Stochastic integration with respect to the fractional
Brownian motion
Elisa Alòs
Departament d’Economia i Empresa
Universitat Pompeu Fabra
c/Ramón Trias Fargas, 25-27
08005 Barcelona, Spain
David Nualart
Facultat de Matemàtiques
Universitat de Barcelona
Gran Via, 585
08007, Barcelona, Spain
Abstract
We develop a stochastic calculus for the fractional Brownian motion
with Hurst parameter H > 12 using the techniques of the Malliavin
calclulus. We establish estimates in Lp , maximal inequalities and a
continuity criterion for the stochastic integral. Finally, we derive an
Itô’s formula for integral processes.
Keywords: Fractional Brownian motion. Stochastic integral. Malliavin calculus. Maximal inequalities. Itô’s formula.
AMS Subject Classification: 60H05, 60H07.
1
Introduction
The fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) is a
centered Gaussian process B = {Bt , t ≥ 0} with the covariance function (see
[15])

1 € 2H
RH (t, s) =
(1)
s + t2H − |t − s|2H .
2
Notice that if H = 12 , the process B is a standard Brownian motion. From
(1) it follows that E|Bt − Bs |2 = |t − s|2H and, as a consequence, B has
α−Hölder continuous paths for all α < H.
If H 6= 21 , 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. Different approaches have been used in order to construct
a stochastic calculus with respect to B and we can mention the following
contributions to this problem:
0
Supported by the DGES grant no BFM2000-0598 and by the INTAS grant no 991-16
1
• Lin [13] and Dai and Heide [5] defined stochastic integrals with respect
to the fractional Brownian motion with parameter H > 21 using a
pathwise Riemann-Stieltjes method. The integrator must have finite
p-variation where p1 + H > 1.
• The stochastic calculus of variations (see [18]) with respect to the
Gaussian process B is a powerful technique that can be used to define
stochastic integrals. More precisely, as in the case of the Brownian
motion, the divergence operator with respect to B can be interpreted
as a stochastic integral. This idea has been developed by Decreusefond
and Üstünel [6, 7], Carmona and Coutin [4], Alòs, Mazet and Nualart
[2, 3], Duncan, Hu and Pasik-Duncan [8] and Hu and Øksendal [10].
The integral constructed by this method has zero mean, and can be
obtained as the limit of Riemann sums defined using Wick products.
• Using the notions of fractional integral and derivative, Zähle has introduced in [25] a pathwise stochastic integral with respect to the fBm
B with parameter H ∈ (0, 1). If the integrator has λ-Hölder continuous paths with λ > 1 − H, then this integral can be interpreted
as a Riemann-Stieltjes integral and coincides with the forward and
Stratonovich integrals studied in [2] and [1].
The purpose of our paper is to develop a stochastic calculus with respect
to the fractional Brownian motion B with Hurst parameter H > 21 using the
techniques of the Malliavin calculus. Unlike some previous works (see, for
instance, [3]) we will not use the integral representation of B as a stochastic
integral with respect to a Wiener process. Instead of this we will rely on the
intrinsic Malliavin calculus with respect to B.
The organization of the paper is as follows. Section 2 contains some
preliminaries on the fBm, and in Section 3 we review the basic facts on
Malliavin calculus that will be used in order to define stochastic integrals.
In Section 4 we show the existence of the symmetric stochastic integral
in the sense of Russo and Vallois [21] under smoothness conditions on the
integrand, in the sense of the Malliavin calculus. This symmetric integral
turns out to be equal to the divergence operator plus a trace term involving
the derivative operator. In Section 5, applying Meyer’s inequalities and the
results of [16], we derive Lp and maximal inequalities for the divergence
integral with respect to fBm. These estimates allow us to deduce continuity
results for the integral process in Section 6. Finally, in Section 7 we establish
an Itô’s formula for the divergence process.
