Besov regularity of stochastic integrals with respect to ...

Besov regularity of stochastic integrals with respect to the fractional Brownian motion with parameter H > 1/2 David Nualart∗ Facultat de Matemàtiques Universitat de Barcelona Gran Via, 585 08007, Barcelona, Spain Youssef Ouknine† Faculté des Sciences Semlalia Département de Mathématiques Université Cadi Ayyad BP 2390, Marrakech, Maroc Abstract Let {Bt , t ∈ [0, 1]} be a fractional Brownian motion with Hurst parameter H > 21 . Using the techniques of the Malliavin calclulus we Rt show that the trajectories of the indefinite divergence integral 0 us δBs α for all q ≥ 1, 1 < α < H, provided belong to the Besov space Bp,q p the integrand u belongs to the space Lp,1 . Moreover, if u is bounded and belongs to Lδ,2 for some even integer p ≥ 2 and for someR δ large t enough, then trajectories of the indefinite divergence integral 0 us δBs H belong to the Besov space Bp,∞ . Keywords: Fractional Brownian motion. Stochastic integrals. Malliavin calculus. AMS Subject Classification: 60H05, 60H07. ∗ †

Supported by the DGES grant BFM2000-0598 Supported by Moroccan Program PARS MI 37

1

1

Introduction

Let B = {Bt , t ∈ [0, 1]} be a fractional Brownian motion with Hurst parameter H. If H 6= 12 , the process B is not a semimartingale and we cannot apply the stochastic calculus developed by Itô in order to define stochastic integrals with respect to B.R Different approaches have been used in order to t define stochastic integrals 0 us dBs . From E|Bt − Bs |2 = |t − s|2H it follows that B has α-Hölder continuous paths for all α < H. As a consequence, if paths with α + β > 1, then we can the process u has β-Hölder continuous Rt define the pathwise integral 0 us dBs using Young’s approach (see [24]). One can weaken the regularity assumptions on the process u using Besov spaces. The trajectories of the fractional Brownian motion belong to the H Besov space Bp,∞ if p1 < H. In [7] Ciesielski, Kerkyacharian and Roynette (see also [20]) have proved that if the process u has trajectories R t in the Besov 1−H space Bp,1 , where p1 < H < 1 − p1 , then the integral process 0 us dBs has its H paths in the Besov space Bp,∞ . A different approach to construct stochastic integrals with respect to the fractional Brownian is based on the Malliavin calculus. In fact, as in the case of the Brownian motion, the divergence operator with respect to B can be interpreted as a stochastic integral called the Skorohod integral [23]. This ¨ unel [9], Carmona and idea has been developed by Decreusefond and Ust¨ Coutin [6], Alòs, Mazet and Nualart [2, 3], Duncan, Hu and Pasik-Duncan [10] and Hu and Øksendal [11]. The integral constructed by this method has zero mean, and can be obtained as the limit of Riemann R t sums defined using Wick products. The divergence integral, denoted by 0 us δBs , turns out to be equal to the symmetric integral in the sense of Russo and Vallois [21] plus an absolutely continuous term (see [4] when H > 12 and [1] for the case H < 21 ). Unlike the pathwise approach, in this method the symmetric (pathwise) integral exists under regularity assumptions of the process u in the sense of the Malliavin calculus, and no regularity on the paths of u is assumed. In this framework, one Rcan study the Besov regularity of the indefinite t divergence integral Xt := 0 us δBs . In the case of the standard Brownian motion (H = 12 ) Lorang [16] has proved that if u ∈ Lδ,2 for some even integer p ≥ 2 and δ > 2(p + 1) and u is bounded then the indefinite Skorohod 1/2 integral X belongs a.s. to the Besov space Bp,∞ . This result generalizes the work by Roynette (see [20]) in the adapted case. A generalization of this

