Lower bounds for the density of the law of locally elliptic Itô processes

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https://hal.inria.fr/inria-00071695

Submitted on 23 May 2006

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Lower bounds for the density of the law of locally elliptic Itô processes

Vlad Bally

To cite this version:

Vlad Bally. Lower bounds for the density of the law of locally elliptic Itô processes. [Research Report]

RR-4887, INRIA. 2003. �inria-00071695�

ISRN INRIA/RR--4887--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Lower bounds for the density of the law of locally elliptic Itô processes

Vlad BALLY

N° 4887

July 2003

0123.%+*,

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a

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^_AxR\[1FGV`‘,T_AxF

a

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a

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‰

F™e(VYXx^cM

a

FdhIR

q−^a MODGFEXx^_MOV\XxR\Q!•7TUV+[ChUVNeEFd^U^,V`‘%T_ACFI‘’VYh_D

X_t^ι =xⁱ₀+ X∞ j=1

Z t 0

U_s^ijdB_s^j+ Z t

0

V_sⁱds, i= 1, ..., q

R\X a Œ,FIR`hUFpMOXYTUFEhUFd^cT_F a NbGT_ACFžQOVNedR`Q a FdXx^cMWTkb+V`‘

XT

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XT

MS^^cDGVNV`TUA R`QS^cV1– ‰ F+YMW‹YF<XxVŒ T_AxF+XCV\X]¤a Fd\FEXxFEhR\e(b

RY^_^_PCDG[]TUMWVYX?– ‰ FnCfR a MŸ¡”FEhUFEXuT_MSR`xQWFeEPCh_‹YF

xt,0≤t≤T ^_PxeAT_AxR`T

x0=X0

R`X a

xT =y ^{R`X a}

^_V\DGF

rt > 0,0≤t≤ T. ^‰ F™R\QO^_VeEV\Xx^_Ma FEhI^cVYDGF

q×q ^a MODGFEXx^_MWVYXxR`Q^_bND<DGFET_hUMOeGR`X a [!VY^_MŸTUMW‹YF a F(nxXCMWT_F#DGR`T_hUMŸf]Fq^

Qt,0≤t≤T ^{R\X a a} FdXCV`TUFub

λt >0T_ACFQOVŒ,FEhFEMO\FdXN‹“R\QWPxFV`‘

Qt. @ACFdX V\PCh ANbN[!V`T_AxFd^_MO^R\h_FŠT_ACFI‘’VYQWQOVŒMOXCx–SRCVYhFE‹YFEhUb

0< t < T ^{R\X a} 0< δ < T−t (H, i) UtU_t^∗ ≥ Qt

(H^ν, ii) kΓδ(t)kk,p,t ≤ K(t)δ^12+ν, ν >0, (H, iii)



 Xq

i=1

X∞ j=1

U_t^ij

2





1/2

≤ K⁰(t)

VYX‚TUACF^_F(T

Q⁻_t ^1/2(X(t)−x(t))≤r(t)^–

—F(TPx^FEf][CQOR\MWXT_AxMO^ a F(nxXCMWT_MOV\X– XCFIŒh_MWT_Fq^

X_t+δⁱ =X_tⁱ+ X∞ j=1

Z t+δ t

U_s^ijdB^j_s+ Γⁱ_δ(t)

ŒMŸTUA

Γⁱ_δ(t) :=

X∞ j=1

Z t+δ t

(Us−Ut)^ijdB_s^j+ Z t+δ

t

V_sⁱds.

@AxFž[xh_MOXxe(MO[xR\Q1TUFEhUD MS^ P_∞

j=1

Rt+δ

t U_t^ijdB_s^j ŒACMSeAMO^§IR\Px^_^_MSR`XeEV\X a MŸTUMWVYXxR`QOQWbGTUV

Ft =σ(B^j_s, s≤ t, j∈N).^{R\X a} AxR\^,TUACFe(V‹R`hUMOR\Xxe(FžD+RTUh_MWf

UtU_t^∗×δ.^KNV (H, i)^UR“bN^TUAxRTT_AxF[Ch_MOXxeEMW[xR\Q”T_Fdh_D MS^

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Γⁱ_δ(t)MO^R~h_FdD+R`MOX a FEhR\X a

(H^ν, ii)

^UR“b]^TUAxRTGT_AxMO^<hUFED+R`MOX a Fdh~D+R“b !FMO\XCVYh_F a ŒMWT_A h_Fq^c[!Fde(T#T_VT_AxF[xh_MOXxe(MO[xR\Q˜TUFEhUD ŒACMSeA$MS^~Fq^k¤

^_FEXuTUMOR\QWQOb›V\‘,VYh a Fdh

δ^1/2. @ACF+XNPCD~1Fdh

ν MS^#R ^cT_hUMOe(T_QOb›[!VY^_MŸTUMW‹YF™XNPCD#!FEhŒAxMOeA a Fd[1FdX a ^VYXT_ACF [xh_VYCQWFdDrRTA1R`X a ¤!MOXTUACF˜‘’hR`DGFV`‘ a MW¡!P1^cMOV\X~[Ch_V]eEFd^U^cFq^

ν= ¹₂ ^{R`X a} ‘’V\hKuTUVNeA1R\^cT_MSe

P DE⁰s, ν=¹₄.

