

0.4 0 0

0 0.4 0

0 0 0.4

H @QG:=5.pgB-@O=59Xcd9F=\@F*:D<@EB:@QDF@X,5,/@

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αδ = 1@X,

αM = 4.5 * @<vw@<,0247G:=\;"GZ;"35.k,/.1;"* lj % .1?AGw;$35@

MB-@m[I4$=\.2@<=aD<;"?A?f@

h−(1+β+αδd′)

4["@FD

% &R

lj 7 MP

β > 1DF.D<@F0V4 B:;$*:*:@i^6+:@

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h−5;"+:34[";"*:3

9F`"4$02@<?f@<*6,D<;"*:3\,/4I,/9^6+:@D<@35@F+-.209<,/4$.1,,5=5;"GjGZ@<3535.1?f.13\,5@GH4$==54$G:GZ;"=M,4$+epj=\9F35+-01,4T,/3*e+-?f9<=5.2^6+:@<3;"b:3\@F=M["9F3Feqt@F3M,024=/4I.235;$*jGw;$+:= 0V4I^e+-@F0201@*:;"+:3 G:=5@<*:;"*:3

αM = 4.5";"+:3 ;"b-,5@F*:;"*-302@<3 =59<35+:0k,4I,53

35+-.1[I4I*6,/3O¡

j

Prix

1 2 3 4 5 6 7

9

10

11

9.5

10.5

Méthode de pénalisationPrix théorique

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

3t 1 !"$#%&')(+*,&- .0/1 ')233!

4'5*&'6(52,!'"7&8j = 7

!9:/1*!; '"<9=!9*9&')(5'>?,!'" &'6(2,'@,AB CD*,@E:/1F'"'"@3,!2;*@*@&8,!- "' G'"IHJK'"':L;K&')(*,&- 0/1 ')235MEN"'"*I/O P'"K!!,.HRQ/1!S:/1ATUA@*V *'3'P 'W+')X"3&= !Y33Z&= 3YZ*'3'

1![& !3Y!"

. .'V .=\*U/1,AB .(∏3

i=1 Si)

13N='"C9!9," .<'). C9!Y]'"'" C%*I/1D&- .9!9= !

^`_a!^b_c!dfe)gUhij*'3'1 k!5A5!"@. 'V ="@*I/l,!B G5#

j

Prix

105

9

10

11

9.5

10.5

11.5HC HC(1,0) Prix théorique

P 0 t 0 t 1 t 2 t 3 t 4 t 5PN,

1t 1 PN,

1t 1 PN,

2t 1 PN,

2t 1 PN,

3t 1 PN,

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s!vstqy!tsv9r.n"s|tvsjws!vw!svpqNpxvs!trs~sqopts~s!vty6"u!ts!zs!qwv|8!ty"s|tvWn|Yty)

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k

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zN,R,Mk (PNtk

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φ : Ω × [0, T ] → RqJD<;"*:3\.2B:9<=59<3

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H2T,β(R

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−dYt = f(t, Yt, Zt)dt− ZtdWt

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T × H2T (Rq)

e J G Xlk]^ E J HaJf.nm % Upo

.

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T (R) R H ξ ∈ L2T

o P E Xl^P C"!

YT = ξ,

−dYt = [φt + Ytδt + Ztγt]dt− ZtdWt

V\J?K'RJ?K Gfe JgR ^ E P J HfG E K (Y, Z) T`VYK ^ H2T,β(R) × H2

T,β(Rq) R H Y R]^ H T E KgK'kiN VYX3P V$# E XZhJ?PSR

Yt = E(ξΓtT +

∫ T

t

Γtsφsds|Ft

)

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dΓts = Γts[δsds+ γ∗sdWs],

Γtt = 1.

