A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4

arXiv:0806.2248v2 [math.PR] 3 Oct 2008

motion of Hurst index bigger or equal to

1/4

by Ivan Nourdin

∗

Université Paris VI

Thisversion: O tober2,2008

Abstra t: Weprovea hange ofvariableformulaforthe2Dfra tionalBrownianmotionofindex

H

^{biggerorequalto}

1/4

^{. For}

H

^{stri tlybiggerthan}

1/4

^{,ourformula oin ideswiththatobtained}

by usingtheroughpaths theory. For

H = 1/4

^(themore interesting ase),there is anadditional term thatis a lassi al Wienerintegralagainstan independent standard Brownian motion.

Key words: Fra tional Brownian motion; weak onvergen e; hange ofvariableformula.

2000 Mathemati s Subje t Classi ation: 60F05,60H05, 60G15,60H07.

1 Introdu tion and main result

In[4℄,CoutinandQianhaveshownthattheroughpathstheoryofLyons[13℄ anbeappliedtothe

2Dfra tionalBrownian motion

B = (B ⁽¹⁾ , B ⁽²⁾ )

^{underthe ondition thatitsHurst parameter}

H

(supposed tobethe same for thetwo omponents) is stri tlybigger than

1/4

^{. Sin ethis seminal}

work,several authors have re overed this fa tby usingdierent routes (see e.g. Feyeland de La

Pradelle [7℄, Friz and Vi toir [8℄ or Unterberger [19 ℄ to ite but a few). On theother hand, it is

still anopen problemto bypassthis restri tion on

H

^.

Rough paths theory is purely deterministi in essen e. A tually, its random aspe t omes

only when it is applied to a single path of a given sto hasti pro ess (like a Brownian motion,

a fra tional Brownian motion, et .). In parti ular, itdoes not allow to produ e a new alea. As

su h,these ondpointofTheorem1.2justbelowshows,in asense,thatitseemsdi ulttorea h

the ase

H = 1/4

^byusingex lusively thetoolsof rough pathstheory.

Before stating our main result, we need some preliminaries. Let

W

^{be a standard (1D)}

Brownianmotion,independentof

B

^{. Weassumethat}

B

^and

W

^{aredenedonthesame}probability spa e

(Ω, F , P )

^with

F = σ{B} ∨ σ{W }

^{. Let}

(X _n )

^{be a sequen e of}

σ{B}

-measurable random variables,andlet

X

^bea

F

-measurablerandom variable. Inthesequel,wewill write

X _n ^stably −→ X

if

(Z, X _n ) −→ (Z, X) ^law

^{for allbounded and}

σ{B}

-measurablerandom variable

Z

^{. In}parti ular,we seethat thestable onvergen e imply the onvergen e inlaw. Moreover,it iseasily he ked that

the onvergen e in probability impliesthe stable onvergen e. We refer to [11 ℄ for an exhaustive

study ofthis notion.

Now, letus introdu e thefollowing obje t:

∗

Laboratoire de Probabilités et ModèlesAléatoires, Université Pierre et Marie Curie, Boîte ourrier

188,4Pla eJussieu,75252ParisCedex5,Fran e,ivan.nourdinupm .fr

Denition 1.1 Let

f : R ² → R

^{be a} ontinuously dierentiable fun tion, and x a time

t > 0

^.

Provided it exists,we dene

R t

0 ∇f(B s ) · dB s

^{to be the limitin} probability, as

n → ∞

^,of

I n (t) :=

⌊nt⌋−1 X

k=0

∂f

∂x (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) + ^∂f _∂x (B ⁽¹⁾ _(k+1)/n , B _k/n ⁽²⁾ )

2 B _(k+1)/n ⁽¹⁾ − B _k/n ⁽¹⁾ )

+

⌊nt⌋−1 X

k=0

∂f

∂y (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) + ^∂f _∂y (B _k/n ⁽¹⁾ , B _(k+1)/n ⁽²⁾ )

2 B _(k+1)/n ⁽²⁾ − B _k/n ⁽²⁾ ).

^(1.1)

If

I _n (t)

^{dened by (1.1) does not onverge in} probability but onverges stably, we denote the limit by

R t

0 ∇f(B s ) · d ^⋆ B _s

^.

Ourmain resultis asfollows:

Theorem 1.2 Let

f : R ² → R

^{beafun tionbelongingto}

C ⁸

andverifying

(H 8 )

^{,see(3.15)below.}

Let also

B = (B ⁽¹⁾ , B ⁽²⁾ )

^{denotea 2Dfra tional Brownianmotionof Hurstindex}

H ∈ (0, 1)

^{, and}

t > 0

^{be a xed time.}

1. If

H > 1/4

^then

R t

0 ∇f(B s ) · dB s

^iswell-dened, andwe have

f (B _t ) = f (0) +

Z _t

0 ∇f(B s ) · dB s .

^(1.2)

2. If

H = 1/4

^{then only}

R t

0 ∇f(B ^s ) · d ^⋆ B s

^iswell-dened, and we have

f (B _t ) ^Law = f (0) +

Z t

0 ∇f(B s ) · d ^⋆ B _s + σ _1/4

√ 2

Z t

0

∂ ² f

∂x∂y (B _s )dW _s .

^(1.3)

Here,

σ _1/4

^{isthe universal onstant dened below by(1.5), and}

R t

0 ∂ ² f / ∂x∂y (B _s )dW _s

^denotes

a lassi al Wiener integral withrespe t to the independent Brownian motion

W

^.

3. If

H < 1/4

^{thentheintegral}

R t

0 B _s · d ^⋆ B _s

^{doesnotexist. Therefore, it isnotpossibletowrite}

a hange of variable formulafor

B ⁽¹⁾ _t B _t ⁽²⁾

^{using the integral dened in Denition1.1.}

Remark 1.3 1. Due to the denition of the stable onvergen e, we an freely move ea h

omponent in(1.3 ) from the right hand side to theleft (or from the left hand side to the

right).

2. Whenever

β

^{denotes a} one-dimensional fra tional Brownian motion with Hurst index in

(0, 1/2)

^{, it is easily he ked, for any xed}

t > 0

^{, that}

P _⌊nt⌋−1

k=0 β _k/n β _(k+1)/n − β k/n

does

not onvergeinlaw. Indeed, on onehand, we have

β _⌊nt⌋/t ² =

⌊nt⌋−1 X

k=0

β _(k+1)/n ² − β k/n ²

= 2

⌊nt⌋−1 X

k=0

β _k/n β _(k+1)/n − β k/n

+

⌊nt⌋−1 X

k=0

β _(k+1)/n − β k/n

2

and,on the otherhand,it iswell-known (seee.g. [12 ℄)that

n ^2H−1

⌊nt⌋−1 X

k=0

β _(k+1)/n − β k/n

2 L ²

n→∞ −→ t.