@AxF~XCVYh_D

k◦kk,p,t

MO^IR KNV\!V\QOFE‹ XxV\hUDŒAxMOeAMWXN‹YV\QO‹\Fd^pT_AxF

L^p XCV\hUD+^ŠV\‘iT_AxF~n1hU^cT

k lmR\QWQOMOR“‹NMOX

a

FEhUMW‹RTUMW‹YFd^žŒACFdh_F

p, k R\h_FG^cVYDGF<MOXuT_Fd\FEh^

a

FE[!FEX

a

MWXxVYX›TUACF

a

MODGFEXx^_MWVYX

q. @AxF+QWVŒ,FEhIMOX

a

F(f t

^_MO\XCMWnxFd^T_AxR`TŠŒ,FŒVYh_¢ŒMŸTUA›e(VYX

a

MWT_MOV\XxR\Q?F(f][!Fde/TRTUMWVYXx^ŒMWT_AmhUFd^_[1Fqe/TTUV

Ft

R\X

a

XCV\TpŒMŸTUA Px^_PxR`Q

FEf][1Fqe/TUR`T_MOV\Xx^ 7ŒFIPx^_FRGe(VYX a MWT_MOV\XxR\Q?lmR`QOQWMSR“‹NMWXeER\QOeEPCQWP1^ A(–

—F(TP1^?XCVŒe(VYD<DGFdXYTT_ACFiQOV]eER\QWMOœdR`T_MOV\X?–>J,V\T_A

Ut =Ut(ω)^{R`X a} kΓδ(t)kk,p,t=kΓδ(t)kk,p,t(ω)

R\h_F#hUR\X a V\D ‹R`hUMSR`CQOFd^d–žKNV‚T_ACF<[ChUV\[!FEh_T_MOFd^

(H, i),(H^ν, ii) ^{R\X a} (H, iii)^ACV\Qa TUh_PCF~V\XCQOb‘’V\h

ω ∈ {Q⁻_t ^1/2(X(t)−x(t))≤r(t)}. @ACMOXC¢‚‘’V\hRGDGV\DGFEXuTTUVGT_ACFFEf]R\DG[CQWFV\‘TUACF a MW¡!P1^cMOV\XZ[ChUV]e(Fd^U^

dXt=σ(t, Xt)dBt+b(t, Xt)dt.^@ACFdX Ut=σ(t, Xt)<sup>R`X a ^cV</sup> (H, i)^_R“b]^T_AxR`T

σσ^∗(t, x)≥Qt

‘’V\h

x ^cPxeAZTUAxRT

Q⁻_t^1/2(x−x(t))≤r(t). K]V™ŒF#XCFEF

a

T_ACFFdQWQOMO[]T_MSe(MWTkb R\^U^cPCDG[]TUMWVYXZV\XCQObVYXmR<T_PC!F

R\h_VYPCX a T_ACF™eEPCh_‹YF

xt. @ACF™^UR`DGFGMO^T_hUPCF<‘’V\hIT_AxF+1VYPCX a F a XCFd^U^RY^_^_PCDG[]T_MOV\X

(H, iii) : Œ,F<XxFEF a

|σσ^∗(t, x)| ≤K⁰(t)VYXCQOb‘’VYh

x MWX TUACFTUPC!F\–ž•¦Xm[xR`h_T_MSe(PxQOR\hT_ACMS^Š[1Fdh_DGMWTU^pT_V‚A1R`X a QOF#PxXu!V\PxX a F a eEVNF ™e(MOFEXuTU^ 7ŒMŸTUAZQWMOXCFdR\h\hUVŒT_A‚‘’V\hF(fCR\D<[xQWFA/–

‰ FInCfXxVŒ ^_V\DGF

h >0^{R`X aZa FEnxXCF} γh(t) : = min{h,3

2r(t), 1

sup_t≤s≤t+hQ^−1/2(t)∂sxs

, λ^(1+ν)/2ν(t) C((K⁰(t) +p

λ(t))K(t))^1/2ν} µh(t) : = sup

0∨(t−h)≤s≤t

1 γh(s)

ŒAxFEhUF

CMS^˜R\X<PxXCMW‹YFEh^_R\Q]F(f][CQOMOeEMŸT,e(V\X1^kTR`XuTd– PChiD+R`MOXGh_Fq^cPCQWT˜MS^ŽT_AxF‘’V\QOQWVŒMOXC1–GX a FEh˜T_ACFR\1V‹YF RY^_^_PCDG[]TUMWVYXx^

(H, i),(H^ν, ii),(H, iii) <sup>TUACF™hR`X a</sup> V\D‹“R\h_MSR`xQWF

XT

AxR\^~R QWV]edR`Q a FEXx^_MŸTkb

pT(x0, y)

ŒAxMOeAMO^pe(V\XuTUMWXNPCVYPx^R`X a DGV\hUFEV‹YFEh

C

T^q+2λ^q+2_T (P(|XT −x0| ≥ |y−x0|))^1/2

≥ pT(x0, y)≥ 1

4e(2πγh(0)λ(0))^q/2 ×exp(−C⁰ Z T

0

µ²_h(t)dt).