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0

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∫ t

0

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|Φ(x1) − Φ(x2)| ≤ C sup0≤t≤T

|x1(t) − x2(t)|

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sup0≤t≤T

|b(t, 0)| + |σ(t, 0)| + |f(t, 0, 0, 0)| + |Φ(0)| ≤ K

E&%0 R]^ H P V$# E K'L H G E K L E K ^ H VYK H Rw=VYP VYK H 0 ^jJ?X [0, T ]

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i=1

E

∫ ti

ti−1

|Zt − Zti−1|2 + |Zt − Zti |2dt ≤ C(1 + |X0|2)|π|

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ξ ∈ L2W

&VYV Us?DC"FcAC1+OU]VYACf : Ω× [0, T ]×R×Rq → R

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R, z, z′ ∈ Rq, p.s.W

&VYVWV2Us?"C1A"bDB(UnP2+ZGWF(St)0≤t≤T

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E(sup0≤t≤T (S+t )2) <∞ mZhCcB?1!"!ABFEÊ?"F

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E(sup0≤t≤T |Yt|2) <∞ 9&d[eVVUYt = ξ +

∫ T

tf(s, Ys, Zs)ds+KT −Kt −

∫ T

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W0 ≤ t ≤ T 9&d[eVYV U

Yt ≥ StW0 ≤ t ≤ T 9&d[eVYVWVVU

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Yt = φ(XT ) +

∫ T

t


t

ZsdWs +KT −Kt,

Yt ≥ g(t,Xt), t ≥ 0,

∫ T

0

(Yt − g(t,Xt))dKt = 0,

Py[TF+φ(·) ≥ g(T, ·) FOU

Y N,0t = φ(XN

T ) +

∫ T

t

f(s,XNs , Y

N,0s , ZN,0

s )ds−∫ T

t

ZN,0s dWs +KN,0

T −KN,0t ,

Y N,0t ≥ g(t,XN

t ), t ≥ 0,

∫ T

0

(Y N,0t − g(t,XN

t ))dKN,0t = 0.

&f.6mRU

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d > 1U¡

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ocb 9dO58<e@ A

g ∈ C1,2([0, T ] × Rd) VTYR]L H E J H R]^ ^ R]^nT!kjX G Yk]R]^ J!K G # E XZ\kjZ\RjK H jE XK'k]R]^ o @?>W:8@ A nVYK ^$ 3 la P ET ^ H V?LjPSRhR]^ H K E=H k h(·, ·) oE J`^hVT E K ^NQXlk# kjX]knZ E T G )ikPSR]^K E=H V H G E K ^*L VYXT`VYK îK E=H X]R*L V=^]a G P=N E J!XXV GIH nVT E=G XiJ?K'R%L E K #J`^ G E KVTYR]LiPSRbN V=^T?R H RjZ%Ng^ h = T

N

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o P Rc G ^ H R3J?K'R L E K ^ H VYK H R3J!K G YRjX]^ RjPIPSR C > 0H RjPIP R e J;R

E sup0≤t≤T

|Yt|2 +

∫ T

0

|Zt|2dt+K2T ≤ C(1 + |X0|2)

E sup0≤t≤T

|Y N,0t |2 +

∫ T

0

|ZN,0t |2dt+ |KN,0

T |2 ≤ C(1 + |X0|2).

?o

E

sup0≤t≤T

[|Y N,0t − Yt|2 + |KN,0

t −Kt|2] +

∫ T

0

|ZN,0t − Zt|2dt

≤ C(1 + |X0|2)√h.

^Tx

.nm ^!m J>DK$K

o p K K E=H RUN E J?X t ∈ [tk, tk+1[

kt =|∂tg(t,Xt)| + |d∑

i=1

∂xig(t,Xt)bi(t,Xt)| +

1

2|

d∑

i,j=1

(σσ∗)i,j(t,Xt)∂2xi,xj

g(t,Xt)|

+ |f(t,Xt, Yt, Zt)|,

kN,0t =|∂tg(t,XNt )| + |

d∑

i=1

∂xig(t,XN

t )bi(tk, XNtk

)| + 1

2|

d∑

i,j=1

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)∂2xi,xj

g(t,XNt )|

+ |f(t,XNt , Y

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t )|. P E Xl^]a VYJ ^ RjK ^nT?R]^3ZR]^J!XlR]^nVYPSk V HE=G XlR]^]a E K V

0 ≤ dKt ≤ ktdt, 0 ≤ dKN,0t ≤ kN,0t dt,∀t ∈ [0, T ], p.s.

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g(t,Xt) = g(0, X0) +

∫ t

0

Usds+

∫ t

0

VsdWs

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∫ T

0

(|Ut| + |Vt|2)dt <∞ p.s.