⌊nt⌋−1 X

k=0

β _k/n β _(k+1)/n − β k/n

= 1

2



β _⌊nt⌋/t ² −

⌊nt⌋−1 X

k=0

β _(k+1)/n − β k/n

2





doesnot onverge inlaw

. Ontheother hand,whenever

H > 1/6

^,thequantity

⌊nt⌋−1 X

k=0

1

2 f (β _k/n ) + f (β _(k+1)/n )

β _(k+1)/n − β k/n

onverges in

L ²

^{for anyregular enough fun tion}

f : R → R

^{,see [9℄ and [3℄. This last fa t}

roughlyexplains whythere isa symmetri  part intheRiemannsum (1.1).

3. Westressthatitisstillanopenproblemtoknowifea hindividualintegral

R t

0

∂f

∂x (B _s )d ^(⋆) B _s ⁽¹⁾

and

R t

0

∂f

∂y (B _s )d ^(⋆) B s ⁽²⁾

^{ouldbe dened separately. Indeed, inthersttwo pointsof Theo-}

rem 1.2,we only prove thattheir sum, thatis

R t

0 ∇f(B s ) · d ^(⋆) B _s

^,iswell-dened.

4. Let us give a qui ker proof of (1.3 ) in the parti ular ase where

f (x, y) = xy

^{. Let}

β

^be

a one-dimensional fra tional Brownian motion of index

1/4

^{. The lassi al} Breuer-Major's theorem[1 ℄ yields:

√ 1

n

⌊n·⌋−1 X

k=0

√ n(β _(k+1)/n −β k/n ) ² −1
Law

= 1

√ n

⌊n·⌋−1 X

k=0

(β _k+1 −β k ) ² −1
stably

n→∞ −→ σ _1/4 W.

^(1.4)

Here, the onvergen e is stable and holds in the Skorohod spa e

D

of àdlàg fun tions on

[0, ∞)

^{. Moreover,}

W

^{still denotes a standard Brownian motion} independent of

β

^(the

independen eisa onsequen eofthe entrallimit theoremformultiple sto hasti integrals

proved in[18 ℄)and the onstant

σ _1/4

^{isgiven by}

σ _1/4 :=

s 1

2

X

k∈Z

p |k + 1| + p

|k − 1| − 2 p

|k|  2

< ∞.

^(1.5)

Now, let

β e

^{be another fra tional Brownian motion of index}

1/4

^, independent of

β

^{. From}

(1.4 ),we get



 1

√ n

⌊nt⌋−1 X

k=0

√ n(β _(k+1)/n − β k/n ) ² − 1

, 1

√ n

⌊nt⌋−1 X

k=0

√ n( e β _(k+1)/n − e β _k/n ) ² − 1




stably

n→∞ −→ σ _1/4 (W, f W )

for

(W, f W )

^{a 2D standard Brownian motion,} independent of the 2D fra tional Brownian motion

(β, e β)

^{. In}parti ular, by dieren e,we have

1

2

⌊n·⌋−1 X

k=0

(β _(k+1)/n − β k/n ) ² − (e β _(k+1)/n − e β _k/n ) ²
stably

n→∞ −→

σ _1/4

2 (W − f W ) ^Law = σ _1/4

√ 2 W.

Now,set

B ⁽¹⁾ = (β + e β)/ √

2

^and

B ⁽²⁾ = (β − e β)/ √

2

^{. Itiseasily he ked that}

B ⁽¹⁾

^and

B ⁽²⁾

are two independent fra tional Brownian motions of index

1/4

^{. Moreover, we an rewrite}

theprevious onvergen e as

⌊n·⌋−1 X

k=0

(B _(k+1)/n ⁽¹⁾ − B ⁽¹⁾ _k/n )(B _(k+1)/n ⁽²⁾ − B _k/n ⁽²⁾ ) ^stably −→

n→∞

σ _1/4

√ 2 W,

^(1.6)

with

B ⁽¹⁾

^,

B ⁽²⁾

^and

W

independent. Onthe other hand,for any

a, b, c, d ∈ R

^:

bd − ac = a(d − c) + c(b − a) + (b − a)(d − c).

Choosing

a = B ⁽¹⁾ _k/n

^,

b = B ⁽¹⁾ _(k+1)/n

^,

c = B _k/n ⁽²⁾

^and

d = B _(k+1)/n ⁽²⁾

^{, and suming for}

k

^over

0, . . . , ⌊nt⌋ − 1

^{,we obtain}

B _⌊nt⌋/n ⁽¹⁾ B _⌊nt⌋/n ⁽²⁾ =

⌊nt⌋−1 X

k=0

B _k/n ⁽¹⁾ B _(k+1)/n ⁽²⁾ − B _k/n ⁽²⁾

+ B _k/n ⁽²⁾ B _(k+1)/n ⁽¹⁾ − B _k/n ⁽¹⁾

+

⌊nt⌋−1 X

k=0

B _(k+1)/n ⁽¹⁾ − B _k/n ⁽¹⁾

B _(k+1)/n ⁽²⁾ − B _k/n ⁽²⁾

.

^(1.7)

Hen e, passingto thelimit using (1.6 ), we get thedesired on lusion in(1.3), in the par-

ti ular ase where

f (x, y) = xy

^{. Note that these ond term inthe right-hand side of (1.7 )}

isthedis rete analogue ofthe2- ovariation introdu ed byErramiand Russoin[6℄.

5. We ouldprove(1.3 )atafun tionallevel(notethatithaspre iselybeendonefor

f (x, y) =

xy

^{intheproofjustbelow). But,inordertokeep thelength ofthispaperwithinlimits, we}

deferto futureanalysis thisrather te hni al investigation.