EpV`T_FTUAxRTTUACFR`!V‹\F,!V\PCX a ^ŽACV\Qa T_hUPCF,‘’V\hFE‹YFEhUb

h >0^{R\X a FE‹\Fdh_b a} F(T_Fdh_DGMOXCMS^kTUMOe[xR`T_A

xt

‘’V\h

ŒAxMOeAGTUACFpFdQWQOMW[CT_MSe(MWTkb<RY^_^_PCDG[]T_MOV\X+AxV\Q

a

^TUh_PCFY–ŽK]VV\XCFpDGR“b~TUR\¢\FTUACFŠ^_PC[ChUFED~PCD V‹YFEhi^_PxeA+[xR`T_Ax^

MOX TUACF#QOVŒFdhp!V\PCX

a – RxV\hžF(fCR`DG[CQOF

£

MWXmT_ACFGeERY^cFV\‘iQWVY`¤7XCV\hUD+R`Q”TkbN[!F

a

MW¡!P1^cMOV\Xx^

£

MŸ‘V\XCFTR`¢YFd^

xt

TUV~!FŠT_ACFI^kTUh_MO\AuT,QWMOXCFIŒACMSeA™MWQOXC¢]^

x0

T_V y,T_ACFIe(VYh_hUFd^_[1VYX

a

MOXCQOVŒFdh˜!V\PCX

a

^R`hUFž§IR`P1^_^_MOR\X‚R`X

a

TUACMS^pMS^XCV\TŠVY[]T_MOD+R`Q¥– M TpT_AxF~e(VYXuT_hR`hUb £ MW‘iV\XCF#TUR\¢\Fd^F(f][1VYXCFEXuTUMOR\Q?TkbN[1F<e(PChU‹\Fq^T_AxFEXmVYXCF#VY]TUR\MWXx^

R<QOVŒFdh1VYPCX a ŒACMSeAMS^V`‘T_ACF^UR`DGFžTkbN[1FRY^,T_ACF a FEXx^_MŸTkb‚V`‘R~QOV\GXCVYh_D+R\Q!QSR“Œ–

@ACFMa FdRmV\‘T_ACF[ChUVuV\‘MO^#T_ACF‚‘’V\QOQWVŒMOXC1– XCF‚eEV\Xx^cT_hUPxe/T^hUFdeEPChU^_MO‹\FEQObmTUACF™TUMWDGF\hUMa

t0 = 0, δ0 =γ_h²(t0), t1 =t0+δ0, ..., δk =γ_h²(tk−1), tk =tk−1+δk, ... ^{R`X a TUR`¢YFd^} N T_V1FGTUACF<n1^kT MOXuT_Fd\FEh^_PxeAT_AxR`T

tN≥T.^{EpVŒ £} ‘’V\hFqR\eA

kVYXCFIŒh_MWT_Fq^

X_tⁱ_k=X_tⁱ_k−1+ X∞ j=1

Z tk−1+δk

tk−1

U_t^ij_k−1dB^j_s+ Γⁱ_δk(tk−1). ^{7 ?*A}

GŠ^cMOXCT_ACF<FEQOQOMW[]TUMOeEMŸTkbR\^U^cPCDG[]TUMWVYX

(H, i) <sup>R`X a</sup> T_ACF+e(VYX a MWT_MOV\XxR\Qi§IR`Px^U^cMSR`X›QOR“Œ V\‘T_ACFGhR`X a V\D

‹R\h_MSR`CQOF P_∞

j=1

Rtk−1+δk

tk−1 U_t^ij_k−1dB_s^{j V\XCF a} Fdh_MO‹\Fq^+RQWVŒ,FEhG!V\PCX a ‘’V\hGT_AxF a FEX1^cMWTkb$V\‘ŠTUACMO^+T_FEhUD R\X a TUACFEX £ Px^_MWXC

(H^ν, ii) VYXCFG[ChUV‹\Fd^T_AxR`TIT_ACFGhUFED+R`MOX a Fdh

Γⁱ_δk(tk−1) D+R“b !FGMWYXCV\hUF a 7MWTIMS^IMOX RQOVŒFdhI^UeER\QWFA/–GKNVVYXCFGV\]TR`MOXx^IRQWVŒ,FEhž1VYPCX a ‘’V\heEV\X a MWT_MOV\X1R`Q a FEX1^cMWTkbmV`‘˜T_ACFGTkbN[!F

P(Xtk ∈ dy|Xtk−1 =x). @AxFnxXxR\QQOVŒFdh!V\PCX

a

ŒMWQOQ!F#VY]TUR\MWXxF

a

R\^T_ACF<e(VYXu‹YV\QOP]T_MOV\XZV\‘T_ACF

N ^QWVŒ,FEh

!V\PxX

a

^,‘’V\hTUACFe(VYX

a

MWT_MOV\XxR\Q

a

FEXx^_MWTkb‚e(VYh_hUFd^_[1VYX

a

MOXCTUV+FdRYeA

tk, k= 0, ..., N.JPCTXCV`TUFŠTUAxRTTUACF R\1V‹YF˜Fd‹“R\QWP1RT_MOV\X1^ŽACV\Qa TUh_PxF,VYXCQObMŸ‘!T_ACF a MW‘’Px^U^cMOV\X

Xt

hUFED+R`MOXx^ŽMOX<RŠT_Px1FV`‘!hUR a MOPx^

rt

R`hUV\PCX a T_ACF

eEPChU‹\F

xt

7!FdeER\Px^_FžVYPChANbN[1V\T_ACFq^cMS^R`hUFIMWX‘’V\he(FžMOXTUACMS^eERY^cFIV\XxQWbA/–i@AxF¢\FEb+[xh_VYCQWFdDMS^,T_VGnxX a