P E X]^Zt = Vt, dP × dt p.p. ^jJ?XP RjK ^ RjZ P R Yt = g(t,Xt)

&f.6mt|6U

R H

0 ≤ dKt ≤ 1Yt=g(t,Xt)[f(t,Xt, g(t,Xt), Vt) + Ut]−dt.

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e J G R]^ H J?K'R ^ RjZ G Z VYX H G Ker`VYPSR NQP J^ r?kjK'kjX VYPSR

g(t,Xt) = g(0, X0) +

∫ t

0

Usds+

∫ t

0

VsdWs + At

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. >

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t a E&% A+t R H A−

t ^ E K H T?R]^bNQX E L]R]^]^jJ`^ G K H k rYXV PSR]^ o P E X]^]a (Ut)t R H (Vt)t ^ E K H T?R]^NQX E L]R]^]^jJ`^ Ft NQX E rYXlR]^]^ G YRjZ\RjK H Z\R]^jJ?X V PSR]^XlR]^ NgR]L H G YRjZ\RjK H s=VYPSRjJ?X]^nT`VYK ^ R R H Rd ^V H G ^#pV G ^VYK H

∫ T

0

|Ut| + |Vt|2dt+ A+T + A−

T <∞ p.s.

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dKt +1

2dLt =1Yt=g(t,Xt)[f(t,Xt, g(t,Xt), Vt) + Ut]

−dt+ dA−t .

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dKt = αt1Yt=g(t,Xt)([f(t,Xt, g(t,Xt), Vt) + Ut]−dt+ dA−

t ).

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tN!dP[N

Y N,1tN

= φ(XNtN

)m

-4?"VWB!A?"Nt ∈ [tk, tk+1[

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¡

Y N,1t = Y N,1

tk+1+

∫ tk+1

t

f(s,XNs , Y

N,1s , ZN,1

s )ds−∫ tk+1

t

ZN,1s dWs.

hC("`X]dC"VYU FZC"B(?"VYUF.!A?"N UA?DUt("PTC"B

[tk, tk+1[WY N,1t = Y N,1

t ∨ g(t,XNt )m

hCcVYUXZNF (?DB5Ê?q qGQP/(dP[U]Ft0 = 0

m G FZB(UcP+OVWGYF=("FC[AVWNÊ?"F

Y N,1tk

= uN,1k (XNtk

)!AT?"N?DC"F/+ZFON(UnP[VWC"FCcAC"+U]VYAC1RSFOB?"N]PIb"GWF

uN,1k

m 6 FGWFOR RSF !m ^/(dPTC"B0#xA<oS?("ATC"C"F.("FZB +ZAC\U]N TGWFZB B?DN

(uN,1k )kWY N,1

FUZN,1

m @!>>@BA p K ^jJON`N E ^ R3PSR]^m ]N E=H mg[]^ R]^ o a o R Ho B^ E K H YkjX G )ik]R]^ o f R]^ # E K'L HfG E K ^ uN,1k ^ E K H4f G Ng^ Lnm GIHQo J?K G # E XjZ\kjZ\RjK H RjK N R H k o P Rc G ^ H RJ?K'R L E K ^ H VYK H R C a G KQT?k NgRjKQT`VYK H RT?R N a H RjP PSR e JgR ∀k VYKQT ∀t ∈ [tk, tk+1[

E

suptk≤t≤tk+1

[|Y N,1t |2 + |ZN,1

t |2]

≤ C(1 + |X0|2)

E|Y N,1t − Y N,1

t |2≤ C(1 + |X0|2)|tk+1 − t|.

6 F Ua"`ZATNXZRSF m:("PTC"B #xA<oSbANC"F GrqFZN(NFZ?DN FOCEUNF(Y N,0, ZN,0, KN,0)

FU(Y N,1, ZN,1, KN,1)

¡465879;:=<?>@BA p K ^jJ NN E ^ R e J;RPSR]^m ]N E H mg[]^ R]^ o a o R Ho B^ E K H XV G R]^ o P E X]^

E

sup0≤t≤T

[|Y N,1t − Y N,0

t |2 + |KN,1t −KN,0

t |2]+

∫ T

0

|ZN,1t − ZN,0

t |2dt

≤ C(1 + |X0|4)h.