2
2
Preliminaries on the fBm
Fix H ∈ (1/2, 1). Let B = {Bt , t ∈ [0, T ]} be a fractional Brownian motion
with parameter H. That is, B is a zero mean Gaussian process with the
covariance (1). We assume that B is defined in a complete probability space
(Ω, F, P ) . We denote by E ⊂ H the set of step functions on [0, T ]. Let H
be the Hilbert space defined as the closure of E with respect to the scalar
product
‹
Š
1[0,t] , 1[0,s] H = RH (t, s).
The mapping 1[0,t] −→ Bt can be extended to an isometry between H and
the Gaussian space H1 (B) associated with B. We will denote this isometry
by ϕ −→ B(ϕ).
It is easy to see that
Z tZ s
(2)
|r − u|2H−2 dudr,
RH (t, s) = αH
0
0
where αH = H(2H − 1). Formula (2) implies that
hϕ, ψiH = αH
Z
Z
T
0
0
T
|r − u|2H−2 ϕr ψ u dudr
(3)
for any pair of step functions ϕ and ψ in E.
For r > u we can write
1
1
(r − u)2H−2 = β(2 − 2H, H − )(ru)H− 2
2
Z u
3
3
1−2H
v
×
(r − v)H− 2 (u − v)H− 2 dv,
(4)
0
where β denotes the Beta function. Let us show the equality (4). By means
r−v
r
of the change of variables z = u−v
and x = uz
, we obtain
Z
u
3
3
v 1−2H (r − v)H− 2 (u − v)H− 2 dv
0
Z ∞
3
2H−2
(zu − r)1−2H z H− 2 dz
= (r − u)
r
u
= (ru)
1
−H
2
2H−2
(r − u)
Z
1
0
1
1
= β(2 − 2H, H − )(ru) 2 −H (r − u)2H−2 .
2
3
3
(1 − x)1−2H xH− 2 dx
Consider the square integrable kernel
Z t
1
3
1
−H
2
KH (t, s) = cH s
(u − s)H− 2 uH− 2 du,
s
where cH =
h
H(2H−1)
β(2−2H,H− 12 )
Notice that
i1/2
and t > s.
∂KH
(t, s) = cH
∂t
’ “H− 1
2
3
t
(t − s)H− 2 ≥ 0.
s
(5)
Taking into account formulas (4) and (2) we deduce that this kernel verifies
Z t∧s
KH (t, u)KH (s, u)du
0
“
Z t∧s ’Z t
2
H− 32 H− 12
y
dy
= cH
(y − u)
u
’Z0 s
“
3
1
×
(z − u)H− 2 z H− 2 dz u1−2H du
u
Z tZ s
1
2
|y − z|2H−2 dzdy
= cH β(2 − 2H, H − )
2 0 0
(6)
= RH (t, s).
Property (6) implies that RH (t, s) is nonnegative definite.
∗ from E to L2 ([0, T ]) defined by
Consider the linear operator KH
Z T
∂KH
∗
(r, s)dr.
(KH
ϕ)(s) =
ϕr
∂r
s
Using equations (6) and (3) we have
=
=
=
=
∗
∗
ϕ, KH
ψiL2 ([0,T ])
hKH
“ ’Z T
“
Z T ’Z T
∂KH
∂KH
ϕr
(r, s)dr
ψu
(u, s)du ds
∂r
∂u
s
0
s
“
Z T Z T ’Z r∧u
∂KH
∂KH
(r, s)
(u, s)ds ϕr ψ u dudr
∂r
∂u
0
0
0
Z TZ T 2
∂ RH
(r, u)ϕr ψ u dudr
0 ∂r∂u
0
Z TZ T
|r − u|2H−2 ϕr ψ u dudr
αH
0
0
= hϕ, ψiH .