TUACF#YVNV a F uPCMWQOMOCh_MOPCD1FETkŒFdFEX

δk

R`X a TUACF~^_MOœEF#V\‘

Xtk−Xtk−1

V\X VYXCFAxR\X a R`X a T_V™Fd‹“R\QWP1RT_F

TUACF<[Ch_VYxR`xMWQOMŸTkbTUAxRTžT_ACFG[ChUV]e(Fq^_^

Xt

h_FdDGR\MWX1^pMOXR™T_PC!FGV`‘˜hR a MOPx^

rt

R\h_VYPCX a TUACFGe(PChU‹\F

xt

VYX

TUACFIV`TUACFEhAxR\X a

.@AxMO^QOFdR a ^,T_V<T_AxF a F(nxXxMŸTUMWVYXV\‘TUACFž‘’PCXxe/TUMWVYX

γh(t). @ACMWXx¢™‘’V\hR<DGV\DGFEXuTTUAxRT

X ^{MS^R a} MW¡!P1^cMOV\X [ChUV]e(Fq^_^^_VZT_AxR`T

Ut =σ(t, Xt). @AxFEXT_AxFEhUF+MO^~RZDGR\XCMW‘’Fd^cT#R`XxR\QWVY\bm!F(TkŒ,FEFdX TUACF a FdeEV\DG[!VY^_MŸTUMWVYX›Px^_F a MWX 7?AŠR\X a TUACFGV\XCF<Px^_F a MWXT_AxF<[1R`hR`DGF(TUh_MWf D<FET_ACV a 7eEV\DG[xR`hUF#ŒMWT_A

(4.1),(4.2)[xR`YFD?EHžMOX"9}<A(–•¦X<^cVYD<FŒ,R“bIŒ,FP1^cFRŠ^cT_V]eAxRY^kTUMOe˜‹\FdhU^_MWVYXV`‘CTUACF[xR`hR`DGFET_hUMŸfIDGF(T_AxV a VYX ^cD+R`QOQ?MOXuT_FEhU‹R`QS^V`‘%T_MODGF\–@ACFEhUFR`hUFžTkŒV+[!V\MOXuTU^VYX ŒACMSeATUACF#^cT_V]eAxRY^kTUMOeIR`[x[Ch_V]eAZAxRY^^cVYDGF

R a ‹R`XuTUR\\Fq^”nxh^cTiV`‘¨R`QOQCMŸT[1Fdh_DGMWTU^ŽTUVQWV]eER\QWMOœEFVYX~TUACF^_F(T˜V\‘1TUhUR`jkFde/TUV\hUMWFq^%ŒACMOeAGhUFED+R`MOX<MWX+RžT_Px1F

R\h_VYPCX a T_AxF a FET_FEhUDGMWXxMO^cT_MSe™e(Pxh_‹YF‚R`X a TUACMS^#[!FEhUDGMŸT^T_V a FdR\Q˜ŒMWT_A$R›e(FEh_TUR\MWX eEQORY^_^#V`‘ a MW¡!Px^_MOV\Xx^

ŒAxMOeAGR`hUFXCV`TPCXCMW‘’V\hUDFdQWQOMO[]T_MSeR`X a<a VXCV\TAxR“‹\F!V\PCX a F a e(VNF ™eEMWFdXYT^E–•¦XG[xR`h_T_MSe(PCQSR`h £ T_AxF^cAxR\[1F

V\‘T_ACFQWVŒ,FEh!V\PxX a MWX)^cPxeAšR›eER\^_F+D+R“b!F a MŸ¡”FEhUFEXuT‘’hUV\D T_ACFZ§IR`Px^U^_MOR\XVYXCF\–mlmV\hUFEV‹\Fdh £ MŸT

[!FEhUDGMŸT^,T_V a FqR`Q¨ŒMWT_AZXCVYX]¤7l›R`hU¢\V‹G[ChUV\CQOFED+^d–

@ACF[1R`[!FEhMS^ŽV\hUYR`XxMWœdF a RY^‘’V\QOQWVŒ^d–•¦X<T_AxFnxhU^cTi^cFqe/TUMWVYX~Œ,F,T_hUFdRTŽTUACF[Ch_VYCQOFED MOX+R`XGR`x^cT_hR\e(T

‘’hR`DGFiŒAxMOeAMS^MOXx^_[CMOh_F a ‘’h_VYD 9vL<7– J,P]TT_ACFiD+R\MWXFd‹R`QOPxRTUMWVYX £ eEV\XuTUR\MWXxF a MOX#@ACFEVYh_FdD ?iMS^%^kTUh_VYXC\Fdh

TUAxR`X,OpVYAxRT^cP"BpMWuRC«^ihUFd^_PCQWTd–Ž•¦X‚[xR\hcTUMOeEPCQSR`h˜MWT[1Fdh_DGMWTU^T_V a FEhUMO‹\Fp^_V\DGFTUPC!Fd^,FE‹R`QOPxR`T_MOV\Xx^ŒACMSeA

R\h_F,e(hUPxe(MSR`QuMOX~VYh a FdhTUVŠVY]TUR\MWX~QWV]edR`Quh_Fq^cPCQWTU^d–Ž@ACMO^Ž^_Fde(T_MOV\X~eEV\XuTUR\MWX1^?TUACFD+R\MWX~T_FqeACXCMSeER`QYT_VNVYQO^ŽR`X a

hUFd^_PCQWTU^d–Ž•¦X™T_AxFI^_FdeEV\X a ^cFqe/T_MOV\XŒF a FqR`Q”ŒMŸTUA•7T~[xh_V]e(Fq^_^_Fd^,R`X a Œ,FŠVY]TUR\MWX‚T_ACFIhUFd^_PCQŸT[ChUFd^_FEXuT_F a