^Tx z

.6m m . 6 *> 6 .>"> P >

4C-]dCWGWFJU]aD`ZAN(XZRSF m(dP[C"B8#xA<oS;!FONRSFOU:(xqI`X[[PTGY?"FZN$GQPvN`<`?"GQP[NVYU`L2

("FZm iFwU]a"`OANXOR F

B?1!"!ABFE^E?"Fcb 9dO58<e@ A

b a σ R H f ^ E K H J?K G # E XZ\kjZ\RjK Hhkj P T?kjX G RjKgK'R]^Tl G KQT G L]R 12RjK H RjZ%Ng^ o

465879;:=<?>@BA p K ^jJ N`N E ^ R e JgRhPSR]^km N E=H m;[]^ R]^ o a o a no )R Hio ^ E K H XV G R]^ o P E Xl^G PbRc G ^ H R3J!K'R L E K ^ H VYK H R C > 0H RjPIPSR e J;R

N−1∑

k=0

sups∈[tk,tk+1]

E

∫ tk+1

tk

|Zt − Zs|2dt ≤ C(1 + |X0|4)√h.

U #$%(')+

C+OFZN(U]PTVWC:C"ATRbDNF(qVWC"`<`\PTGYVYU`ZBBATCEU-+ZATC"B(U]PTR RSFOCEU ?DU]VYGWVYB(`ZFOBm 7 7 8@39 l

∀ γ > 0,∀ (a, b) ∈ R2, (a+ b)2 ≤ (1 + γh)a2 + (1 +1

γh)b2.

&/.6m <"U @/@!>>@ 8@ :=9/l 1:O@\

@!>>@BA CFE=G RjK H T > 0 R H tk = kh N E J!X 0 ≤ k ≤ N = ThJ?K'R T G ^ LjXlk H G ^V HfG E K T?R

[0, T ] T?R_N V=^ h oE J!X HE J H R]^ ^jJ GIH R]^_N E ^ GIHfG YR]^ (ak)0≤k≤N , (bk)0≤k≤N R H (ck)0≤k≤NH RjPIPSR]^ e JgR

ak−1 + ck−1 ≤ (1 + γh)ak + bk−1 t.+ GIH m γ > 0 u a E KdV

ak +N−1∑

i=k

ci ≤ eγ(T−tk)[aN +

N−1∑

i=k

bi].

&/.6mRU

U )+# $%('*)

AT?"B GYVWB U]AC"BwV+OV ?"C +OFZN UnPTVYC C"ATRbDNF("F (D`<]dC"VMU]VYAC"BHFOU N(`ZB?DGYUnP&U]BkF p1!ATB`ZB/(dP[C"B #ENON x\^yS mhC !FO?DUk`F`PTGWFOR FOCEU.+ZATC"B?"GMU]FONHB?DN0+ZFSB? (FOU # - AG z |ISW # 6 % LFSW # ! % IS A? # ?1( % % S m AT?"BNFOR FON +OVWACDB nmx*Pn\P[U]VYB!A?DNwGWPsU]NP("?"+U]VYACicNPTC#"ZPTVWB(F>("FZBJUFZNRSFZBJGWVY`ZBJP[?p C"ARjb"N(FZB("F +OA?_[FZN U]?"N(FYm

%$ & 2('*) 0W #67* $ 2 (,+ 0?!( 0W - 7/.l /9 A CFE=G RjK H ε > 0 a G J!K'R LjP V=^]^ R T?R # E K'L HfG E K ^ Rd → R a 1 ≤ p < ∞ a R H ν J?K'RZ\R]^jJ?X]RnT?RiNQX ET V G P GIH k_^jJ?X Rd

oE J?XJ?K'R # E K'L HfG E K g : Rd → R a E KdT?k)K GIH

‖g‖Lp(ν) := ∫

|g(z)|pdν 1p .