2

result has been obtained by Berkaoui and Ouknine in [5] where it is required that u ∈ Lp+1,2 for some even integer p ≥ 4. The aim of this paper is to apply the methodology introduced by Lorang in [16] to the case of the fBm with Hurst parameter H > 12 . In order to carry out this program we make use of the stochastic calculus for the divergence integral with respect to the fBm with parameter H > 21 developed in [4]. We Rt show in Theorem 4 that the indefinite divergence integral 0 us δBs belong α to the Besov space Bp,q for all q ≥ 1, p1 < α < H, if the integrand u belongs Rt to the space Lp,1 . In Theorem 5 we prove that 0 us δBs belongs to the Besov H space Bp,∞ if the process u is bounded and belongs to the space Lδ,2 for some 2 even integer p ≥ 4 and δ > 2(p−1) ∨ 2H−1 . The Besov regularity of the 1−H indefinite divergence integral with respect to the fBm with parameter H < 12 is studied in the paper [14]. The paper is organized as follows. In Section 2 we recall some definitions on the Besov norms and spaces. Section 3 is devoted to present the main facts of the Malliavin calculus and stochastic integrals with respect to the fractional Brownian motion. Finally Section 4 contains our main results and their proofs.

2

Preliminaries on Besov spaces

The modulus of continuity of a function f : [0, 1] → R in the Lp norm, where 1 ≤ p < ∞, is defined as ω p (f, t) = sup |h|≤t

Z

p1 |f (s + h) − f (s)| ds , p

Ih

α where Ih = {s ∈ [0, 1] : s+h ∈ [0, 1]}. The Besov spaces Bp,q , where α ∈ [0, 1] and 1 ≤ p, q < ∞ are Banach spaces of functions on [0, 1] equipped with the norm q 1q p1 Z 1 Z 1 ω (f, t) dt p + . kf kα,p,q = |f (t)|p dt α t t 0 0 α is the set of functions f in Lp ([0, 1]) If q = ∞, then the Besov space Bp,∞ such that ω p (f, t) < ∞. sup tα 0 1 p,1 the set of stochastic processes u = {ut , t ∈ [0, 1]} such that we denote by LH kukpLp,1 := E kukpL1/H ([0,1]) + E kDukpL1/H ([0,1]2 ) < ∞. H

Notice as a consequence of Meyer inequalities (see for example [18]) for all p p > 1 the space Lp,1 H is included in the domain of the divergence in L and p,1 for all u ∈ LH we have (see [4]) E |δ (u)|p ≤ Cp,H E kukLp 1/H ([0,1]) + E kDukpL1/H ([0,1]2 ) . (5) The symmetric integral introduced by Russo and Vallois in [21] of a stochastic process u = {ut , t ∈ [0, 1]} with integrable trajectories is defined as the limit in probability as ε tend sto zero of Z 1 −1 (2ε) us (Bs+ε − Bs−ε )ds, 0

6

R1 and it is denoted by 0 ut dBt . By convention we will assume that all processes and functions vanish outside the interval [0, 1]. The following proposition (see [4]) shows that the divergence operator can also be interpreted as a stochastic integral. We recall that the in the case of the classical Brownian motion (i.e. if H = 21 ) then the divergence operator coincides with an extension of the Itô integral for anticipating processes introduced by Skorohod (see [23]). Proposition 1 Let u ∈ L2,1 H , and assume that a.s. Z 1Z 1 |Ds ut | |t − s|2H−2 dsdt < ∞. 0

(6)

0

Then the symmetric integral of u in any inteval [0, t] ⊂ [0, 1] exists and we have Z t Z tZ 1 Z t us δBs + αH Dr us |r − s|2H−2 drds, us dBs = (7) where

Rt 0

0

0

0

0

us δBs = δ(1[0,t] u) denotes the indefinite divergence integral.