R\1V‹YF\–•¦X~TUACFTUACMWh a ^_Fde/TUMWVYXGŒF a MS^Ue(Px^U^ŽT_AxFF(fCR`DG[CQOFV`‘ a MW¡!P1^cMOV\XG[ChUV]e(Fd^U^d–•¦XGV\h a FdhŽT_VMOQWQOPx^cT_hRT_F

VYPChDGF(TUACV a Œ,Fe(VYXx^cMa Fdh,QOV\\¤¥XxV\hUDGR\Q!Tkbu[!F a MW¡!P1^cMOV\Xx^ 7T_AxR`TMO^

σ(t, x) =α(t, x)x) ^{R\X a} eEV\DG[xR\h_F VYPChiQOVŒFdh1VYPCX a ^iŒMŸTUAGT_ACF[xh_Fqe(MS^cFQOV\\¤¥XCVYh_D+R\Q a FEXx^_MWTkb\– RMOXxR\QWQOb#MOXGT_ACFpQORY^kT^_Fde(T_MOV\XGŒ,Fe(V\X1^cMa FEh

TUACFF(fCR\D<[xQWFIV`‘KuT_V]eAxRY^kTUMOe

P DE⁰s ⁷ŒACMSeAZAxR\^,nxh^cT!FEFdXT_hUFdRTUF a MWX 9v*<A/–

#Ž'&%

!#""$

•¦X‚T_ACMS^^cFqe/TUMWVYX™ŒFŠŒMWQOQ”Px^_FŠ[xR\hcTUMOR\Q!l›R`QOQWMSR“‹NMWX™edR`QSe(PCQOPx^^cV~ŒFŠAxR“‹\FT_V<MWXuT_hUV a PxeEFŠ^_V\DGFpXCV\TUR`T_MOV\X?–

‰ FGe(V\X1^cMa FEhIR`X›MWX]n1XCMŸTUF a MWDGFdXx^cMOV\X1R`Q JhUVŒXCMSR`X DGV`TUMWVYX

B = (B^j)j∈N

R\X a›a FdXCV`TUF#Nb

Ft

T_ACF

n1QŸTUhUR`T_MOV\X \FdXCFEhRTUF a ub

B. ^‰ FmhUF(‘’FdhGT_V EpPxR`QSR`h_T 9@?dz<‘’VYh+T_AxFmXCV`TRTUMWVYX?–:•¦X R`QOQTUACMS^™^cFqe/TUMWVYX

Œ,F nCf ^_V\DGF

t ≥ 0, δ > 0. ^‰ AxRTMO^^c[!FdeEMŸn1e›MWX:V\PCh+‘’hR`DGFmMS^‚T_AxR`TŒ,FmFEDG[CQOVb$TUACF›[1R`h_T_MSR`Q l›R`QOQWMSR“‹NMWX eER\QOeEPCQOPx^VYXT_ACF™MOXuT_Fdh_‹R`Q

[t, t+δ]. @ACMS^DGFdR\Xx^TUAxRT#Œ,F‚eEV\Xx^_Ma FEh a Fdh_MO‹RTUMW‹YFd^ŒMWT_A hUFd^_[!Fde/T#T_V

Bs, s ∈ [t, t+δ] <sup>V\XCQOb R`X a</sup> Œ,F‚FdDG[CQWVbT_AxFe(VYh_hUFd^_[!V\X a MOXC›KNVY1VYQWFd‹XxV\hUDG^ £ PCX a Fdh TUACFe(VYX a MWT_MOV\XxR\Q!FEf][1Fqe/TUR`T_MOV\XŒMŸTUAhUFd^_[1Fqe/T,T_V

Ft

–RxV\hF(fCR\D<[xQWF £

D^1,2_t ^a Fq^cMO\XCFq^iTUACF^c[1R\e(FžV`‘T_ACF

‘’PxXxe/TUMWVYXxR`QS^#ŒAxMOeA R`hUF‚V\XxF™T_MODGFd^ a MW¡!Fdh_FdXuT_MSR`CQOFMOX

L², MOX)lmR\QWQOMOR“‹NMOX?«^#^_FEXx^_F £ VYX T_AxFMOXuT_Fdh_‹R`Q

[t, t+δ]. l V\hUF#[xh_Fqe(MS^cFdQWb £ TUACMS^ž^_[xRYe(F#MS^Ie(VYXx^kTUh_P1e/T_F a MOX›TUACF#‘’VYQWQOVŒMOXC‚ŒR“b\– XCFGe(V\X1^cMa FEh^pT_ACF

^_MOD<[xQWF‘’PxXxe/TUMWVYXxR`QS^

F =f(G, Bs1−Bs0, ..., Bsk−Bsk−1)^ŒACFEhUF t=s0< s1< ... < sk =t+δ, G^{MS^R`X} Ft

DGFdR\^_PChR`CQOF,hUR\X a V\D‹“R\h_MSR`xQWFR`X a

f(G, y1, ..., yk)MS^iRI^_D<VNV\T_A~‘’PCXxe/TUMWVYXGŒMŸTUA<hUFd^_[!Fde/T TUV (y1, ..., yk). RxV\h^_PxeAR~‘’PCX1e/T_MOV\X1R`Q?VYXCF a F(nxXxFd^TUACFlmR`QOQOMOR“‹NMOX a Fdh_MO‹RT_MO‹\FINb

DsF = Xk i=1

∂f

∂yi(G, Bs1−Bs0, ..., Bs_k−Bsk−1)1[si,si−1)(s).