^Tx %

. >

oQE J H R #pVYZ G PIPSRi)K G R T?R# E K'L HfG E K ^ g1, · · · , gN : Rd → RH RjPIPSR e J;R%N E J!X HE J H R # E K'L HfG E K

g ∈ G a G P Rc G ^ H R j = j(g) ∈ 1, · · · , N H RjP e JgR‖g − gj‖Lp(ν) < ε

R]^ H VjN`NgRjPSk]R J!K'R ε L E JeYRjX H J?XlR6T?R G N VYX XVpN`N E X H ‖ · ‖Lp(ν)

o?o p K K E=H R N (ε,G, ‖ · ‖Lp(ν)) P V H V G PIPSR6T?RP VNQP J`^UNgR H GIH R ε L E JeYRjX H J?XlR6T?R G N VYX3XVjN`N E X H

‖ · ‖Lp(ν)

op K NQXlRjKQT N (ε,G, ‖ · ‖Lp(ν)) = ∞ ^d G P K Rc G ^ H R N V=^\T?R ε L E JeYRjX H J!XlR o4 P E Xl^N (ε,G, ‖ · ‖Lp(ν)) R]^ H VjN`NgRjPSk3K E Z XlRnT?R ε L E J@YRjX H J!XlR6T?R G N VYX3XVjN`N E X H ‖ · ‖Lp(ν)

o o3CFE=GIH zM1 = (z1, · · · , zM ) M N E=G K H ^ ) c!R]^ T?R Rd

obCFE=GIHνM P VZ\R]^jJ!XlRhRjZ%N G X Gfe JgRhL E XXlR]^

N E KQT`VYK H R]a Gfo R o

νn(A) =1

M

M∑

m=1

1A(zm), (A ⊂ Rd).

P E Xl^

‖g‖Lp(νM ) = 1

M

M∑

m=1

|g(zm)|p 1p

R H HE J H R ε L E JeYRjX H J?X]R T?R G N VYXX VjN`N E X H ‖ · ‖Lp(νM ) ^ RjXV VjN`NgRjPSkJ?K'R Lp ε L E JeYRjX H J?XlRT?R G ^jJ?X zM1 R H P RK E Z XlRnT?R ε L E JeYRjX H J?XlR6T?R G NWVYX XVjN`N E X H ‖ · ‖Lp(νM ) ^ RjXV\K E=H k

Np(ε,G, zM1 ).

K Tl VYJ H XlR]^ H RjXjZ\R]^]a Np(ε,G, zM1 ) R]^ H P RjK H G RjX N Z G K G Z VYP H RjP e J G P&Rc G ^ H R*T?R]^# E K'L H G E K ^g1, · · · , gN : Rd → R VTYRLhP V NQX E NQX G k H k e J;R N E J?X H E J H R g ∈ G a G P Rc G ^ H R j = j(g) ∈1, · · · , N H RjP e JgR

1

M

M∑

m=1

|g(zm) − gj(zm)|p 1p < ε.

%$ .- 2U $' #" !87f,#$ ! ' #" ' 7% ' #"*! 2"

- 7/.l /9 A CFE=GIH A J!K'R LjP V ^l^ RnT?R ^ E J`^ RjK ^ RjZ PSR]^3T?R Rd R H ^ E=GIH M ∈ No

o E J?X z1, · · · , zM ∈ Rd a E K T?k)K GIH

s(A, z1, · · · , zM) =∣∣A ∩ z1, · · · , zM : A ∈ A

∣∣,

Ld R]^ H TT G XlR e JgR s(A, z1, · · · , zM) R]^ H PSR K E Z X]R T?Rd^ E J`^ RjK ^ RjZ PSR]^ T G ^ H G K'L H ^MT?Rz1, · · · , zM RjK G K H RjXl^ R]L H VYK H VTYR]LnT?R]^ kjPSkjZ\RjK H ^nT?R A o

^@Lx

.nm'| m K,wK$ >D> h h _ -&.K . K P

?o3CFE=GIHG J?K ^ E J`^ RjK ^ RjZ PSRMT?R Rd T?R L VYXT G KQVYP M ocp K T GIH e JgR A #XV rYZ\RjK H R G ^ G

s(A, G) = 2M a Gfo R o ^ G H E J H ^ E J`^ RjK ^ RjZ PSR\T?R G NgRjJ H jH X]R X]R NQXlk]^ RjK H kh^ E J^ P V # E XZ\RA ∩G N E J!X3J?KkjPSkjZ\RjK H A ∈ A o

oJf R [jZ\R L E R L G RjK H Tl k]LjP V H RjZ\RjK H S(A,M) T?R A R]^ H T?k)K G L E Z Z\RS(A,M) = max

z1,··· ,zM⊂Rds(A, z1, · · · , zM).