For any p > 1 we define the seminorm Z Z 1 p p kukLp,1 = E |us | ds + E

0

0

1

Z

0

1

|Dr us |p dsdr,

and we define the space Lp,1 as the set of processes u = {ut , t ∈ [0, 1]} such that kukLp,1 < ∞. Notice that if pH ≥ 1 kukLp,1 ≤ kukLp,1 H

and Lp,1 ⊂ Lp,1 If u belongs to Lp,1 and pH > 1 then the indefinite H . R t divergence integral 0 us δBs possesses a continuous version (see [4]). We also denote by Lp,2 the space of processes such that kukLp,2 < ∞, where Z 1 Z 1Z 1 p p |us | ds + E kukLp,2 = E |Dr us |p dsdr 0 0 0 Z +E |Dθ Dr us |p dsdrdθ. [0,1]3

The following Itô’s formula has also been proved in [4]: 7

Theorem 2 Let F be a function of class C 2 (R). Assume that u = {ut , t ∈ [0, 1]} is a process in the space L2,2 . Assume that kuk2 belongs to H. Then for each t ∈ [0, T ] the following formula holds Z t F (Xt ) = F (0) + F 0 (Xs )us δBs Z 1 Z s Z t 0 2H−2 00 + αH |s − σ| F (Xs ) us Dσ uθ δBθ dσ ds 0 0 0 Z s Z t 2H−2 00 + αH F (Xs )us uθ (s − θ) dθ ds. 0

0

The above Itô’s formula holds assuming only that the process u belongs locally to the space D2,2 (|H|) introduced in [4] which contains the space L2,2 .

4

Besov regularity of the divergence integral

Fix pH ≥ 1, and consider a process u ∈ Lp,1 . Define Xt = each j ≥ 1, and 1 ≤ k ≤ 2j , set Z t Xjk (t) = χjk (s)us δBs .

Rt 0

us δBs , and for

0

We denote by 1jk the indicator function of the set use of the following technical lemma. Lemma 3 For all p >

1 H

k−1 2j

, 2kj . We will make

and δ ≥ 1 we have

E |Xjk (t)|p ≤ Cp,H 2jp( +E

Z

0

1 −H+ H 2 δ

1Z

0

( Z ) E

1

1 0

pH δ 1jk (s)|us | ds δ H

δ H

1jk (s)|Dr us | dsdr

where Cp,H is the constant appearing in (5).

8

) pH δ

,

Proof. Using the estimate (5) for the moment of order p of the divergence operator we can write Z t p p E |Xjk (t)| = E χjk (s)us δBs 0 ( Z pH 1 1 1 H ≤ Cp,H E χjk (s) |us | H ds 0

+E

Z

0

1Z

0

1

1 χjk (s) H |Dr us | H1 dsdr

Assume δ > 1. Applying Hölder inequality with p

E |Xjk (t)|

≤ Cp,H

Z

0

Z

+E

1

0

1

δ0 χjk (s) H ds

Z

0

1 δ

1 δ0

+

( Z pH δ0 E

1

0

1

δ H

1jk (s)|Dr us | dsdr

pH δ

pH )

.

= 1 yields

pH δ 1jk (s)|us | ds δ H

)

.

Finally, the desired result follows from the equality Z

0

1

pH δ0 δ0 1 H H χjk (s) ds = 2jp( 2 −H+ δ ) .

The case δ = 1 is obvious because χjk (s) ≤ 2j/2 . As a consequence of Lemma 3 we get the following inequality: E |Xjk (t)|p ≤ Cp,H 2jp( 2 −H+ δ ) kukpLδ/H,1 . H

1

(8)

On the other hand, taking δ = pH, we obtain j

2 X k=1

E |Xjk (t)|p ≤ Cp,H 2jp( 2 −H+ p ) kukpLp,1 . 1

1

(9)

Theorem 4 Let 0 < α < H, α1 < p < ∞, and u ∈ Lp,1 . Then the trajectoRt ries of the indefinite divergence integral 0 us dBs belong almost surely to the α for all 1 ≤ q ≤ ∞. Besov space Bp,q 9

Proof. Fix 0 < β < H − α. By Thebychev inequality and using (9) we obtain   2j X 1 1 P 2−jp( 2 −α+ p ) |Xjk (1)|p > 2−jpβ  k=1

j

−jp(−β+ 21 −α+ p1 )

≤ 2

E

2 X k=1

|Xjk (t)|p ≤ cp 2jp(H−α−β) kukLp p,1 .