6. In the very re ent work [16℄, Réveilla and I proved the following result (see also Burdzy

and Swanson [2℄ for similar results in the ase where

β

^{is repla ed by the solution of the}

sto hasti heatequationdrivenbyaspa e/timewhitenoise). If

β

^denotesaone-dimensional fra tional Brownian motionof index

1/4

^andif

g : R → R

^{isregular enough, then}

√ 1

n

n−1 X

k=0

g(β _k/n ) √

n(β _(k+1)/n −β k/n ) ² −1
stably

n→∞ −→

1

4

Z ₁

0

g ^′′ (β _s )ds + σ _1/4

Z ₁

0

g(β _s )dW _s

^(1.8)

for

W

^{astandardBrownianmotion}independent of

β

^{. Comparewith}Proposition3.3below.

Inparti ular, by hoosing

g

identi ally one in(1.8 ),itagrees with(1.4).

7. The fra tional Brownian motion of index

1/4

^{has a remarkable physi al} interpretation in terms of parti le systems. Indeed, if one onsider an innitenumber of parti les, initially

pla edonthereallinea ordingtoaPoissondistribution,performingindependentBrownian

motions and undergoing elasti  ollisions, then the traje tory of a xed parti le (after

res aling) onverges to a fra tional Brownian motion of index

1/4

^{. See Harris [10℄ for}

heuristi arguments, andDürr,Goldstein andLebowitz[5 ℄ for pre iseresults.

Now, the rest of the note is entirely devoted to the proof of Theorem 1.2 . The Se tion 2

ontainssome preliminariesandxthe notation. Somete hni al resultsarepostponedinSe tion

3. Finally,the proof ofTheorem 1.2is done inSe tion4.

We shallnow providea shortdes ription ofthe toolsof Malliavin al ulus thatwill be neededin

thefollowing se tions. Thereaderisreferredtothemonographs[14℄and [17℄foranyunexplained

notion or result.

Let

B = (B _t ⁽¹⁾ , B _t ⁽²⁾ ) _{t∈[0,T ]}

^{bea2Dfra tionalBrownianmotionwithHurstparameterbelonging}

to

(0, 1/2)

^{. We denoteby}

H

^{the Hilbertspa e dened asthe losureof thesetof step}

R ²

^-valued

fun tionson

[0, T ]

^{,withrespe tto thes alarprodu tindu edby}

1 [0,t 1 ] , 1 [0,t 2 ]

, 1 [0,s 1 ] , 1 [0,s 2 ]



H = R _H (t ₁ , s ₁ ) + R _H (t ₂ , s ₂ ), s _i , t _i ∈ [0, T ], i = 1, 2,

where

R H (t, s) = ¹ ₂ t ^2H + s ^2H − |t − s| ^2H

. The mapping

(1 [0,t 1 ] , 1 [0,t 2 ] ) 7→ B t ⁽¹⁾ 1 + B _t ⁽²⁾ ₂

^{an be}

extended to an isometry between

H

^{and the Gaussian spa e asso iated with}

B

^{. Also,}

H

^will

denotetheHilbertspa edenedasthe losureofthesetofstep

R

^{-valuedfun tionson}

[0, T ]

^,with

respe tto thes alarprodu tindu edby

1 [0,t] , 1 [0,s]



H = R _H (t, s), s, t ∈ [0, T ].

The mapping

1 [0,t] 7→ B t ⁽ⁱ⁾

⁽

i

^equals

1

^or

2

^{) an be extendedto an isometry between}

H

^{and the}

Gaussian spa easso iated with

B ⁽ⁱ⁾

^.

Consider the set ofall smooth ylindri al random variables,i.e. of theform

F = f B(ϕ ₁ ), . . . , B(ϕ _k )

, ϕ _i ∈ H, i = 1, . . . , k,

^(2.9)

where

f ∈ C ^∞

^{is bounded with bounded} derivatives. The derivative operator

D

^{of a smooth}

ylindri al random variableofthe aboveform isdened asthe

H

^{-valued randomvariable}

DF =

X k

i=1

∂f

∂x _i B(ϕ ₁ ), . . . , B(ϕ _k )

ϕ _i =: D _B (1) F, D _B (2) F

.

Inparti ular, we have

D _B (i) B _t ^(j) = δ _ij 1 [0,t]

^for

i, j ∈ {1, 2}

^,and

δ _ij

^{theKrone ker symbol.}

Byiteration,one andenethe

m

^thderivative

D ^m F

^{(whi hisasymmetri elementof}

L ² (Ω, H ^⊗m ))

for

m > 2

^{. As usual, for any}

m > 1

^{, the spa e}

D ^m,2

^{denotes the losure of the set of smooth}

random variables withrespe tto the norm

k · k ^m,2

^{dened bytherelation}

kF k ² m,2 = E|F | ² +

X m

i=1

EkD ⁱ F k ² _H ^⊗i .

The derivative

D

^{veries the hain rule. Pre isely, if}

ϕ : R ⁿ → R

^{belongs to}

C ¹

withbounded derivativesand if

F _i

^,

i = 1, . . . , n

^,arein

D ^1,2

^,then

ϕ(F ₁ , . . . , F _n ) ∈ D ^1,2

^and

Dϕ(F ₁ , . . . , F _n ) =

X n

i=1

∂ϕ

∂x _i (F ₁ , . . . , F _n )DF _i .

The

m

^thderivative

D _B ^m _(i)

⁽

i

^equals

1

^or

2

^{)veriesthe followingLeibnitzrule: forany}

F, G ∈ D ^m,2

su h that

F G ∈ D ^m,2

^{,we have}

D _B ^m _(i) F G

t 1 ,...,t m = X

D _B ^r _(i) F

s 1 ,...,s r D ^m−r

B ⁽ⁱ⁾ G

u 1 ,...,u m−r , t _i ∈ [0, T ], i = 1, . . . , m,

^(2.10)

where thesumrunsoveranysubset

{s 1 , . . . , s _r } ⊂ {t 1 , . . . , t _m }

^{andwhere we write}

{t 1 , . . . , t _m } \

{s 1 , . . . , s _r } =: {u 1 , . . . , u _m−r }

^.

The divergen e operator

δ

^{is the adjoint of the derivative operator. If a random variable}

u ∈ L ² (Ω, H)

^belongsto

domδ

^{,the domainofthedivergen eoperator,then}

δ(u)

^{isdenedbythe}

dualityrelationship

E F δ(u)

= EhDF, ui H

for every

F ∈ D ^1,2

^.