*G K ^ G aUPSR [jZ\R\L E R L G RjK H Tl k]LjP V H RjZ\RjK H R]^ H PSR K E Z XlR Z V c G Z VYP T?R6^ E J`^ RjK ^ RjZ PSR]^T G kjXlRjK H ^3Tl J?KdRjK ^ RjZ PSR T?R M N E=G K H ^ e J G NgRjJeYRjK H jH X]R ETH RjKgJ`^ N VYX G K H RjXl^ R]L H G E KBVTYR]LPSR]^ kjPSkjZ\RjK H ^nT?R A o

- 7/.l /9 A CFE=GIH A J!K'R LjP V=^]^ R T?Rn^ E J`^ RjK ^ RjZ PSR]^6T?R Rd VTYR]L A 6= ∅o f V T G Z\RjK ^ G E K t E JBT G Z\RjK ^ G E K T?RWVjNQK G mgRjX E K'RjK =G ^ u VA T?R A R]^ H T?k)K G R N VYX

VA = supM ∈ N : S(A,M) = 2M,Gfo R o VA R]^ H PSR NQP J`^wrYXVYKQTBRjK HfG RjX M H RjP e J G P Rc G ^ H R6J?K RjK ^ RjZ PSR T?R M N E=G K H ^ e J G NgRjJ H jH XlRk]LjP V H kUN VYX A o

- 7/.l /9 A CFE=GIH G J!K&R LjP V=^]^ R T?R # E K'L H G E K ^T?R Rd NQX]RjKQVYK H ^ R]^s=VYPSRjJ!Xl^ T`VYK ^ RoHp K

K E=H R G+ PSR_^ E J^ RjK ^ RjZ PSR3T?R Rd+1 T?k)K G N VYXG+ =

(z, t) ∈ Rd × R; t ≤ g(z), g ∈ G

^ E=GIH P RjK ^ RjZ PSRT?R]^_^ E J`^rYX VjNdmgR]^T?R4# E K'L H G E K ^T?R G o

465879;:=<?>@BA CFE=GIH G J!K'R LjP V=^]^ R T?R # E K'L HfG E K ^ g : Rd → [0, B] VTYR]L 2 ≤ VG+ < ∞ a ^ E=GIHp ≥ 1 a ^ E=GIH ν J?K'RZ\R]^jJ?XlRnT!RiNQX ET V G P GIH k]^%^jJ?X Rd aiR H ^ E=GIH 0 < ε @BA CFE=GIH G J!K R]^ N V?L]RqYR]L HE X G RjP T?R\T G Z\RjK ^ G E K r T?R$# E K'L HfG E K ^\T?R Rd → R R H^ E=GIH

A =z : g(z) ≥ 0 : g ∈ G

.

P E X]^VA ≤ r.

^@LL

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7% n2"*$Q! 2" 7* 0 , 0 #!? 2">@?"!1!ABACDBKê?DF GrqAC A"b"BFON\[F

MN` P[GWVWBP[U]VYATC"B VYC"("`!FOC"(dP[CEU]FOB("? +ZA?1!"GWF

(X,Y )C"ATU`ZFOB

(Xm, Ym)1≤m≤MmhC C"ATUF

FM?"C FZB !dP+ZF [F+U]AN(VWFOG=(DF cATC"+OUVWAC"BuÊ?"V0!FZ?DU3("`!FOC"("NF("F

(X1 · · · , XM)mZhC(D`<]dC"VMU

mM

WdGWPjcATC"+OUVWAC ("FHN(`F`TNFZB(BVWATCcFZB(UVWRS`ZFIW +OARSR Fj¡

mM(·) = arg infψ∈FM

1

M

M∑

m=1

|ψ(Xm) − Ym|2.hCcVWC\U]N(A ("?"VYUGQP C"ATU]P[U]VYAC

‖ψ‖2M =

1

M

M∑

m=1

|ψ(Xm)|2..GYANB<W"AC PvGQP U]aD`ZAN(XZRSFwB?"Vk[[PTC\U8&Ua"`ZATNXZRSFqLTL1mYL.!"P$`FqL z | (dPTC"B0# x\ôSVU¡465879;:=<?>@BA C J N`N E ^ E K ^ e JgR

σ2 = supx∈Rd

V ar(Y |X = x) <∞.