Then the desired result follow by Borel-Cantelli lemma. The following is the main result of this paper. 2 Theorem 5 Let p ≥ 2 be an even integer and let δ > 2(p−1) ∨ 2H−1 . If the 1−H δ,2 process u is bounded and belongs Rto the space L , then the trajectories of t the indefinite divergence integral 0 us dBs belong almost surely to the Besov H space Bp,∞ .

Rt Proof. In order to show that the paths of the indefinite integral 0 us dBs H belong almost surely to the Besov space Bp,∞ we need to show that j

−jp( 21 −H+ p1 )

sup 2 j

2 X k=1

p (1) < ∞ Xjk

(10)

almost surely. Using the Itô’s formula for the divergence integral (see (2)) we can write Z t p p−1 Xjk (t) = p Xjk (s)χjk (s)us δBs (11a) 0 Z t p−2 Xjk (s)χjk (s)us +αp,H 0 Z 1 Z s 2H−2 × |s − σ| χjk (θ)Dσ uθ δBθ dσ ds 0 0 Z s Z t p−2 2H−2 +αp,H Xjk (s)χjk (s)us dθ ds, χjk (θ)uθ |s − θ| 0

0

where αp,H = p(p − 1)H(2H − 1). The proof of the estimate (10) will be done in several steps: 10

Step 1. Set (p) γ jk (t)

=

Z

t 0

p−1 Xjk (s)χjk (s)us δBs .

We are going to show that 2j X (p) −jp( 21 −H+ p1 ) sup 2 γ jk (1) < ∞ j k=1

(12)

almost surely. The estimate (12) is a consequence of Borel-Cantelli lemma and the inequality  2 2j X (p) E γ jk (1) ≤ C2jθ , (13) k=1

where θ < 2 + p(1 − 2H). Along the proof C will denote a constant depending on p, H and kuk∞ and that may vary from one formula to another one. (p) (p) In order to show (13) we need to evaluate terms of the form E γ jk (1)γ jk (1) ,

where 1 ≤ k, l ≤ 2j . Applying the estimate (4) for the expectation of two divergences we obtain Z 1Z 1 (p) (p) χjk (s)χjl (t) X p−1 (s)X p−1 (t) E γ jk (1)γ jk (1) ≤ CE jk jl 0

:

0 2H−2

×|s − t| dsdt Z p−1 Dr X (θ)uθ Ds X p−1 (t)ut +CE jk jl [0,1]4 × χjk (θ)χjl (t) |r − t|2H−2 |θ − s|2H−2 drdθdsdt (1)

(2)

= Bjkl + Bjkl .

For the first term we have Z 1Z 1 12 (1) 2(p−1) 2(p−1) j Bjkl ≤ C2 1jk (s)1jl (t) EXjk (s)EXjl (t) |s − t|2H−2 dsdt. 0

0

(14)

From (8) with the exponent 2(p − 1) > 2(p−1)

E Xjk

1 H

we deduce the estimate

(s) ≤ C2j(p−1)(1−2H+ 11

2H δ

) kuk2(p−1) ,

Lδ/H,1

(15)

for any δ ≥ 1. Substituting (15) into (14) yields Z 1Z 1 (1) 2(p−1) j ((p−1)(1−2H+ 2H )+1) δ Bjkl ≤ C2 kukLδ/H,1 1jk (s)1jl (t)|s − t|2H−2 dsdt. 0

0

Hence, j

2 X

k,l=1

where

Bjkl ≤ C2j (p(1−2H)+2H+ (1)

cH =

Z

1

0

Z

0

2(p−1)H δ

) kuk2(p−1) c , Lδ/H,1 H

1

|s − t|2H−2 dsdt.

This implies the desired inequality provided 2H + δ>

2(p−1)H δ

< 2, which means

H (p − 1). 1−H

(16)

(2)

In order to estimate the term Bjkl let us compute Z s p−1 p−2 Dr Xjk (s)us = (p − 1)Xjk (s)us χjk (θ)Dr uθ δBθ 0

p−2 +(p − 1)Xjk (s)us χjk (r)ur 1[0,s] (r)

p−1 +Xjk (s)Dr us .

