Approximation of solutions of SDEs driven by a fractional Brownian motion, under pathwise uniqueness

0.13 MB·

by  Oussama El Barrimi

ModernStochastics:TheoryandApplications3(2016)303–313
DOI:10.15559/16-VMSTA69
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n
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J Approximation of solutions of SDEs driven by

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a fractional Brownian motion, under pathwise

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R uniqueness
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t OussamaElBarrimia,1,∗,YoussefOukninea,b
a
m
aCadiAyyadUniversity,FacultyofSciencesSemlalia,
[ Av.MyAbdellah,2390,Marrakesh,Morocco

1 bHassanIIAcademyofSciencesandTechnologyRabat,Morocco
v
4 [email protected](O.ElBarrimi),[email protected](Y.Ouknine)
4
2 Received:29July2016,Revised:12December2016,Accepted:13December2016,
1 Publishedonline:20December2016
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. Abstract Ouraiminthispaperistoestablishsomestrongstabilitypropertiesofasolution
1
of a stochastic differential equation driven by a fractional Brownian motion for which the
0
pathwiseuniquenessholds.TheresultsareobtainedusingSkorokhod’sselectiontheorem.
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Keywords FractionalBrownianmotion,Stochasticdifferentialequations
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v 2010MSC 60G15,60G22
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X
r
a
1 Introduction
Considera fractionalBrownianmotion(fBm), a self-similar Gaussian processwith
stationary increments. It was introduced by Kolmogorov [5] and studied by Man-
delbrotand Van Ness [6]. The fBm with Hurst parameter H ∈ (0,1) is a centered
Gaussianprocesswithcovariancefunction
1
R (t,s)=E BHBH = t2H +s2H −|t−s|2H .
H t s 2
(cid:0) (cid:1) (cid:0) (cid:1)
∗Correspondingauthor.
1Thisauthorissupported bytheCNRST“Centre National pourlaRecherche Scientifique etTech-
nique”,grantNo.I003/034,Rabat,Morocco.
©2016TheAuthor(s).PublishedbyVTeX.OpenaccessarticleundertheCCBYlicense.
www.i-journals.org/vmsta
304 O.ElBarrimi,Y.Ouknine
If H = 1/2, then the process B1/2 is a standard Brownian motion. When H 6= 1,
2
BH isneitherasemimartingalenoraMarkovprocess,sothatmanyofthetechniques
employedinstochasticanalysisarenotavailableforanfBm.Theself-similarityand
stationarityofincrementsmakethefBmanappropriatemodelformanyapplications
indiversefieldsfrombiologytofinance.Wereferto[7]fordetailsonthesenotions.
Considerthefollowingstochasticdifferentialequation(SDE)
dX =b(t,X )dt+dBH,
t t t (1)
(X0 =x∈Rd,
where b : [0,T]×Rd → Rd is a measurablefunction,andBH is a d-dimensional
fBm with HurstparameterH < 1/2whose componentsare one-dimensionalinde-
pendentfBmsdefinedonaprobabilityspace(Ω,F,{F } ,P),wherethefiltra-
t t∈[0,T]
tion{F } isgeneratedbyBH,t∈[0,T],augmentedbytheP-nullsets.Ithas
t t∈[0,T] t
beenprovedin[2]thatifbsatisfiestheassumption
b∈L1,∞ :=L∞ [0,T];L1 Rd ∩L∞ Rd , (2)
∞
forH < 1 ,thenEq.(1)hasa(cid:0)uniquestr(cid:0)ong(cid:1)solutio(cid:0)n,w(cid:1)h(cid:1)ichwillbeassumed
2(3d−1)
throughoutthispaper.
NoticethatifthedriftcoefficientisLipschitzcontinuous,thenEq.(1)hasaunique
strongsolution,whichis continuouswithrespectto theinitialcondition.Moreover,
thesolutioncanbeconstructedusingvariousnumericalschemes.