CFE=GIHKM P V T G Z\RjK ^ G E K T?R FM a VYP E Xl^

E

(

‖mM −m‖2M

∣∣∣∣X1, . . . , XM

)

≤ σ2KM

M+ min

ψ∈FM

‖ψ −m‖2M .

C;G T?RUNQP J`^]a FM R]^ H G KQT?k NgRjKQT`VYK H T?R]^3Xlk VYP G ^V H G E K ^ (Xm, Ym)1≤m≤M a E KdVE

(

‖mM −m‖2M

)

≤ σ2KM

M+ min

ψ∈FM

E|ψ(X) −m(X)|2.6 FJU]a"ÒANXORSFqLL1m_^/!dP$`FvL z /(dPTCDB=# x\^yS C"AT?"B-(DAC"C"F465879;:=<?>@BA CFE=GIH G J!K'R LjP V=^]^ R T?Ri# E K'L H G E K ^ g : Rd → R

jE XK'k]R*N VYX B R H ^ E=GIH ε > 0o

P E X]^P(∃g ∈ G : ‖g‖ − 2‖g‖M > ε

)≤ 3EN2(

√2

24ε,G, X2M

1 ) exp(− Mε2

288B2)

E&%X2M

1 = (X1, · · · , XM , XM+1, · · · , X2M ) VTYR]L (XM+m)1≤m≤M J?K'RUL E N G R GfoSGfo T_T?R (Xm)1≤m≤Mo

%$ $ 1"*, +!,<7* g2U k7%"465879;:=<?>@BA e CFE=G RjK H X1, · · · , XM T?R]^*=VYX G V PSR]^ VYPSk V HE=G XlR]^Xlk]RjPIPSR]^ G KQT?k NgRjKQT`VYK H R]^ oCFE=G RjK Ha1, b1, · · · , aM , bM T?R]^ Xlk]RjP ^ R H E K ^jJON`N E ^ R e JgR Xm ∈ [am, bm] VTYR]L NQX ET V G P GIH k 1 N E J?X1 ≤ m ≤M

o P E X]^]a ∀ε > 0

P

(∣∣∣∣

1

M

M∑

m=1

(Xm − E(Xm))

∣∣∣∣> ε

)

≤ 2 exp

(

− 2Mε2

1M

∑Mm=1 |bm − am|2

)

^@L ^

.nm <!m . . 6 >D . K 6;6 #) ) )

. !"GY?"BVYFZ?"N(B NF!"NVYBFOBFW ATC3("AVMU:!"N(`+OVWBFONÊ?"FOGWGWFHB(AGW?DUVWACcATC3+,aDAVWB(VYU-("? !DNA"b"GYXZRSF0("FkRSAVWC"(DNFZB+ P[NNÒBE¡

infα

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(Xm)1≤m≤MFU

("F + PTN ("VWCdP[GK

&AC6B(?"!"!AB(FK ≤ M

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M= Id

A10BFZB(U$GQP R P[U]N(V+ZF5(DFwU]PTVWGYGYF

M ×K("AC\U$GWFOB GYV2`CDFZB B(AC\U

p(Xm)∗m

G Cxqtn P !dPTB8(DF !DNA"b"GYXZRSF*("PTC"B GWF*+ZPTB A10 GQPcRsP&U]NV+ZFB∗B

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p(·) (qAN(Ua"AC"ATNR PTGWVMBFONlGWFOB +ZAGYAC"C"FOB'("FiGWP$RsP[UNV+ZFBm

- PTN=+ZATCEU]N(F$W (dP[C"B GWF*+ZPTB A10 VWG FXp@VYB(U]F ("FOB/+OAGWVYC"` P[NVMU]ÒBHFZC\U]NFsGYFZB=+ZAGYAC"C"FOB8("FB

W VYG +ZACR[@VYFZC\U("F +,aDAVWB(VWN ?"C"F:R PTC"VYXZN(F ("FNÒBA?"(DNF:GWF !"N(A"b"GYXZRSF ("FR ATVWC"("N(FZB/+ZPTNN(`ZB &V mFYm'("F +,aDAVWB(VWN ?"C"FBATGW?DUVWAC !dPTN(UV+O?"GWVYXONFyUê?DVBAVMU@+OAR!dP[U]V1b"GYFPy[F+wGWFcPTVMU ("FJB(?"!"!AB(FZN B∗B