4
(7)
∗ provides an isometry between the Hilbert
As a consequence, the operator KH
2
spaces H and L ([0, T ]). Hence, the process W = {Wt , t ∈ [0, T ]} defined
by
∗ −1
Wt = B((KH
) (1[0,t] ))
is a Wiener process, and the process B has an integral representation of the
form
Z t
KH (t, s)dWs ,
(8)
Bt =
0
€ ∗

because KH
1[0,t] (s) = KH (t, s).
The elements of the Hilbert space H may not be functions but distributions of negative order (see, for instance, [3, Proposition 6] or the recent
paper by Taqqu [20]). In fact, from (5) and (7) it follows that H coincides
H− 1
1
with the space of distributions f such that s 2 −H I0+ 2 (fH− 1 )(s) is a square
integrable function, where fH− 1 (s) = f (s)s
2
H− 12
, and
H− 1
I0+ 2
2
is the left-sided
1
2
fractional integral of order H − (see [22]).
We can find a linear space of functions contained in H in the following
way. Let |H| be the linear space of measurable functions ϕ on [0, T ] such
that
“2
Z T ’Z T
∂KH
2
(r, s)dr ds < ∞.
(9)
kϕk|H| :=
|ϕr |
∂r
s
0
From the above computations, it is easy to check that
Z TZ T
2
(10)
= αH
|ϕr | |ϕu | |r − u|2H−2 drdu.
kϕk|H|
0
0
It is not difficult to show that |H| is a Banach space with the norm k · k|H|
and E is dense in |H|. On the other hand, it has been shown in [20] that the
space |H| equipped with the inner product hϕ, ψiH is not complete and it is
isometric to a subspace of H. We will identify |H| with this subspace.
The following estimate has been proved in [16]
kϕk|H| ≤ bH kϕk
1
L H ([0,T ])
,
(11)
1
for some constant bH > 0. This estimate implies the inclusion L H ([0, T ]) ⊂
|H|. For the sake of completeness we reproduce here the main arguments
of the proof of (11). Using Hölder’s inequality with exponent q = H1 in (10)
we get

1−H
1
“ 1−H
“H Z T ’Z T
’Z T
1

.
|ϕu | (r − u)2H−2 du
kϕk2|H| ≤ αH
dr
|ϕr | H dr
0
0
0
5
The second factor in the above expression, up to a multiplicative constant,
2H−1
1
norm of the left-sided fractional integral I0+
it is equal to the 1−H
|ϕ|
(see [22]). Finally is suffices to apply the Hardy-Littlewood inequality (see
[24, Theorem 1, p. 119])
 α 
I0+
f Lq (0,∞) ≤ cH,p kf kLp (0,∞) ,
(12)
where 0 < α < 1, 1 < p < q < ∞ satisfy
1
, and p = H.
values α = 2H − 1, q = 1−H
3
1
q
=
1
p
− α, with the particular
Malliavin Calculus
The process B = {Bt , t ∈ [0, T ]} is Gaussian and, hence, we can develop a
stochastic calculus of variations (or Malliavin calculus) with respect to it.
Let us recall the basic notions of this calculus.
Let S be the set of smooth and cylindrical random variables of the form
F = f (B(φ1 ), . . . , B(φn )),
(13)
where n ≥ 1, f ∈ Cb∞ (Rn ) (f and all its partial derivatives are bounded),
and φi ∈ H. The derivative operator D of a smooth and cylindrical random
variable F of the form (13) is defined as the H-valued random variable
n
X
∂f
(B(φ1 ), . . . , B(φn ))φi .
DF =
∂xi
i=1
The derivative operator D is then a closable operator from Lp (Ω) into
Lp (Ω; H) for any p ≥ 1. For any integer k ≥ 1 we denote by Dk the iteration of the derivative operator. For any p ≥ 1 the Sobolev space Dk,p is
the closure of S with respect to the norm
kF kpk,p = E|F |p + E
k
X
 j p
D F  ⊗j .
H
j=1
In a similar way, given a Hilbert space V we denote by Dk,p (V ) the coresponding Sobolev space of V -valued random variables.