For every

q > 1

^,let

H q

^{be the}

q

^{th Wiener haos of}

B

^{,that is, the losed linear subspa e of}

L ² (Ω, A, P )

^{generated bythe random variables}

{H q (B (h)) , h ∈ H, khk H = 1}

^{, where}

H _q

^{is the}

q

^{th Hermitepolynomial given by}

H _q (x) = (−1) ^q e ^{x 2 /2 d} _dx ^q q e ^{−x 2 /2}

. The mapping

I _q (h ^⊗q ) = H _q (B (h))

^(2.11)

providesalinearisometrybetweenthesymmetri tensorprodu t

H ^⊙q

^{(equippedwiththemodied}

norm

√ 1

q! k · k _H ^⊗q

^{) and}

H q

^{. Thefollowing dualityformula holds}

E (F I _q (f )) = E (hD ^q F, f i _H ^⊗q ) ,

^(2.12)

for any

f ∈ H ^⊙q

^and

F ∈ D ^q,2

^{. In} parti ular,we have

E 

F I _q ⁽ⁱ⁾ (g) 

= E D

D _B ^q _(i) F, g E

H ^⊗q

 , i = 1, 2,

^(2.13)

for any

g ∈ H ^⊙q

^and

F ∈ D ^q,2

^{,where,for simpli ity, we write}

I _q ⁽ⁱ⁾ (g)

^wheneverthe orresponding

q

^{th multipleintegral isonly withrespe tto}

B ⁽ⁱ⁾

^.

Finally, we mention the following parti ular ase (a tually, theonly one we will need in the

sequel) ofthe lassi al multipli ationformula: if

f, g ∈ H

^,

q > 1

^and

i ∈ {1, 2}

^,then

I _q ⁽ⁱ⁾ (f ^⊗q )I _q ⁽ⁱ⁾ (g ^⊗q ) =

X q

r=0

r!

 q

r

 2

I _2q−2r ⁽ⁱ⁾ (f ^⊗q−r ⊗ g ^⊗q−r )hf, gi ^r H .

^(2.14)

3 Some te hni al results

Inthisse tion,we olle tsome ru ialresultsfortheproofof(1.3 ),theonly asewhi hisdi ult.

Here andinthe rest of the paper, we set

∆B _k/n ⁽ⁱ⁾ := B _(k+1)/n ⁽ⁱ⁾ − B _k/n ⁽ⁱ⁾ , δ _k/n := 1 [k/n,(k+1)/n]

^and

ε _k/n := 1 [0,k/n] ,

for any

i ∈ {1, 2}

^and

k ∈ {0, . . . , n − 1}

^.

In thesequel,for

g : R ² → R

^belongingto

C ^q

,we will need assumption ofthe type:

(H q ) sup

s∈[0,1]

E

 

  ∂ ^a+b g

∂x ^a ∂y ^b (B _s ⁽¹⁾ , B _s ⁽²⁾ )

 

 

p

< ∞

^forall

p > 1

^{andall integers}

a, b > 0

^s.t.

a + b 6 q.

Webeginbythe followingte hni al lemma:

Lemma 3.1 Let

β

^{be a 1Dfra tional Brownian motionof Hurst index}

1/4

^{. We have}

(i)

 E β r (β _t − β s )
 6 p

|t − s|

^{for any}

0 6 r, s, t 6 1

^,

(ii)

n−1 X

k,l=0

 



ε _l/n , δ _k/n 

H

 

 = _n→∞ O(n),

(iii)

n−1 X

k,l=0

 



δ _l/n , δ _k/n 

H

 

 ^r =

n→∞ O(n ^1−r/2 )

^{for any}

r > 1

^,

(iv)

n−1 X

k=0

 

 

ε _k/n , δ _k/n 

H + 1

2 √ n

 

  = _n→∞ O(1)

^,

(v)

n−1 X

k=0

 

 

ε _k/n , δ _k/n  2

H − 1

4n

 

  = _n→∞ O(1/ √

n)

^.

Proof of Lemma 3.1.

(i) We have

E β _r (β _t − β s )

= 1

2

√ t − √

s

+ 1

2

p |s − r| − p

|t − r| 

.

Using the lassi alinequality

  p

|b| − p

|a| 

 ≤ p

|b − a|

^{,thedesiredresult follows.}

(ii) Observethat

ε _l/n , δ _k/n 

H = 1

2 √

n

√ k + 1 − √

k − p

|k + 1 − l| + p

|k − l| 

.

Consequently,for any xed

l ∈ {0, . . . , n − 1}

^{,we have}

n−1 X

k=0

 



ε _l/n , δ _k/n 

H

 

 ≤ 1

2 + 1

2 √

n

X l−1

k=0

√ l − k − √

l − k − 1

+1 +

n−1 X

k=l+1

√ k − l + 1 − √

k − l

!

= 1

2 + 1

2 √

n

√ l + √

n − l

fromwhi h we dedu ethat

sup

06l6n−1

n−1 X

k=0

 



ε _l/n , δ _k/n 

H

 

 = n→∞ O(1)

^{. It follows that}

n−1 X

k,l=0

 



ε _l/n , δ _k/n 

H

 

 6 n sup

06l6n−1

n−1 X

k=0

 



ε _l/n , δ _k/n 

H

 

 = _n→∞ O(n).

(iii) We have,by noting

ρ(x) = ¹ ₂ p

|x + 1| + p

|x − 1| − 2 p

|x|

:

n−1 X

k,l=0

 



δ _l/n , δ _k/n 

H

 

 ^r = n ^−r/2

n−1 X

k,l=0

|ρ ^r (l − k)| 6 n ^1−r/2 X

k∈Z

|ρ ^r (k)| .

Sin e

P

k∈Z |ρ ^r (k)| < ∞

^if

r > 1

^{,the desired on lusion follows.}

(iv) isa onsequen e ofthe following identity ombined withateles opi sumargument:

 

 

ε _k/n , δ _k/n 

H + 1

2 √

n

 

  = 1

2 √

n

√ k + 1 − √

k

.

(v) We have

 

 

ε _k/n , δ _k/n  2

H − 1

4n

 

  =

1

4n

√ k + 1 − √

k    √

k + 1 − √

k − 2    .

Thus,the desiredboundisimmediately he ked by ombininga teles opingsumargument

withthefa tthat

 

 √

k + 1 − √

k − 2    =

 

 

√ 1

k + 1 + √

k − 2

 

  6 2.