Our purpose in this paper is to establish some stability results under the path-
wiseuniquenessofsolutionsandunderweakregularityconditionsonthedriftcoef-
ficientb.Wementionthataconsiderableresultinthisdirectionhasbeenestablished
in[1]whenanfBmisreplacedbyastandardBrownianmotion.
The paper is organized as follows. In Section 2, we introduce some properties,
notation,definitions,andpreliminaryresults.Section3isdevotedtothestudyofthe
variationof solutionwith respectto the initial data. In the last section, we dropthe
continuityassumptiononthedriftandtrytoobtainthesameresultasinSection3.
2 Preliminaries
Inthissection,wegivesomepropertiesofanfBm,definitions,andsometoolsused
intheproofs.
ForanyH <1/2,letusdefinethesquare-integrablekernel
KH(t,s)=cH t H−12 − H − 1 s12−H t(u−s)H−21uH−32 du , t>s,
s 2
"(cid:18) (cid:19) (cid:18) (cid:19) Zs #
wherec =[ 2H ]1/2,t>s.
H (1−2H)β(1−2H,H+1))
2
Notethat
H−1
∂KH(t,s)=cH H − 1 t 2(t−s)H−32.
∂t 2 s
(cid:18) (cid:19)(cid:18) (cid:19)
ApproximationofsolutionsofSDEsdrivenbyfBm,underpathwiseuniqueness 305
Let BH = {BH, t ∈ [0,T]} be an fBm defined on (Ω,F,{F } ,P). We
t t t∈[0,T]
denotebyζ thesetofstepfunctionson[0,T].LetHbetheHilbertspacedefinedas
theclosureofζ withrespecttothescalarproduct
h1 ,1 i =R (t,s).
[0,t] [0,s] H H
Themapping1 →BH canbeextendedtoanisometrybetweenHandtheGaus-
[0,t] t
sian subspace of L2(Ω) associated with BH, and such an isometry is denoted by
ϕ→BH(ϕ).
NowweintroducethelinearoperatorK∗ fromζ toL2([0,T])definedby
H
b ∂K
K∗ϕ (s)=K (b,s)ϕ(s)+ ϕ(t)−ϕ(s) H(t,s)dt.
H H ∂t
Zs
(cid:0) (cid:1) (cid:0) (cid:1)
TheoperatorK∗ isanisometrybetweenζ andL2([0,T]),whichcanbeextendedto
H
theHilbertspaceH.
DefinetheprocessW ={W ,t∈[0,T]}by
t
W =BH K∗ −11 .
t H [0,t]
ThenW isaBrownianmotion;moreov(cid:0)e(cid:0)r,BH(cid:1)hasthe(cid:1)integralrepresentation
t
BH = K (t,s)dW(s).
t H
Z0
WeneedalsotodefineanisomorphismK fromL2([0,T])ontoIH+12(L2)associ-
H 0+
atedwiththekernelK (t,s)intermsofthefractionalintegralsasfollows:
H
(KHϕ)(s)=I02+Hs21−HI012+−HsH−21ϕ, ϕ∈L2 [0,T] .
Note that, for ϕ ∈ L2([0,T]), Iα is the left fractionalRiema(cid:0)nn-Lio(cid:1)uville integral
0+
operatoroforderαdefinedby
1 x
Iα ϕ(x)= (x−y)α−1ϕ(y)dy,
0+ Γ(α)
Z0
whereΓ isthegammafunction(see[3]fordetails).
TheinverseofK isgivenby
H
KH−1ϕ (s)=s21−HD012+−HsH−21D02+Hϕ(s), ϕ∈I0H++12 L2 ,
where for(cid:0)ϕ ∈ I(cid:1)H+21(L2), Dα is the left-sided Riemann Liouvil(cid:0)le d(cid:1)erivative of
0+ 0+
orderαdefinedby
1 d x ϕ(y)
Dα ϕ(x)= dy.