M= Id

BPTC"B@!FZN U]F("F`ÒC"`ZNPTGWVMU]` m $C"F.!ABBV1b"VYGWVMU`FZB U GWPvB(?"V1[[P[CEU]F mhC FXv F+OU]?DF ?DC"F (D`+OAR*!ATBVYUVWAC6FZC [[PTGYFZ?"N(BSBVYC:`?"GYVWXONFZB& >6aW[AVYN#Q 6 % TS U/("F B√

M

Ê?"VBFq`+ONVMU B√

M= UΣV ∗ P[F+

U?DC"FR P[U]N(V+ZF$AN U]a"A"`TACdPTGYF/&

U ∗U = IdU (DFUnP[VWGYGYF

M ×MWΣ?"CDF

R P[U]N(V+OF(DFHGWPjcATNRSF

Σ =

σ1 0 · · · 0

0 σ2 · · · 0mmm0

m m m0

0 . . . 0 σK0 . . . . . . 0mmm

. . . . . .

mmm0 . . . . . . 0

A10σ1 ≥ σ2 ≥ · · · σK ≥ 0

BATCEU GWFOB[[P[GWFO?"NB B(VWC:`?DGWVYXZNFOB ("F B√M

FUV?"C"FwR P[UNV+ZFAN(Ua"A"ÀTCdPTGYF

("FsUnPTVYGWGMFK × K

m4>@V U]A?@U]FZB GWFOBa[[PTGYFZ?"N(B BVWC-`?"GWVYXZN(FZB B(ACEUvB(UNV+OU]FOR FOCEU/!ABVMU]Vk[FZBvPTGYANBkAC6BFNFU]N(A?-[F (dPTC"B GYF+ZP("NF (q?"C"F BATGW?DUVWAC8?"C"V2Ê?"F mi>@V FZGYGWFZBqBAC\UsC"ATC Ce?DGWGYFZB (?"B\Ê?q ?"C NPTC:`r < K

!D?"VWBσr+1 = · · · = σK = 0

WPTGYANBsAC +na"AVWB(VYU ("F1NFZR!"GWP+ZFONB

!dPTN[u1, · · · , ur]GWFOB

r!"NFORSVWXZN(FZB +OAGWATC"C"FOB(DF

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Bm44C A?DUNFIW4VYGiFZB U PTGWATNBacP+OVWGYF ("Fs[`ZN(V1]dFONÊ?"F:GQP1B(AGY?DU]VYAC

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^@L

. >

FOUCÊ?"F GrqAC !A?-[[P[VYUEb"VWFOC B(?"!"!AB(FZNaÊ?"F GWFOB0+ZATGWAC"CDFZB0("F B√M

B(ACEUkAN(Ua"AC"ATNR PTGYFZB<W Ê?"VYU(U]F NFOR*!DGQP+OFZN

K!dP[N

r < Km

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F

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Lnm V=^ H G L T G RjXlRjK H G VYP*R e J V H G E K ^ t VYX G ^]a= nu aiT GIH R T o P6VYX E J GVYKQT f o V o P G V a E P J!Z\R E # GIH Z VYK ! R]^ o E=H R]^ V H m o C RjX o a N Vr?R]^ A o f8E KerYZ V1K!a h VYXP E + L % % ?m#tPTN]x^ m m"P[NGWA 3m?. (DV1v ?"BVYACkR A (DFZGANlFZGYF+OUNV+ZVYU.!DNV+OFZBpm dV H mgRjZ V HfG LlVYP G KQVYK'L]R L ^s¡ ^ !\#"D^ % !$d^TxTx\^!m

#t wx\^ myPTGYG%& m_Um P 'dPTGWGYFZN(AF%m dFZN(CdPTC"("F& PTC1( m 4G wPTNA?DV m[K$F,dF+OU]F(a >)(B*-)(IBP[C"(,+[PTN(VQP[UVWATCdPTG VWCDF*-E?dPTGYVYU]VMFZBpm $! ( ! R N E X H A ^[xx\^!m#t xDL. - miNVWPTC"(*m FZG%ATC P[C"(0/ m `OR VYCFm ATC"B5TFON1.U !F Ua"FZAN(FZR2ANf > 4Bm PSR]L H X E K o E Z Z o X ET V o3 g¡YL4"L65d^TxxDLYm

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