@AxFEXV\XCFeEV\Xx^_M

a

FEh^˜TUACFXCVYh_D

kFk²1,2,t:=Et(|F|²) +Et( Z t+δ

t |DsF|²ds).

BpFEhUFŠR`X

a

MWX+TUACFŠ^_FuPCFdQ

Et

a

Fd^_MWYXCFd^ŽTUACFŠeEV\X

a

MŸTUMWVYXxR`Q1F(f][1Fqe/TRT_MOV\X‚ŒMŸTUA™h_Fq^c[!Fde(TiT_V

Ft

TUAxRT,MO^

Et(Φ) :=E(Φ|Ft).

KNV kFk1,2,t

MO^R\X

Ft

D<FqR\^_PChR`CQOFIhUR\X

a

V\D‹“R\h_MSR`xQWFR`X

a

XCV\TpR+e(VYXx^kTR`XuTd–

‰ F a

FEnxXCF

D_t^{1,2 T_V}

!FTUACFŠeEQWVu^cPChUFV`‘”TUACFp^_MWDG[CQOF‘’PCX1e/T_MOV\X1R`QS^˜^_[xR\eEFŒMŸTUA™h_Fq^c[!Fde(TT_VTUACMS^˜XCVYh_DZ–KNV

F ∈D_t^{1,2 DGFqR`Xx^}

TUAxRTpT_ACFdh_FFEfNMS^cTU^R+^_FuPCFdXxe(FV\‘Ž^_MWDG[CQOF‘’PCX1e/T_MOV\X1R`QS^

Fn, n∈N, ^_PxeAT_A1RT

kFn−Fmk1,2,t→0

R\X a

E|Fn−F|²→0R`QOD<Vu^kT^_PCh_FdQWb™RY^

n, m→ ∞. •¦XT_ACF^UR`DGFIŒ,R“bGV\XxF a F(nxXCFq^ACMO\ACFdh,VYh a FEh

a FEhUMW‹RTUMW‹YFd^R\X a T_AxF hUFEXx^cT_FdMWX GpQWFdD#xRYe¢‚V\[!FEhRTUV\h

L^{– RCVYh} F ∈(D^1,2_t )^q, F = (F¹, ..., F^q) ^T_ACF

l›R`QOQWMSR“‹NMWXe(V‹R`hUMOR\Xxe(FžD+RTUh_MWf‚MS^ a F(nxXxF a Nb

φ^ij_F,t= Z t+δ

t

DsFⁱDsF^jds.

‰ F a FEXCV\T_Fub

λ_F,t(hUFd^_[1Fqe/T_MO‹\FdQWb

λF,t)TUACFQWVŒ,FEh 7hUFd^_[1Fqe/T_MO‹\FdQWbžTUACFQOR\h_YFEhA”FEMO\FdXu‹R\QWPCFV`‘

φF,t.

@AxFEb‚R\h_FI\MO‹\FdX‚Nb

λ_F,t= inf

|ξ|=1

Xq i,j=1

ξiξjφ^ij_F,t, λF,t= sup

|ξ|=1

Xq i,j=1

ξiξjφ^ij_F,t.

•¦XZV\h a FdhTUV a F(n1XCFIACMWYACFEhVYh a FEh a Fdh_MO‹RT_MO‹\Fq^,ŒF a FEnxXCFIT_AxF#KNV\!V\QOFE‹™XxV\hUDG^

kFk^pk,p,t:=Et(|F|^p) + X

1≤|α|≤k

Et((

Z t+δ t

...

Z t+δ t

D^α_s1,...,s|α|F

2

ds1...ds_|α|)^p/2)

ŒAxFEhUF £ ‘’VYhŽRžD~PCQŸTUMŸ¤7MOX a F(f

α= (α1, ..., αm)^{ŒF a FdXCV`TUF} |α|=m.^{@ACF a} Fdh_MO‹RT_MO‹\Fq^

D_s^α_1,...,s|α|F ^R`hUF

n1hU^cT a F(n1XCF a VYX+^cMODG[CQWF‘’PCXxe/TUMWVYXxR`QS^˜R\X a T_AxFEX+F(fNTUFEX a F a T_VYFEXCFdhUR\Q]‘’PCXxeEMWVYXxR`QS^Nb<[xR\^U^cMOXCITUVT_ACF

QOMOD<MWTMOX

k◦kk,p,t.^{‰ F a} FEXCV\T_FpNb

D^k,p_t T_AxFŠe(QOVY^_PChUFV`‘”TUACFž^_MOD<[xQWF‘’PxXxe/TUMWVYXxR`QS^^_[xRYe(FŒMWT_A™hUFd^_[!Fde/T TUV k◦kk,p,t. @ACF<xR\^_MSe‘RYe/TIŒACMOeAŒ,F~ŒMOQWQŽPx^_FGMO^ŠTUACFGMWXuT_Fd\hRTUMWVYXmNb [xR`h_TU^Š‘’V\hUD#PxQOR‚ŒACMOeAŒF