0+ Γ(1−α)dx (x−y)α
Z0
Ifϕisabsolutelycontinuous(see[8]),then
K−1ϕ (s)=sH−21I12−Hs21−Hϕ′(s). (3)
H 0+
(cid:0) (cid:1)
306 O.ElBarrimi,Y.Ouknine
Definition2.1. Onagivenprobabilityspace(Ω,F,P),aprocessXiscalledastrong
solutionto(1)if
(1) Xis{F } adapted,where{F } isthefiltrationgeneratedbyBH,t∈
t t∈[0,T] t t∈[0,T] t
[0,T];
(2) X satisfies(1).
Definition2.2. Asextuple(Ω,F,{F } ,P,X,BH)iscalleda weaksolution
t t∈[0,T]
to(1)if
(1) (Ω,F,P) is a probabilityspace equipped with the filtration {F } that
t t∈[0,T]
satisfiestheusualconditions;
(2) X isan{F } -adaptedprocess,andBH isan{F } -fBm;
t t∈[0,T] t t∈[0,T]
(3) X andBH satisfy(1).
Definition 2.3 (Pathwise uniqueness). We say that pathwise uniqueness holds for
Eq.(1)ifwhenever(X,BH)and(X,BH)aretwoweaksolutionsofEq.(1)defined
on the same probability space (Ω,F,(F ) ,P), then X and X are indistin-
t t∈[0,T]
guishable. e
Themaintoolusedinthe proofsisSkorokhod’sselectiontheoreme givenbythe
followinglemma.
Lemma 2.4. ([4], p. 9) Let (S,ρ) be a complete separable metric space, and let
P, P , n = 1,2,..., be probabilitymeasureson (S,B(S)) suchthatP converges
n n
weakly to P asn → ∞. Then,ona probabilityspace (Ω,F,P), we canconstruct
S-valuedrandomvariablesX,X ,n=1,2,...,suchthat:
n
e e e
(i) P = PXn,n = 1,2,...,andP = PX,wherePXn andPX arerespectively
n
thelawsofX andX;
n
e e e e
(ii) X convergestoX P-a.s.
n
We will also make use of the following result, which gives a criterion for the
e
tightnessofsequencesoflawsassociatedwithcontinuousprocesses.
Lemma 2.5. ([4], p. 18) Let {Xn, t ∈ [0,T]}, n = 1,2,..., be a sequence of d-
t
dimensionalcontinuousprocessessatisfyingthefollowingtwoconditions:
(i) ThereexistpositiveconstantsM andγ suchthatE[|Xn(0)|γ] ≤ M forevery
n=1,2,...;
(ii) thereexistpositiveconstantsα,β,M ,k =1,2,...,suchthat,foreveryn≥1
k
andallt,s∈[0,k],k =1,2,...,
E Xn−Xn α ≤M |t−s|1+β.
t s k
(cid:2)(cid:12) (cid:12) (cid:3)
Then, there exist a subsequenc(cid:12)e(n ), a p(cid:12)robability space (Ω,F,P), and d-dimen-
k
sionalcontinuousprocessesX,Xnk,k =1,2,...,definedonΩsuchthat
e e e
(1) ThelawsofXnk andXnk coincide;
e e e
(2) Xnk convergestoX uniformlyoneveryfinitetimeintervalP-a.s.
t e t
e e e
ApproximationofsolutionsofSDEsdrivenbyfBm,underpathwiseuniqueness 307
3 Variationofsolutionswithrespecttoinitialconditions
The purpose of this section is to ensure the continuousdependence of the solution
withrespecttotheinitialconditionwhenthedriftbiscontinuousandbounded.Note
that,inthecase ofordinarydifferentialequation,thecontinuityofthecoefficientis
sufficienttoensurethisdependence.
Next,wegiveatheoremthatwillbeessentialinestablishingthedesiredresult.
Theorem 3.1. Let b be a continuous bounded function. Then, under the pathwise
uniquenessforSDE(1),wehave
2
lim E sup X (x)−X (x ) =0.
t t 0
x→x0 0≤t≤T
h (cid:12) (cid:12) i
(cid:12) (cid:12)
BeforeweproceedtotheproofofTheorem3.1,westatethefollowingtechnical
lemma.