(17)

We are going to use this decomposition in order to deduce an estimate for p−1 2 . E Dr Xjk (s)us We have, by Hölder’s inequality with λ1 + λ10 = 1 2 ! Z s 2(p−2) (s) E Xjk χjk (θ)Dr uθ δBθ 0

λ1 ≤ E |Xjk (s)|2(p−2)λ

Z 2λ0 ! λ10 s χjk (θ)Dr uθ δBθ . E

From (8) with the exponent 2(p − 2)λ >

E |Xjk (s)|2(p−2)λ

λ1

(18)

0

1 H

we deduce the estimate

≤ C2j(p−2)(1−2H+ 12

2H δ

) kuk2(p−2) ,

Lδ/H,1

(19)

for all δ ≥ 1. Applying the estimate (8) to the process {Dr us , s ∈ [0, 1]} with the exponent 2λ0 > H1 yields Z 2λ0 ! λ10 s 2H χjk (θ)Dr uθ δBθ E (20) ≤ C2j (1−2H+ δ ) kDr ukL2 δ/H,1 , 0

for all δ ≥ 1. Substituting (19) and (20) into (18) we obtain 2 ! Z s 2(p−2) χjk (θ)Dr uθ δBθ (s) E Xjk 0

2(p−1)H (p−1)(1−2H)+ δ

≤ C2j (

) kuk2(p−2) kD uk2 δ/H,1 . (21) Lδ/H,1 r L p−1 (s)us we have using (8) For the second term in the expression of Dr Xjk with the exponent 2(p − 2) > H1 , 2(p−2) +1) χjk (r)2 EX 2(p−2) (s) ≤ C2j ((p−2)(1−2H+ 2H δ ) (22) kukLδ/H,1 1jk (r). jk Finally, by Hölder’s inequality with λ1 + λ10 = 1 λ1 10 0 λ E|Dr us |2λ , E |Xjk (s)|2(p−1) |Dr us |2 ≤ E |Xjk (s)|2(p−1)λ

and by (8) with the exponent 2(p − 1)λ > H1 , we obtain 1 2(p−1) 2(p−1) 2λ0 λ0 j(p−1)(1−2H+ 2H 2 ) δ . kukLδ/H,1 E |Dr us | E |Xjk (s)| |Dr us | ≤ C2 (23) Using the inequalities (21), (22) and (23) into (17) we deduce n p−1 2 2(p−1)H 2(p−2) E Dr Xjk (s)us ≤ C2jp(1−2H) 2j (2H−1+ δ ) kukLδ/H,1 kDr uk2Lδ/H,1 2(p−2)H δ

) kuk2(p−2) 1 (r) Lδ/H,1 jk 1 2(p−1)H 2(p−1) 2λ0 λ0 j (2H−1+ ) δ +C2 kukLδ/H,1 E |Dr us | (24) .

+C2j (4H−1+

We are going to use this estimate in order to treat the second term given by: (2) Bjkl

j

≤ C2

Z

[0,1]4

2 21 2 p−1 p−1 E Dr Xjk (θ)uθ E Ds Xjl (t)ut

×1jk (θ)1jl (t)|r − t|2H−2 |θ − s|2H−2 drdθdsdt. 13

(25)

Applying the inequality (24) we can write 6 X 2 21 2 (i) p−1 (θ)uθ E Ds Xjlp−1 (t)ut ≤ C2jp(1−2H) Cjkl , E Dr Xjk i=1

where Cjkl = 2j (2H−1+ (1)

2(p−1)H δ

Cjkl = 2j (4H−1+

(3)

2(p−2)H δ

) kuk2(p−2) 1 (r)1 (s), jl Lδ/H,1 jk 10 2(p−1)H 0 0 2λ 2(p−1) , = 2j (2H−1+ δ ) kukLδ/H,1 E |Dr uθ |2λ E |Ds ut |2λ (2)