✷

Also thefollowing lemmawill beuseful inthesequel:

Lemma 3.2 Let

α > 0

^and

q > 2

^{be two positive integers,}

g : R ² → R

^{be any fun tion belonging}

to

C ^2q

andverifying

(H 2q )

^{dened by (3.15), and}

B = (B ⁽¹⁾ , B ⁽²⁾ )

^{be a 2Dfra tional Brownian}

motion of Hurst index

1/4

^{. Set}

V n = n ^−q/4

n−1 X

k=0

g(B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B _k/n ⁽¹⁾
α

H q n ^1/4 ∆B _k/n ⁽²⁾

,

where

H _q

^{denotes the}

q

^{th Hermite polynomial dened by}

H _q (x) = (−1) ^q e ^{x 2 /2 d} _dx ^q q e ^{−x 2 /2}

. Then,

the following bound isin order:

E |V n | ²

= O(n ^{1−q/2−α/2} )

^as

n → ∞.

^(3.16)

E |V n | ²

= n ^−q/2

n−1 X

k,l=0

E 

g(B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )g(B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) ∆B _k/n ⁽¹⁾
α

∆B _l/n ⁽¹⁾
α

×H q n ^1/4 ∆B _k/n ⁽²⁾

H _q n ^1/4 ∆B _l/n ⁽²⁾


=

(

^2.11

)

n−1 X

k,l=0

E 

g(B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )g(B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) ∆B _k/n ⁽¹⁾
α

∆B _l/n ⁽¹⁾
α

I _q ⁽²⁾ (δ _k/n ^⊗q )I _q ⁽²⁾ (δ ^⊗q _l/n ) 

=

(

^2.14

)

X q

r=0

r!

 q

r

 2 n−1 X

k,l=0

E 

g(B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )g(B ⁽¹⁾ _l/n , B _l/n ⁽²⁾ )

× ∆B _k/n ⁽¹⁾
α

∆B _l/n ⁽¹⁾
α

I _2q−2r ⁽²⁾ (δ _k/n ^⊗q−r ⊗ δ ^⊗q−r _l/n ) 

hδ k/n , δ _l/n i ^r H

=

(

^2.13

)

X q

r=0

r!

 q

r

 2 n−1 X

k,l=0

E D

D ^2q−2r _{B (2)} 

g(B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )g(B _l/n ⁽¹⁾ , B ⁽²⁾ _l/n )

× ∆B _k/n ⁽¹⁾
α

∆B _l/n ⁽¹⁾
α 

, δ _k/n ^⊗q−r ⊗ δ ^⊗q−r _l/n E

H ^⊗2q−2r hδ k/n , δ _l/n i ^r H

=

(

^2.10

)

X q

r=0

r!

 q

r

 2 X

a+b=2q−2r

(a + b)!

a!b!

n−1 X

k,l=0

E

 d ^a g

dy ^a (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) d ^b g

dy ^b (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )

× ∆B ⁽¹⁾ _k/n
α

∆B _l/n ⁽¹⁾
α 

(2q − 2r)! D

ε ^⊗a _k/n ⊗ε e ^⊗b _l/n , δ _k/n ^⊗q−r ⊗ δ _l/n ^⊗q−r E

H ^⊗2q−2r hδ k/n , δ _l/n i ^r H .

(3.17)

Now, observe that,uniformlyin

k, l ∈ {0, . . . , n − 1}

^:

D ε ^⊗a _k/n ⊗ε e ^⊗b _l/n , δ ^⊗q−r _k/n ⊗ δ _l/n ^⊗q−r E

H ^⊗2q−2r =

n→∞ O(n ^−(q−r) ),

^{see Lemma 3.1(i)}

,

 

 E

 d ^a g

dy ^a (B _k/n ⁽¹⁾ , B ⁽²⁾ _k/n ) d ^b g

dy ^b (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) ∆B _k/n ⁽¹⁾
α

∆B _l/n ⁽¹⁾
α    = _n→∞ O(n ^−α/2 ),

^use

(H 2q ),

and,also:

n−1 X

k,l=0

hδ k/n , δ _l/n i ^r H = O(n ^1−r/2 )

^{for anyxed}

r > 1

^{,seeLemma 3.1(iii)}

.

Finally,thedesired on lusionisobtainedbypluggingthesethreeboundsinto(3.17 ),afterhaving

separatedthe ases

r = 0

^and

r = 1

^.

✷

The independent Brownian motion appearing in(1.3) omes fromthefollowing proposition.

Proposition 3.3 Let

(β, e β)

^{be a 2D fra tional Brownian motion of Hurst index}

1/4

^{. Consider}

two fun tions

g, eg : R ² → R

^{belonging in}

C ⁴

, and assume that they both verify

(H 4 )

^{dened by}

(3.15). Then

(G n , e G n ) := 1

√ n

n−1 X

k=0

g(β _k/n , e β _k/n ) √

n(∆β _k/n ) ² − 1

, 1

√ n

n−1 X

k=0

eg(β k/n , e β _k/n ) √

n(∆ e β _k/n ) ² − 1
!

stably

n→∞ −→



σ _1/4

Z 1

0

g(β s , e β s )dW s + 1

4

Z 1

0

∂ ² g

∂x ² (β s , e β s )ds, σ _1/4

Z 1

0 eg(β ^s , e β s )df W s + 1

4

Z 1

0

∂ ² eg

∂y ² (β s , e β s )ds



,

where

(W, f W )

^{is a 2D standard Brownian motion} independent of

(β, e β)

^{, and}

σ _1/4

^{is dened by}

(1.5).

In theparti ular ase where

g(x, y) = g(x)

^and

eg(x, y) = eg(y)

^{, the on lusion of the}proposition followsdire tlyfrom(1.8 ). Inthe general ase, theproofonly onsiststoextendliteralytheproof

of (1.8 ) ontained in[16℄. Detailsareleft to the reader.

4 Proof of Theorem 1.2

We arenowinpositionto prove our mainresult, thatis Theorem 1.2.

Proof of the third point ( ase

H < 1/4

^{). Firstly, observe that (1.4 ) is a tually a parti u-}

lar aseof the following result, whi h is valid for anyfra tional Brownian

β

^{with Hurstindex}

H

belongingto

(0, 3/4)

^:

√ 1

n

⌊n·⌋−1 X

k=0

n ^2H (∆β _k/n ) ² − 1
stably

n→∞ −→ σ _H W

with

W

^an independent Brownian motionand

σ _H > 0

^{an (expli it) onstant. By mimi king the}

proof ontained inthe fourthpoint ofRemark 1.3,weget, here, for any

H ∈ (0, 3/4)

^,

n ^2H−1/2

⌊n·⌋−1 X

k=0

∆B _k/n ⁽¹⁾ ∆B _k/n ^{(2) stably} −→

n→∞

σ _H

√ 2 W.