Lemma 3.2. Let Xn be the solution of (1) corresponding to the initial condition
x . Then, for every p > 1 , there exists a positive constant C such that, for all
n 2H p
s,t∈[0,T],
E Xn−Xn 2p ≤C |t−s|2pH.
t s p
Proof. Fixs<tin[0,T].(cid:2)W(cid:12) ehave (cid:12) (cid:3)
(cid:12) (cid:12)
2p
t
Xn−Xn 2p ≤C b u,Xn du + BH −BH 2p .
t s p u t s
Duetothes(cid:12)(cid:12)tationarity(cid:12)(cid:12)oftheinc"r(cid:12)(cid:12)(cid:12)(cid:12)eZmsen(cid:0)tsandt(cid:1)hesc(cid:12)(cid:12)(cid:12)(cid:12)aling(cid:12)(cid:12)propertyof(cid:12)(cid:12)an#fBmandthe
(cid:12) (cid:12)
boundednessofb,wegetthat
E Xn−Xn 2p ≤C |t−s|2p+|t−s|2pH
t s p
(cid:12) (cid:12) ≤Cp|(cid:2)t−s|2pH, (cid:3)
(cid:12) (cid:12)
whichfinishestheproof.
LetusnowturntotheproofofTheorem3.1.
Proof. Suppose that the result of the theorem is false. Then there exist a constant
δ >0andasequencex convergingtox suchthat
n 0
2
infE sup X (x )−X (x ) ≥δ.
t n t 0
n 0≤t≤T
h (cid:12) (cid:12) i
LetXn(respectively,X)bethesol(cid:12)utionof(1)corresp(cid:12)ondingtotheinitialcondition
x (respectively,x ).AccordingtoLemma3.2,thesequence(Xn,X,BH)satisfies
n 0
conditions(i)and(ii)ofLemma2.5.Then,bySkorokhod’sselectiontheoremthere
existasubsequence{n ,k ≥ 1},aprobabilityspace(Ω,F,P),andstochasticpro-
k
cesses(X,Y,BH),(Xk,Yk,BH,k),k ≥1,definedon(Ω,F,P)suchthat:
e e e
(α) foreeaechek ≥1,fthelfawseof(Xk,Yk,BH,k)and(Xenk,eX,eBH)coincide;
f f e
308 O.ElBarrimi,Y.Ouknine
(β) (Xk,Yk,BH,k) convergesto (X,Y,BH) uniformly on every finite time in-
tervalP-a.s.
e e e e e e
Thankstoproperty(α),wehave,fork≥1andt>0,
e
2
t
E Xk−x − b s,Xk ds−BH,k =0.
t k s t
(cid:12)(cid:12) Z0 (cid:12)(cid:12)
(cid:12)(cid:12)e (cid:0) e (cid:1) e (cid:12)(cid:12)
Inotherwords,Xk sa(cid:12)tisfiesthefollowingSDE: (cid:12)
t
t
e Xk =x + b s,Xk ds+BH,k.
t k s t
Z0
(cid:0) (cid:1)
Similarly, e e e
t
Yk =x + b s,Yk ds+BH,k.
t 0 s t
Z0
(cid:0) (cid:1)
Using(β),wededucetheat e e
t t
lim b s,Xk ds= b(s,X )ds
s s
k→∞Z0 Z0
(cid:0) (cid:1)
and e e
t t
lim b s,Yk ds= b(s,Y )ds
s s
k→∞Z0 Z0
(cid:0) (cid:1)
inprobabilityanduniformlyint∈[0e,T]. e
Thus,theprocessesX andY satisfythesameSDEon(Ω,F,P)withthesame
driving noise BH and the initial condition x . Then, by pathwise uniqueness, we
t 0
concludethatX =Y foerallt∈e[0,T],P-a.s. e e e
t t
Ontheotheerhand,byuniformintegrabilitywehavethat
e e e
2
δ ≤ liminfE max X (x )−X (x )
t n t 0
n 0≤t≤T
= liminfEh max (cid:12)(cid:12)Xk−Yk 2 (cid:12)(cid:12) i
t t
k 0≤t≤T
h (cid:12) (cid:12) i
≤ E maxe|X −Y(cid:12)e|2 , e (cid:12)
t t
0≤t≤T
h i
whichisacontradiction.Theenthedesieredreesultfollows.