Cjkl

) kuk2(p−2) kD uk δ/H,1 kD uk δ/H,1 , s L Lδ/H,1 r L

(2p−3)H 2(p−2) (4) Cjkl = 2j (3H−1+ δ ) kukLδ/H,1 × [kDr ukLδ/H,1 1jk (s) + kDs ukLδ/H,1 1jl (r)] ,

2(p−1)H (5) Cjkl = 2j (2H−1+ δ ) kuk2p−3 Lδ/H,1 1 1 2λ0 2λ0 2λ0 2λ0 × kDr ukLδ/H,1 E |Ds ut | + kDs ukLδ/H,1 E |Dr uθ |

and (2p−3)H (6) Cjkl = 2j (3H−1+ δ ) kukL2p−3 δ/H,1 1 1 2λ0 2λ0 2λ0 2λ0 × 1jk (r) E |Ds ut | + 1jl (s) E |Dr uθ | .

Replacing these six terms into (25) yields j

2 X

k,l=1

Z 6 X 2 X j

(2) Bjkl

j(p(1−2H)+1)

≤ C2

:

i=1 k,l=1

[0,1]4

(i)

Cjkl

×1jk (θ)1jl (t)|r − t|2H−2 |θ − s|2H−2 drdθdsdt 6 X = di . i=1

14

We have j (p(1−2H)+2H+

d1 ≤ C2

2(p−1)H δ

Z ) kuk2(p−1) sup Lδ/H,2 r

1 2H−2

|r − t|

0

Z 2 X

2 dt ,

j

2(p−2)H p(1−2H)+4H+ δ

d2 ≤ C2j (

) kuk

2(p−2)

Lδ/H,1

k,l=1

1jk (r)1jl (s)

[0,1]4

×1jk (θ)1jl (t)|r − t|2H−2 |θ − s|2H−2 drdθdsdt. We claim that for all k, l Z 1Z 1 1jk (r)1jl (t)|r − t|2H−2 drdt ≤ C2−2Hj . 0

(26)

0

Indeed, the maximum value of this integral corresponds to the case k = l 1 where it is equal to H(2H−1) 2−2Hj . As a consequence we deduce the inequality d2 ≤ C2j (p(1−2H)+2H+

2(p−2)H δ

We have d3 ≤ C2

j (p(1−2H)+2H+

2(p−1)H δ

) kuk2(p−1)

Lδ/H,1

2(p−1)H p(1−2H)+2H+ δ

d3 ≤ C2j (

and taking λ0 =

λ 2

=

δ 2H

) kuk

Z

1 2H−1

2(p−1)

Lδ/H,1

Z

and assuming δ >

d3 ≤ C2j (p(1−2H)+2H+

Lδ/H,1

[0,1]4

×|r − t|2H−2 |θ − s|2H−2 drdθdsdt. Applying Hölder’s inequality with λ >

) kuk2(p−2) .

E |Dr uθ |

2λ0

2λ0

E |Ds ut |

1 2λ0

we obtain

[0,1]2

E |Ds ut |

2 we 2H−1

2(p−1)H δ

2λ0

λ 2λ0

λ2 , dsdt

get

) kuk2p

Lδ/H,1 .

We have (2p−3)H d4 ≤ C2j (p(1−2H)+3H+ δ ) kuk2p−4 Lδ/H,1 j Z 2 X × [kDr ukLδ/H,1 1jk (s) + kDs ukLδ/H,1 1jl (r)]

k,l=1

[0,1]4

×1jk (θ)1jl (t)|r − t|2H−2 |θ − s|2H−2 drdθdsdt. 15

(27)

Using the estimate supθ,k Z 2 X j

k,l=1

≤ C2

[0,1]4

j(1−2H)

R1 0

1jk (s)|θ − s|2H−2 ds ≤ C2j(1−2H) we can write

kDr ukLδ/H,1 1jk (θ)1jl (t)1jk (s)|r − t|2H−2 |θ − s|2H−2 drdθdsdt Z

1

0

kDr ukLδ/H,1 dr.