YMW‹YF+MWXšT_ACF+‘’VYQWQOVŒMOXCx– ‰ Fe(VYXx^_Ma Fdh

F = (F1, ..., Fq)^£ Fi, G ∈ D^k,p_t ^{R\X a} Œ,F‚R\^U^cPxD<FGTUAxRT

F

MS^XCVYX a FE\FdXCFEhRTUF a 7TUAxRTMS^

λ_F,t >0 R`QOD<Vu^kT#^_PChUFEQObA/– RCVYhR‘’PCXxe(T_MOV\X

f : R^q → R ŒACMSeAMS^

MOX]nxXxMŸTUFEQOb a MW¡”FEhUFEXuT_MSR`CQOFR`X a ‘’V\hpRGD~PCQŸTUMŸ¤7MOX a F(f

α= (α1, ..., αk) ^{a FEXCV\T_F} D^α T_ACF~e(V\hUhUFd^_[1VYX a MOXC

a FEhUMW‹RTUMW‹YFpTUAxRTMS^

D^αf =∂^kf /∂xα1...∂xαk

–i@ACFdX

(IPi) Et(∂f

∂xi

(F)G) = Et(f(F)Hi(F, G)) with Hi(F, G) = −

Xq j=1

(Gφb^ji_F,tL(Fj) + Z t+δ

t

DsFjDs(φb^ji_F,tG)ds

ŒAxFEhUF

φbF,t

a Fq^cMO\XxFEFd^žTUACFGMWXN‹YFEh^cF<V`‘

φF,t

R`X a

L ^a Fd^_MWYXCFd^žTUACF h_X1^kTUFEMOX GpACQOR\D#!Fde¢mVY[1FdhUR`T_VYh

ŒMWT_AZhUFd^_[1Fqe/TT_V

Bs, s∈[t, t+δ]. @ACMS^,‘’V\hUD#PxQOR<D+R“b+1FMWT_FdhUR`T_F a MOXV\h a FEhTUV<VY]TUR\MWX

(IPα) Et(D^αf(F)G) = Et(f(F)Hα(F, G)) with

Hα(F, G) = Hαk(F, H_{(α1,...,αk−1)}(F, G)) f or α= (α1, ..., αk).

‰ FD+R`¢\FžTkŒ,VGh_FdDGR\h_¢]^e(VYXxe(Fdh_XxMWXC<TUACFR`!V‹\FžMWXuTUFE\hRTUMWVYX‚Nb‚[1R`h_TU^,‘’V\hUD#PxQORx–Ž@AxFInxhU^cTV\XCF

eEV\XxeEFEhUXx^ŽRžQWV]eER\QWMOœdR`T_MOV\X~[Ch_V]eEF a PChUF\– ‰ FRY^c¢YF a

φF,t

T_V!F,MOXN‹\FdhcTUMWCQOF\– J,P]TŽTUR\¢uMOXCRžQWVNVY¢T_V

(IPi)

Œ,F+^cFdF<T_AxR`TžTUACF+D+RTUh_MWf

φF,t

AxRY^žT_VZ1F+MOXN‹\FdhcTUMWCQOF<MWXT_AxF+h_Fd\MOV\X›ŒAxFEhUF

G MO^XCV`TXNPCQOQ£ V\hI[CP]T MWTŠV\T_ACFdh_ŒMS^_F £

G A1R\^TUV™!F#XNPCQOQMOXZT_ACF~h_Fd\MOV\XZMOXmŒACMSeA

φF,t

MS^pXCV\TpMOXu‹YFEh_T_MOCQOF\–@ACMS^pMS^TUh_PxF‘’V\h

TUACFGnxh^kTITUFEhUD|MWX

Hi(F, G). •¦XT_ACF™^_FdeEV\X a T_Fdh_D R\[C[!FdR`h^

Ds(φb^ji_F,tG) =Dsφb^ji_F,tG+φb^ji_F,tDsG.

‰ FpXxFEF a T_AxR`T

DsGMS^R\QO^_VXNPCQOQ”MWX+TUACFžh_Fd\MOV\X+MOX™ŒACMSeA

φF,t

MS^ a FEYFEXCFdhUR`T_F a –%oVYPC\AxQWb+^_[1FqR`¢NMWXx

TUACMS^ŒMOQWQ?!FTUh_PCF#MŸ‘

GMO^XuPxQWQMŸ‘

λ_F,t MS^XNPCQWQ/– ‰ F a V+XCV\T‘’V\hUD+R`QOMWœdFID<VYh_FITUACMS^Ma FdRGŒACMSeA ŒMOQWQ

!FIPx^_F a MWXT_ACF^_FuPCFdQ¨MWXZR#DGVYh_Fž[ChUFde(MS^_Fp‘’hR`DGFIMWXŒACMSeA‚TUACFR`!V‹\FŠh_VYPC\A a Fd^Ue(hUMW[CT_MOV\X‚MS^D#P1eA

DGVYh_F<e(QOFdR`hq–I@ACF<^_FdeEV\X a h_FdDGR\h_¢Ze(VYXxe(Fdh_X1^TUACF~D+R\MWXmFd‹R`QOPxRTUMWVYXm[ChUV a Pxe(F a NbT_ACF<MWXuTUFE\hRTUMWVYX

Nb[xR`h_TU^‘’V\hUD#PCQSRC– @ACFdh_F™FEf]MO^cT~^_V\DGF™e(VYXx^kTR`XuTU^

pm, km

^_PxeAT_AxR`T#‘’VYh~RZD#PxQŸTUMŸ¤7MWX a F(f

α ^ŒMWT_A

|α| ≤m

|Et(D^αf(F)G)| ≤bφ^ji_F,t

p_m,tkFkp_m,k_m,tkGkp_m,k_m,t.