4 Thecaseofdiscontinuousdriftcoefficient
In this section, we drop the continuity assumption on the drift coefficient and only
assumethatbisbounded.Thegoalofthissectionistogeneratethesameresultasin
Theorem3.1withoutthecontinuityassumption.
Next, in order to use the fractional Girsanov theorem given in [8,Thm.2], we
shouldfirstcheckthattheconditionsimposedinthelatteraresatisfiedinourcontext.
Thiswillbedoneinthefollowinglemma.
ApproximationofsolutionsofSDEsdrivenbyfBm,underpathwiseuniqueness 309
Lemma4.1. SupposethatXisasolutionofSDE(1),andletbbeaboundedfunction.
Thentheprocessv =K−1( ·b(r,X )dr)enjoysthefollowingproperties:
H 0 r
(1) v ∈L2([0,T]), P-a.Rs.;
s
(2) E[exp{1 T |v |2ds}]<∞.
2 0 s
Proof. (1)InligRhtof(3),wecanwrite
|vs|= sH−21I012+−Hs12−H b(s,Xs)
1 s
=(cid:12)(cid:12)Γ(1 −H)sH−21 (cid:12)(cid:12) (s−r)(cid:12)(cid:12)−(cid:12)(cid:12)12−Hr12−H b(r,Xr) dr
2 Z0
≤ kbk∞Γ(11−H)sH−12 s(s−r)−21−H(cid:12)(cid:12)r12−Hdr(cid:12)(cid:12)
2 Z0
Γ(3 −H)
= kbk∞ 2 s12−H
Γ(2−2H)
Γ(3 −H)
≤ kbk∞ 2 T12−H,
Γ(2−2H)
wherek·k denotesthenorminL∞([0,T];L∞(Rd)).
∞
Asaresult,wegetthat
T
|v |2ds<∞, P-a.s.
s
Z0
(2)Theseconditemisobtainedeasilybythefollowingestimate:
1 T 1
E exp |v |2ds ≤exp C T2(1−H)kbk2 ,
2 s 2 H ∞
" ( Z0 )# (cid:26) (cid:27)
whereC = Γ(32−H)2,whichfinishestheproof.
H Γ(2−2H)2
Next, we will establish the following Krylov-type inequality that will play an
essentialroleinthesequel.
Lemma4.2. SupposethatX isasolutionofSDE(1).Then,thereexistsβ >1+dH
suchthat,foranymeasurablenonnegativefunctiong :[0,T]×Rd 7→Rd,wehave
+
1/β
T T
E g(t,X )dt≤M gβ(t,x)dxdt , (4)
t
Z0 Z0 ZRd !
whereM isaconstantdependingonlyonT,d,β,andH.
Proof. LetW bead-dimensionalBrownianmotionsuchthat
t
BH = K (t,s)dW .
t H s
Z0
310 O.ElBarrimi,Y.Ouknine
FortheprocessvintroducedinLemma4.1,letusdefineP by
dP =exp − T v dW − 1 T v2dt b:=Z−1.
dP t t 2 t T
( Z0 Z0 )
b
Then,inlightofLemma4.1togetherwiththefractionalGirsanovtheorem[8,Thm.2],
wecanconcludethatP isaprobabilitymeasureunderwhichtheprocessX−xisan
fBm.
Now,applyingHöblder’sinequality,wehave
T T
E g(t,X )dt=E Z g(t,X )dt
t T t
Z0 ( Z0 )
1/ρ
b T
≤C E Zα 1/α E gρ(t,X )dt , (5)
T t
( Z0 )
(cid:8) (cid:2) (cid:3)(cid:9)
b b
where1/α+1/ρ=1,andC isapositiveconstantdependingonlyonT,α,andρ.