^(4.18)

But, see(1.7), theexisten e of

R _·

0 B s · d ^⋆ B s

^{would implyinparti ular that}

P _{⌊n·⌋−1}

k=0 ∆B _k/n ⁽¹⁾ ∆B _k/n ⁽²⁾

onverges inlaw as

n → ∞

^{,whi h isin} ontradi tion with(4.18 )for

H < 1/4

^{. Theproof of the}

third point is done.

Proof of the se ond point ( ase

H = 1/4

^{). For the simpli ity of the} exposition, we assume from now that

t = 1

^{, the general ase being of ourse similar up to umbersome notation. For}

any

a, b, c, d ∈ R

^{,bythe lassi al Taylorformula, we an expand}

f (b, d)

^{as( omparewith(1.7 )):}

f (a, c) + ∂ ₁ f (a, c)(b − a) + ∂ 2 f (a, c)(d − c) + 1

2 ∂ ₁₁ f (a, c)(b − a) ² + 1

2 ∂ ₂₂ f (a, c)(d − c) ²

+ 1

6 ∂ ₁₁₁ f (a, c)(b − a) ³ + 1

6 ∂ ₂₂₂ f (a, c)(d − c) ³ + 1

24 ∂ ₁₁₁₁ f (a, c)(b − a) ⁴ + 1

24 ∂ ₂₂₂₂ f (a, c)(d − c) ⁴

+ ∂ ₁₂ f (a, c)(b − a)(d − c) + 1

2 ∂ ₁₁₂ f (a, c)(b − a) ² (d − c) + 1

2 ∂ ₁₂₂ f (a, c)(b − a)(d − c) ²

+ 1

6 ∂ ₁₁₁₂ f (a, c)(b − a) ³ (d − c) + 1

4 ∂ ₁₁₂₂ f (a, c)(b − a) ² (d − c) ² + 1

6 ∂ ₁₂₂₂ f (a, c)(b − a)(d − c) ³

(4.19)

plus a remainder term. Here, as usual, the notation

∂ _1...12...2 f

^{(where theindex 1 is repeated}

k

times and the index 2 is repeated

l

^{times) means that}

f

^isdierentiated

k

^{times w.r.t. therst}

omponent and

l

^{times w.r.t. the se ond one. By ombining (4.19 ) with the following identity,}

available for any

h : R → R

^belongingto

C ⁴

:

h ^′ (a)(b − a) + 1

2 h ^′′ (a)(b − a) ² + 1

6 h ^′′′ (a)(b − a) ³ + 1

24 h ^′′′′ (a)(b − a) ⁴

= h ^′ (a) + h ^′ (b)

2 (b − a) − 1

12 h ^′′′ (a)(b − a) ³ − 1

24 h ^′′′′ (a)(b − a) ⁴ +

^{some remainder}

we getthat

f (b, d)

^{an also be expandedas}

f (a, c) + 1

2 ∂ ₁ f (a, c) + ∂ ₁ f (b, c)

(b − a) − 1

12 ∂ ₁₁₁ f (a, c)(b − a) ³ − 1

24 ∂ ₁₁₁₁ f (a, c)(b − a) ⁴

+ 1

2 ∂ ₂ f (a, c) + ∂ ₂ f (a, d)

(d − c) − 1

12 ∂ ₂₂₂ f (a, c)(d − c) ³ − 1

24 ∂ ₂₂₂₂ f (a, c)(d − c) ⁴

+ ∂ ₁₂ f (a, c)(b − a)(d − c) + 1

2 ∂ ₁₁₂ f (a, c)(b − a) ² (d − c) + 1

2 ∂ ₁₂₂ f (a, c)(b − a)(d − c) ²

+ 1

6 ∂ ₁₁₁₂ f (a, c)(b − a) ³ (d − c) + 1

4 ∂ ₁₁₂₂ f (a, c)(b − a) ² (d − c) ² + 1

6 ∂ ₁₂₂₂ f (a, c)(b − a)(d − c) ³

(4.20)

plusa remainder term.

By setting

a = B _k/n ⁽¹⁾

^,

b = B _(k+1)/n ⁽¹⁾

^,

c = B _k/n ⁽²⁾

^and

d = B _(k+1)/n ⁽²⁾

^{in(4.20 ), and bysuming the}

obtained expression for

k

^over

0, . . . , n − 1

^{, we dedu e that the on lusion in Theorem 1.2 is a}

onsequen e ofthe following onvergen es:

S _n ⁽¹⁾ :=

n−1 X

k=0

∂ ₁₁₁ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B _k/n ⁽¹⁾
3 L ²

n→∞ −→ − 3

2

Z 1

0

∂ ₁₁₁₁ f (B _s ⁽¹⁾ , B ⁽²⁾ _s )ds

^(4.21)

S _n ⁽²⁾ :=

n−1 X

k=0

∂ ₁₁₁₁ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B _k/n ⁽¹⁾
4 L ²

n→∞ −→ 3

Z 1

0

∂ ₁₁₁₁ f (B _s ⁽¹⁾ , B _s ⁽²⁾ )ds

^(4.22)

S _n ⁽³⁾ :=

n−1 X

k=0

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B _k/n ⁽²⁾
3 L ²

n→∞ −→ − 3

2

Z 1

0

∂ ₂₂₂₂ f (B _s ⁽¹⁾ , B ⁽²⁾ _s )ds

^(4.23)

S _n ⁽⁴⁾ :=

n−1 X

k=0

∂ ₂₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B _k/n ⁽²⁾
4 L ²

n→∞ −→ 3

Z 1

0

∂ ₂₂₂₂ f (B _s ⁽¹⁾ , B _s ⁽²⁾ )ds

^(4.24)

S _n ⁽⁵⁾ :=

n−1 X

k=0

∂ 12 f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∆B _k/n ⁽¹⁾ ∆B ⁽²⁾ _k/n ^stably −→

n→∞

σ _1/4

√ 2

Z 1

0

∂ 12 f (B _s ⁽¹⁾ , B _s ⁽²⁾ )dW s

+ 1

4

Z 1

0

∂ ₁₁₂₂ f (B ⁽¹⁾ _s , B _s ⁽²⁾ )ds

^(4.25)