(28)

Substituting (28) into (27) yields d4 ≤ C2j (p(1−2H)+H+1+

(2p−3)H δ

) kuk2p−3 . Lδ/H,2

For the fifth term we can write the estimate 2(p−1)H d5 ≤ C2j (p(1−2H)+2H+ δ ) kuk2p−3 Lδ/H,1 Z 1 1 2λ0 2λ0 2λ0 2λ0 kDr ukLδ/H,1 E |Ds ut | + kDs ukLδ/H,1 E |Dr uθ | × [0,1]4

×|r − t|2H−2 |θ − s|2H−2 drdθdsdt.

1

Hence, using the continuous embedding of H into L H ([0, 1]) we obtain taking δ λ0 = 2H 2(p−1)H d5 ≤ C2j (p(1−2H)+2H+ δ ) kuk2p−3 Lδ/H,1 Z Z H1 !H Z 1 1 1 2λ0 2λ0 ds dt × E |Ds ut |

0

0

0

≤ C2j (p(1−2H)+2H+

2(p−1)H δ

1

1 H δ/H,1

kDr ukL

dr

H

) kuk2p−1 . Lδ/H,2

Finally (2p−3)H d6 ≤ C2j (p(1−2H)+3H+ δ ) kuk2p−3 Lδ/H,1 j Z 2 1 1 X 2λ0 2λ0 2λ0 2λ0 1jk (r) E |Ds ut | + 1jl (s) E |Dr uθ | ×

k,l=1

[0,1]4

×1jk (θ)1jl (t)|r − t|2H−2 |θ − s|2H−2 drdθdsdt. R1 Using the estimate supt,k 0 1jk (r)|r − t|2H−2 dr ≤ C2j(1−2H) we can write (2p−3)H d6 ≤ C2j (p(1−2H)+H+1+ δ ) kuk2p−3 Lδ/H,1 Z 1Z 1 λ 0 2λ0 × dsdt. E |Ds ut |2λ

0

0

16

Finally choosing λ0 =

δ 2H

we get (2p−3)H δ

d6 ≤ C2j (p(1−2H)+H+1+

) kuk2p−2 . Lδ/H,1

As a consequence we deduce j

2 X

k,l=1

Bjkl ≤ C2j (p(1−2H)+H+1+ (2)

2(p−1)H δ

) kuk2p

L

δ/H,2

∨1 .

This inequality provides the desired estimation if H +1+

2(p − 1)H 2(p−1) ∨ 2H−1 . 1−H Step 2. Define Z s Z 1 Z t (p) p−2 2H−2 ψ jk (t) = |s − σ| χjk (θ)Dσ uθ δBθ dσ ds. Xjk (s)χjk (s)us 0

0

0

We want to show that 2j X (p) ψ jk (1) < ∞

−jp( 12 −H+ p1 )

sup 2 j

k=1

almost surely, and by Borel-Cantelli lemma it suffices to check that 2j X (p) ψ (1) E ≤ C2jθ , jk k=1

where θ < 1 + p( 12 − H). We have

2j 2j Z X j X (p) E ψ jk (1) ≤ C2 2 k=1

k=1

1

0

Z

0

1

1jk (s) |s − σ|2H−2

Z s p−2 χjk (θ)Dσ uθ δBθ dσds. ×E Xjk (s) 0

17

(29)

By Hölder’s inequality with λ1 + λ10 = 1 we can write Z s p−2 E Xjk (s) χjk (θ)Dσ uθ δBθ 0

(p−2)λ ≤ E Xjk (s)

1 λ

(30)

Z 0! 1 s λ λ0 . E χjk (θ)Dσ uθ δBθ 0

Applying the estimate (8) to the processes u and {Dr us , s ∈ [0, 1]} yields (p−2)λ λ1 1 H E Xjk (s) ≤ C2j(p−2)( 2 −H+ δ ) kukLδ/H,1 (31) and

Z λ0 ! λ10 s 1 H χjk (θ)Dσ uθ δBθ E ≤ C2j ( 2 −H+ δ ) kDσ ukLδ/H,1

(32)

0

and substituting (31) and (32) into (30) we obtain Z s p−2 E Xjk (s) χjk (θ)Dσ uθ δBθ

(33)

0

≤ C2j ((p−1)( 2 −H )+ 1

(p−1)H δ

) kukp−2

Lδ2 /H,1 kDσ ukLδ1 /H,1 .