From[2,Lemma4.3]wecanseethatE[Zα]satisfiesthefollowingproperty:
T
E ZTα ≤CH,d,bT kbk∞ <∞, (6)
whereC isacontinuou(cid:2)sinc(cid:3)reasingfun(cid:0)ctiond(cid:1)ependingonlyonH,d,andT.
H,d,T b
Ontheotherhand,applyingagainHölder’sinequalitywith1/γ+1/γ′ = 1and
γ >dH +1,weobtain
T T
E gρ(t,X )dt= gρ(t,y) 2πt2H −d/2exp−ky−xk2/2t2H dydt
t
Z0 Z0 ZRd (cid:0) (cid:1) 1/γ′
b ≤ T 2πt2H −dγ′/2exp−γ′ky−xk2/2t2H dydt
Z0 ZRd !
(cid:0) (cid:1) 1/γ
T
× gργ(t,y)dydt . (7)
Z0 ZRd !
Adirectcalculationgives
2πt2H −dγ′/2exp−γ′ky−xk2/2t2H dy =(2π)d/2−dγ′/2 γ′ −d/2t(1−γ′)dH.
Rd
Z
(cid:0) (cid:1) (cid:0) (cid:1)
Pluggingthisinto(7),weget
1/γ′
T T
E gρ(t,X )dt≤ (2π)d/2−dγ′/2 γ′ −d/2t(1−γ′)dHdt
t
Z0 Z0 !
(cid:0) (cid:1) 1/γ
b T
× gργ(t,y)dydt
Z0 ZRd !
1/γ′
≤ (2π)d/2−dγ′/2 γ′ −d/2 1/γ′ T t(1−γ′)dHdt
Z0 !
(cid:0) (cid:0) (cid:1) (cid:1)
ApproximationofsolutionsofSDEsdrivenbyfBm,underpathwiseuniqueness 311
1/γ
T
× gργ(t,y)dydt
Z0 ZRd !
1/γ
T
≤C γ′,T,d,H gργ(t,y)dydt .
Z0 ZRd !
(cid:0) (cid:1)
Finally,combiningthiswith(5)and(6),wegetestimate(4)withβ =ργ.Theproof
isnowcomplete.
Nowweareabletostatethemainresultofthissection.
Theorem4.3. IfthepathwiseuniquenessholdsforEq.(1),thenwithoutthecontinu-
ityassumptiononthedriftcoefficient,theconclusionofTheorem3.1remainsvalid.
Proof. TheproofissimilartothatofTheorem3.1.Theonlydifficultyistoshowthat
t t
lim b s,Xk ds= b(s,X )ds
s s
k→∞Z0 Z0
(cid:0) (cid:1)
inprobability.Inotherwords,forǫ>e0,wewillshowtheat
t
limsupP b s,Xk −b(s,X ) ds >ǫ =0. (8)
s s
k→∞ "(cid:12)(cid:12)Z0 (cid:12)(cid:12) #
Letusfirstdefine (cid:12)(cid:12) (cid:0) (cid:0) e (cid:1) e (cid:1) (cid:12)(cid:12)
(cid:12) (cid:12)
bδ(t,x)=δ−dφ(x/δ)∗b(t,x),
where∗denotestheconvolutiononRd,andφisaninfinitelydifferentiablefunction
withsupportintheunitballsuchthat φ(x)dx=1.