•7‘?VYXCFTR`¢\Fq^˜RQOVNV\¢~T_VTUACFphUFdeEPCh^cMO‹\F

a

FEnxXCMWT_MOV\X™V\‘

Hα

T_ACFdX‚VYXCF!FEQOMWFd‹\Fd^iT_AxR`T˜TUACFŠR\1V‹YFFE‹R`QOPxR¤

TUMWVYX™ACVYQa ^iT_hUPCF £ CPCTF(f][!Fde(TU^˜T_AxR`TTUACFpXCVYh_D V`‘

Le(VYD<Fq^iVYX‚R\^˜ŒFdQWQ¥–Ž•¦X+‘R\e(TV\XxFpD+R“b a V\DGMWX1RT_F TUACFmXCVYh_D+^GV\‘ŠTUACF hUFEXx^cT_FdMWX GQOFED~1FqR\e¢šVY[1FdhUR`T_VYh

L ub$T_ACF›XCV\hUD+^GV`‘žT_ACF a FEhUMW‹RTUMW‹YFd^ 7T_ACF K]V\!V\QOFE‹GXCV\hUD+^ A˜Px^_MWXx<l Fdb\Fdhd«^iMOXCFuPxR\QWMWT_MOFd^d–Ž•7‘VYXCF a V~XCV\T,ŒR`XuTT_V<Px^cFŠT_AxFd^_FŠMOXCFuPxR\QWMWT_MOFd^ £ V\XCF

A1R\^žT_V R aCa MWXT_AxF a F(nxXCMWT_MOV\XV\‘

kFkp,k,t

TUACF+XCV\hUD+^IV`‘R e(FEh_TUR\MWXXNPCD#!FEhV\‘VY[1FdhUR`T_MOV\Xx^

L ^{R\^}

Œ,FEQOQ 7^_V<T_ACMS^MS^XCV\TR<hUFdR\QWQOb™MOD<[!V\h_TUR\XuT[!V\MOXuT £ xP]TpR`QODGVY^cTR<DGF(T_T_FdhV`‘%XCV`TRT_MOV\X4A/–

E @ M @• E •¦X R`QOQT_AxF+‘’V\QOQWVŒMOXCZŒ,F+ŒMWQOQ˜Œ,V\hU¢mŒMWT_A

q^{¤ a} MWDGFEX1^cMOV\XxR\Q‘’PCXxe(T_MOV\XxR\QO^ŒACFEhUF

q

MS^IRnCf]F a MWXuT_Fd\FdhžR\X a Œ,F<ŒMWQOQPx^_F<MOXuT_Fd\hRT_MOV\X›ub›[xR`h_TU^

q+ 1 T_MODGFd^d–GKNVŒ,F<ŒMOQOQFEDG[CQOVbZT_ACF R\1V‹YFIFE‹R`QOPxRTUMWVYX ŒMŸTUA

m=q+ 1. ^{‰ F a FEXCV\T_F} k=kq+1

R\X a

p=pq+1

R`X a Px^_FTUACMO^ŠXCV\TURTUMWVYX

TUAChUV\PCYACV\P]T<T_AxF ŒACV\QOFZ[xR`[!FEhq– @VnxXCMS^_A)Œ,FZ\MO‹\FT_ACF‘’VYQWQOVŒMWXx^_MOD<[xQWF‘R\e(T‚e(VYXxe(Fdh_XCMOXCT_ACF

l›R`QOQWMSR“‹NMWXe(V‹R`hUMOR\Xxe(FžD+RTUh_MWf

±˜±

ƒ„

±

F, G∈(D^1,2)^q .

(detφF+G,t)^1/q≥ 1

2λ_F,t−λG,t.

… ±?± Gp^_MWXC<TUACFFEQOFEDGFEXuTR`hUb™MWXxFuPxR`QOMWTkb

(x+y)²≥¹2x²−y² VYXCFIV\]TR`MOXx^

(detφF+G,t)^1/q ≥ λ_F+G,t= inf

|ξ|=1

Z t+δ t

Xq i=1

ξiDs(Fⁱ+Gⁱ)

!2

ds

≥ 1 2 inf

|ξ|=1

Z t+δ t

Xq i=1

ξiDsFⁱ

!2

ds−sup

|ξ|=1

Z t+δ t

Xq i=1

ξiDsGⁱ

!2

ds

= 1

2λ_F,t−λG,t.

# $ # " $

‰

FIe(V\X1^cM

a

FEhXCVŒ ^_V\DGFDGFdRY^cPChR`xQWF[xh_V]e(Fq^_^_Fd^

h^ij(s), s∈[t, t+δ], i= 1, ..., q, j∈N ^_PxeA+TUAxRT

h^ij(s)^{MO^} Ft

D<FqR\^_PChR`CQOFžR\X aZa

FEXxV`T_F

khk²t := 1 δ

Xq i=1

X∞ j=1

Z t+δ t

h^ij(s)²ds.