S _n ⁽⁶⁾ :=

n−1 X

k=0

∂ ₁₁₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B _k/n ⁽¹⁾
2

∆B _k/n ⁽²⁾ −→ ^{L 2}

n→∞ − 1

2

Z 1

0

∂ ₁₁₂₂ f (B _s ⁽¹⁾ , B _s ⁽²⁾ )ds

^(4.26)

S _n ⁽⁷⁾ :=

n−1 X

k=0

∂ ₁₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∆B _k/n ⁽¹⁾ ∆B _k/n ⁽²⁾
2 L ²

n→∞ −→ − 1

2

Z 1

0

∂ ₁₁₂₂ f (B _s ⁽¹⁾ , B _s ⁽²⁾ )ds

^(4.27)

S _n ⁽⁸⁾ :=

n−1 X

k=0

∂ ₁₁₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B ⁽¹⁾ _k/n
2

∆B _k/n ⁽²⁾
2 L ²

n→∞ −→

Z 1

0

∂ ₁₁₂₂ f (B _s ⁽¹⁾ , B _s ⁽²⁾ )ds

^(4.28)

S _n ⁽⁹⁾ :=

n−1 X

k=0

∂ ₁₁₁₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B ⁽¹⁾ _k/n
3

∆B ⁽²⁾ _k/n ^Prob −→

n→∞ 0

^(4.29)

S _n ⁽¹⁰⁾ :=

n−1 X

k=0

∂ ₁₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∆B _k/n ⁽¹⁾ ∆B _k/n ⁽²⁾
3 Prob

n→∞ −→ 0.

^(4.30)

Notethattheterm orrespondingto theremainderin(4.20 ) onverges inprobabilityto zerodue

to the fa tthat

B

^{hasa nitequarti variation.}

Proofof (4.21), (4.23),(4.26) and(4.27). ByLemma 3.2with

q = 3

^and

α = 0

^,andbyusing

thebasi fa tthat

∆B ⁽²⁾ _k/n
3

= n ^−3/4 H ₃ (n ^1/4 ∆B _k/n ⁽²⁾ ) + 3

√ n ∆B ⁽²⁾ _k/n ,

^(4.31)

we immediately seethat(4.23 )is a onsequen e of thefollowing onvergen e:

√ 1 n

n−1 X

k=0

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∆B _k/n ⁽²⁾ −→ ^{L 2}

n→∞ − 1

2

Z 1

0

∂ ₂₂₂₂ f (B _s ⁽¹⁾ , B _s ⁽²⁾ )ds.

^(4.32)

So,let usprove (4.32 ). We have, onone hand:

E

 

 



√ 1 n

n−1 X

k=0

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∆B _k/n ⁽²⁾

 

 



2

= 1

n

n−1 X

k,l=0

E 

∂ 222 f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ 222 f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) ∆B _k/n ⁽²⁾ ∆B _l/n ⁽²⁾ 

= 1

n

n−1 X

k,l=0

E 

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ ₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) I ₂ ⁽²⁾ (δ _k/n ⊗ δ l/n ) 

+ 1

n

n−1 X

k,l=0

E 

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ ₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) 

hδ k/n , δ _l/n i ^H

= 1

n

n−1 X

k,l=0

E 

∂ ₂₂₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ ₂₂₂ f (B ⁽¹⁾ _l/n , B _l/n ⁽²⁾ ) 

hε k/n , δ _k/n i ^H hε k/n , δ _l/n i ^H

+ 1

n

n−1 X

k,l=0

E 

∂ 2222 f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ 2222 f (B ⁽¹⁾ _l/n , B _l/n ⁽²⁾ ) 

hε k/n , δ _k/n i ^H hε l/n , δ _l/n i ^H

+ 1

n

n−1 X

k,l=0

E 

∂ ₂₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ ₂₂₂₂ f (B ⁽¹⁾ _l/n , B _l/n ⁽²⁾ ) 

hε l/n , δ _k/n i ^H hε k/n , δ _l/n i ^H

+ 1

n

n−1 X

k,l=0

E 

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ ₂₂₂₂₂ f (B ⁽¹⁾ _l/n , B _l/n ⁽²⁾ ) 

hε l/n , δ _k/n i ^H hε l/n , δ _l/n i ^H

+ 1

n

n−1 X

k,l=0

E 

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ ₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) 

hδ k/n , δ _l/n i ^H

= a ⁽ⁿ⁾ + b ⁽ⁿ⁾ + c ⁽ⁿ⁾ + d ⁽ⁿ⁾ + e ⁽ⁿ⁾ .

Using Lemma 3.1 (i) and (ii), we have that

a ⁽ⁿ⁾

^,

c ⁽ⁿ⁾

^and

d ⁽ⁿ⁾

^{tends to zero as}

n → ∞

^{. Using}

Lemma 3.1(iii),wehave that

e ⁽ⁿ⁾

^{tendsto zeroas}

n → ∞

^{. Finally,observe that}

b ⁽ⁿ⁾ = 1

4n ²

n−1 X

k,l=0

E 

∂ 2222 f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ 2222 f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) 

− 1

2n √

n

n−1 X

k,l=0

E 

∂ ₂₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∂ ₂₂₂₂ f (B ⁽¹⁾ _l/n , B _l/n ⁽²⁾ )  

hε l/n , δ _l/n i ^H + 1

2 √

n



+ 1

n

n−1 X

k,l=0

E 

∂ ₂₂₂₂ f (B _k/n ⁽¹⁾ , B ⁽²⁾ _k/n ) ∂ ₂₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )  

hε k/n , δ _k/n i ^H + 1

2 √ n



hε l/n , δ _l/n i ^H .

Therefore,using Lemma 3.1(i) and(iv), we have

E

 

 



√ 1

n

n−1 X

k=0

∂ 222 f (B ⁽¹⁾ _k/n , B _k/n ⁽²⁾ )∆B _k/n ⁽²⁾

 

 



2

= E

 

 



1

2n

n−1 X

k=0

∂ 2222 f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )

 

 



2

+ o(1).

^(4.33)

Onthe otherhand,we have

E 1

√ n

n−1 X

k=0

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∆B _k/n ⁽²⁾ × −1

2n

n−1 X

l=0

∂ ₂₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )

!