ApplyingChebyshev’sinequality,weobtain
R
t
P b s,Xk −b(s,X ) ds >ǫ
s s
"(cid:12)(cid:12)Z0 (cid:12)(cid:12) #
≤(cid:12)(cid:12)(cid:12) 1(cid:0)E(cid:0) et b(cid:1)s,Xk −e b(cid:1)(s,X(cid:12)(cid:12)(cid:12) ) 2ds
ǫ2 s s
"Z0 #
(cid:12) (cid:0) (cid:1) (cid:12)
≤ 4 E (cid:12)t b se,Xk −bδ es,X(cid:12)k 2ds
ǫ2 s s
( "Z0 #
(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
t (cid:12) e e (cid:12)
+E bδ s,Xk −bδ(s,X ) 2ds
s s
"Z0 #
(cid:12) (cid:0) (cid:1) (cid:12)
t(cid:12) e e (cid:12)
+E bδ(s,X )−b(s,X ) 2ds
s s
"Z0 #)
(cid:12) (cid:12)
4 (cid:12) e e (cid:12)
= (J +J +J ).
ǫ2 1 2 3
Fromthecontinuityofbδ inxandfromtheconvergenceofXk toX uniformlyon
s s
everyfinitetimeintervalP a.s.itfollowsthatJ convergesto0ask →∞forevery
2
δ >0. e e
e
312 O.ElBarrimi,Y.Ouknine
On the otherhand,let θ : Rd → R be a smoothtruncationfunctionsuchthat
+
θ(z)=1intheunitballandθ(z)=0for|z|>1.
ByapplyingLemma4.2weobtain
t
J =E θ Xk/R bδ s,Xk −b s,Xk 2ds
1 s s s
Z0
t(cid:0) (cid:1)(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
+E 1e−θ X(cid:12) k/R ebδ s,Xk e−b(cid:12)s,Xk 2ds
s s s
Z0
(cid:0) (cid:0) (cid:1)(cid:1)(cid:12)t (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
≤N bδ−b +e2CE (cid:12) 1−eθ Xk/R des, (cid:12) (9)
β,R s
Z0
(cid:13) (cid:13) (cid:0) (cid:0) (cid:1)(cid:1)
whereN doesnotde(cid:13)pendon(cid:13)δ andk,andk·k deenotesthenorminLβ([0,T]×
β,R
B(0,R)).
The last expressionin the right-handside of the last inequalitysatisfies the fol-
lowingestimate:
t
E 1−θ Xk/R ds≤supP sup Xk >R . (10)
s s
Z0 (cid:0) (cid:0) (cid:1)(cid:1) k≥1 hs≤t(cid:12) (cid:12) i
e (cid:12)e (cid:12)
Butweknowthatsup E[sup |Xk|p]<∞forallp>1,andthus
k≥1 s≤t s
lim supPesup Xk >R =0. (11)
s
R→∞k≥1 s≤t
h (cid:12) (cid:12) i
Substitutingestimate(10)into(9),lettingδ(cid:12)e→(cid:12)0,andusing(11),wededucethatthe
convergenceofthetermJ follows.
1
Finally,sinceestimate(10)alsoholdsforX,itsufficestousethesamearguments
asbeforetoobtaintheconvergenceofthetermJ ,whichcompletestheproof.
3
e
Acknowledgements
We thankthereviewerforhisthoroughreviewandhighlyappreciatethe comments
andsuggestions,whichsignificantlycontributedtoimprovingthequalityofthepaper.
References
[1] Bahlali,K.,Mezerdi,B.,Ouknine,Y.:Pathwiseuniquenessandapproximationofsolutions
ofstochasticdifferentialequations.In:Azéma,J.,Yor,M.,Émery,M.,Ledoux,M.(eds.)
Séminaire de Probabilités XXXII. Springer, Berlin (1998). MR1655150. doi:10.1007/
BFb0101757
[2] Banos,D.,Nilssen,T.,Proske,F.:StrongexistenceandhigherorderFréchetdifferentia-
bilityof stochastic flows of fractional Brownian motion driven SDEwith singular drift.
arXiv:1509.01154(2015)
[3] Decreusefond, L.,Üstünel, A.S.:Stochasticanalysis of thefractional Brownian motion.
PotentialAnal.10,177–214(1998). MR1677455.doi:10.1023/A:1008634027843
[4] Ikeda,N.,Watanabe,S.:StochasticDifferentialEquationsandDiffusionProcesses.North-
Holland,Amsterdam(1981).MR1011252