Finally, replacing (33) into (29) we obtain 2j X (p−1)H 1 (p) p−1 E ψ (1) jk ≤ C2j (p( 2 −H )+H+ δ ) kukLδ2 /H,1 . k=1

Hence, we need

H+

(p − 1)H < 1, δ

that means, δ > (p−1)H . 1−H Step 3. In order to complete the proof of the theorem it remains to show that 2j X (p) −jp( 12 −H+ p1 ) sup 2 (34) η jk (1) < ∞, j

where

(p) η jk (t)

=

Z

0

t

p−2 (s)χjk (s)us Xjk

k=1

Z 18

s 2H−2

0

χjk (θ)uθ |s − θ|

dθ ds.

We have Z tZ s (p) p−2 Xjk (s) χjk (s)χjk (θ) |s − θ|2H−2 dθds η jk (t) ≤ C Z0 t 0 p−2 = C Xjk (s)ρjk (s)ds, 0

where j

ρjk (s) = 2

Z

0

s

1jk (s)1jk (θ) |s − θ|2H−2 dθ.

Notice that the term ρjk (s) is deterministic and Z 1 1 j(1−2H) 2 . ρjk (s)ds = 2αH 0

(35)

In order to show (34) we have to show the estimate 2 Z X j

sup 2

−jq ( 12 −H+ 1q )

j

k=1

1

0

q−2 Xjk (s)ρjk (s)ds < ∞,

(36)

almost surely, for q = p. We are going to show the estimate (36) for q = 2, 4, ..., p by a recurrence argument. For q = 2 it is immediate using (35). Suppose it holds for q and let us show that 2 Z X j

sup 2

−j ((q+2)( 21 −H )+1)

j

k=1

1

0

q Xjk (s)ρjk (s)ds < ∞

(37)

almost surely. By Itô’s formula (11a) we can write (q)

(q)

(q)

q Xjk (s) = pγ jk (s) + αp,H ψ jk (s) + αp,H η jk (s).

(38)

By the same arguments as in the proof of Step 2 we can show that sup 2 j,s

Moreover, 2 Z X j

k=1

0

1

−j (q ( 21 −H )+1)

2j X (q) ψ jk (s) < ∞. k=1

2j Z X (q) j(1−2H) η jk (s) ρjk (s)ds ≤ C2 k=1

19

0

1

q−2 Xjk (θ)ρjk (θ)dθ

(39)

and applying the induction hypothesis we obtain 2 Z X j

−j ((q+2)( 12 −H )+1)

sup 2 j

k=1

0

(q)

1

(q) η (s) jk ρjk (s)ds < ∞.

(40)

In order to handle the term γ jk (s) in (38) we use the following integration by parts formula Z 1 Z 1 (q) (q) γ jk (s)ρjk (s)ds γ jk (1) ρjk (s)ds = 0 0 Z s Z 1 q−1 ρjk (θ)dθ χjk (s)us δBs . + Xjk (s) 0

0

From Step 1 we konw that X 2j 1 (q) sup 2−j (q( 2 −H )+1) γ jk (1) < ∞. j k=1

(41)

Rs On the other hand, 0 ρjk (θ)dθ is bounded by a constant. Hence, the proof of Step 1 also provides the estimate 2j Z 1 Z s X 1 q−1 Xjk (s) ρjk (θ)dθ χjk (s)us δBs < ∞. sup 2−j (q( 2 −H )+3−2H ) j 0 k=1 0 (42) Finally, from (39), (40), (41) and (42) we deduce (37). Acknowledgement 6 The work was carried out during a stay of Youssef Ouknine at the“Centre de Recerca Matemàtica” from Barcelona. He would like to thank the CRM for hospitality and generous support.

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