= − 1

2n √

n

n−1 X

k,l=0

E ∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∂ ₂₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ ) ∆B _k/n ⁽²⁾

= − 1

2n √ n

n−1 X

k,l=0

E ∂ ₂₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∂ ₂₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )

hε k/n , δ _k/n i ^H

− 1

2n √

n

n−1 X

k,l=0

E ∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B ⁽²⁾ _k/n )∂ ₂₂₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )

hε l/n , δ _k/n i ^H

= 1

4n ²

n−1 X

k,l=0

E ∂ 2222 f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∂ 2222 f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )

− 1

2n √

n

n−1 X

k,l=0

E ∂ ₂₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∂ ₂₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )


hε k/n , δ _k/n i ^H + 1

2 √

n



− 1

2n √ n

n−1 X

k,l=0

E ∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B ⁽²⁾ _k/n )∂ ₂₂₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )

hε l/n , δ _k/n i ^H

termsintheprevious expressiontends to zeroas

n → ∞

^{. Thatis}

E 1

√ n

n−1 X

k=0

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∆B _k/n ⁽²⁾ × −1

2n

n−1 X

l=0

∂ ₂₂₂₂ f (B _l/n ⁽¹⁾ , B _l/n ⁽²⁾ )

!

= E

 

 



1

2n

n−1 X

k=0

∂ ₂₂₂₂ f (B _k/n ⁽¹⁾ , B ⁽²⁾ _k/n )

 

 



2

+ o(1).

^(4.34)

Wehave proved,see(4.33 ) and(4.34 ), that

E

 

 



√ 1

n

n−1 X

k=0

∂ ₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )∆B _k/n ⁽²⁾ + 1

2n

n−1 X

k=0

∂ ₂₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ )

 

 



2

n→∞ −→ 0.

Thisimplies(4.32 ).

The proof of (4.21 ) follows dire tly from (4.23 )by ex hanging the roles played by

B ⁽¹⁾

^and

B ⁽²⁾

^{. Ontheother hand,by ombining Lemma 3.2withthefollowing basi identity:}

∆B _k/n ⁽²⁾
2

= 1

√ n H 2 (n ^1/4 ∆B _k/n ⁽²⁾ ) + 1

√ n ,

weseethat(4.27 )isalsoadire t onsequen e of(4.32 ). Finally,(4.26 )isobtained from(4.27 )by

ex hangingthe rolesplayed by

B ⁽¹⁾

^and

B ⁽²⁾

^.

Proof of (4.22), (4.24) and(4.28). By ombining Lemma 3.2withthe identity

∆B _k/n ⁽¹⁾
4

= 1

n H ₄ (n ^1/4 ∆B _k/n ⁽¹⁾ ) + 6

n H ₂ (n ^1/4 ∆B _k/n ⁽¹⁾ ) + 3

n ,

we see that (4.24) is easily obtained through a Riemann sum argument. We an use the same

arguments in order to prove (4.22 ). Finally, to obtain (4.28 ), it su es to ombine Lemma 3.2

withtheidentity

∆B _k/n ⁽¹⁾
2

∆B _k/n ⁽²⁾
2

= 1

n + 1

√ n ∆B _k/n ⁽¹⁾
2

H ₂ (n ^1/4 ∆B _k/n ⁽²⁾ ) + 1

n H ₂ (n ^1/4 ∆B _k/n ⁽¹⁾ ).

Proof of (4.29) and (4.30). We only prove (4.30 ), the proof of (4.29 ) being obtained from

(4.30 )byex hangingthe rolesplayed by

B ⁽¹⁾

^and

B ⁽²⁾

^{. By ombining (4.31) withLemma 3.2 , it}

su esto prove that

√ 1

n

n−1 X

k=0

∂ ₁₂₂₂ f (B _k/n ⁽¹⁾ , B _k/n ⁽²⁾ ) ∆B _k/n ⁽¹⁾ ∆B _k/n ^{(2) Prob} −→

n→∞ 0.

Butthislast onvergen e followsdire tlyfromLemma3.3. Therefore,theproofof(4.30 )isdone.

thatwe have addressedjustafter the statement of Theorem 1.2. Indeed,wehave

n−1 X

k=0

∂ ₁₂ f B _k/n ⁽¹⁾ , B _k/n ⁽²⁾

∆B _k/n ⁽¹⁾ ∆B ⁽²⁾ _k/n

= 1

2 √

n

n−1 X

k=0

∂ ₁₂ f β _k/n + e β _k/n

√ 2 , β _k/n − e β _k/n

√ 2

√ n ∆β _k/n
2

− 1 

− 1

2 √

n

n−1 X

k=0

∂ ₁₂ f β _k/n + e β _k/n

√ 2 , β _k/n − e β _k/n

√ 2

√ n ∆ e β _k/n
2

− 1 

for

β = (B ⁽¹⁾ + B ⁽²⁾ )/ √

2

^and

β = (B e ⁽¹⁾ − B ⁽²⁾ )/ √

2

^{. Note that}

(β, e β)

^{is also a 2D fra tional}

Brownian motion of Hurst index

1/4

^{. Hen e, using} Proposition 3.3 with

g(x, y) = eg(x, y) =

f 

x+y √

2 , ^x−y √

2



,we get

n−1 X

k=0

∂ ₁₂ f B _k/n ⁽¹⁾ , B _k/n ⁽²⁾

∆B _k/n ⁽¹⁾ ∆B _k/n ⁽²⁾

stably

n→∞ −→

σ _1/4

2

Z 1

0

∂ 12 f (B _s ⁽¹⁾ , B _s ⁽²⁾ )d(W − f W ) s + 1

4

Z 1

0

∂ 1122 f (B _s ⁽¹⁾ , B _s ⁽²⁾ )ds

Law = σ _1/4

√ 2

Z ₁

0

∂ ₁₂ f (B _s ⁽¹⁾ , B _s ⁽²⁾ )dW _s + 1

4

Z ₁

0

∂ ₁₁₂₂ f (B _s ⁽¹⁾ , B ⁽²⁾ _s )ds,

for

(W, f W )

^{a 2DstandardBrownian motion}independent of

(β, e β)

^{. Theproof of(4.25 )is done.}

Proof of the rst point ( ase

H > 1/4

^{). The proof an be done by following exa tly the}

same strategy than inthestep above. The only dieren e is that, using a versionof Lemma 3.2

together with omputations similar to that allowing to obtain (4.32 ), the limits in(4.21 )(4.28 )

are, here,all equal to zero(forthe sake ofsimpli ity,the te hni al details areleftto thereader).

Therefore,we an dedu e(1.2) byusing(4.20 ).

✷

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