The divergence operator δ is the adjoint of the derivative operator, defined by means of the duality relationship
E(F δ(u)) = E hDF, uiH ,
6
where u is a random variable in L2 (Ω; H). We say that u belongs to the
domain of the operator δ, denoted by Dom δ, if the above expression is
continuous in the L2 norm of F . A basic result says that the space D1,2 (H)
is included in Dom δ.
The following are two basic properties of the divegence operator:
i) For any u ∈ D1,2 (H)
Eδ(u)2 = E kuk2H + E hDu, (Du)∗ iH⊗H ,
(14)
where (Du)∗ is the adjoint of (Du) in the Hilbert space H ⊗ H.
ii) For any F in D1,2 and any u in the domain of δ such that F u and
F δ(u) + hDF, uiH are square integrable, then F u is in the domain of
δ and
δ(F u) = F δ(u) + hDF, uiH .
(15)
We denote by |H| ⊗ |H| the space of measurable functions ϕ on [0, T ]2
such that
Z
Œ
Œ
ŒŒ
2
2
Œϕr,θ Œ Œϕu,η Œ |r − u|2H−2 |θ − η|2H−2 drdudθdη < ∞.
kϕk|H|⊗|H| = αH
[0,T ]4
As before, |H| ⊗ |H| is a Banach space with respect to the norm k · k|H|⊗|H| .
Furthermore, equipped with the inner product
Z
2
hϕ, ψiH⊗H = αH
ϕr,θ ψ u,η |r − u|2H−2 |θ − η|2H−2 drdudθdη
[0,T ]4
the space |H|⊗|H| is isometric to a subspace of H⊗H and it will be identified
with this subspace.
Lemma 1 If ϕ belongs to |H| ⊗ |H| then
kϕk|H|⊗|H| ≤ bH kϕk
1
L H ([0,T ]2 )
.
Proof. A slight extension of (11) implies that
h|ϕ|, |ψ|iH ≤ b2H kϕk
1
L H ([0,T ])
kψk
1
L H ([0,T ])
.
for all ϕ and ψ in H. As a consequence, applying twice this inequality yields
the desired result.
7
For any p > 1 we denote by D1,p (|H|) the subspace of D1,p (H) formed
by the elements u such that u ∈ |H| a.s., Du ∈ |H| ⊗ |H| a.s., and
E kukp|H| + E kDukp|H|⊗|H| < ∞.
Note that the space D1,2 (|H|) ⊂ D1,2 (H) is included in the domain of δ, and
from (14) we have
Eδ(u)2 ≤ E kuk2H + E kDuk2|H|⊗|H| .
4
Stochastic integrals with respect to the fractional Brownian motion
In the case of an ordinary Brownian motion, the adapted processes in
L2 ([0, T ] × Ω) belong to the domain of the divergence operator, and on
this set the divergence operator coincides with the Itô stochastic integral.
Actually, the divergence operator coincides with an extension of the Itô
stochastic integral introduced by Skorohod in [23]. We can ask in which
sense the divergence operator with respect to a fractional Brownian motion
B can be interpreted as a stochastic integral. Note that the divergence operator provides an isometry between the Hilbert Space H associated with
the fBm B and the Gaussian space H1 (B), and gives rise to a notion of
stochastic integral in the space of determinstic functions |H| included in H.
Let us recall the definition of the symmetric integral introduced by Russo
and Vallois in [21]. By convention we will assume that all processes and
functions vanish outside the interval [0, T ].
Definition 2 Let u = {ut , t ∈ [0, T ]} be a stochastic process with integrable
trajectories. The symmetric integral of u with respect to the fBm B is defined
as the limit in probability as ε tend sto zero of
Z T
−1
(2ε)
us (Bs+ε − Bs−ε )ds,
0
provided this limit exists, and it is denoted by
RT
0
ut dBt .
Let ST be the set of smooth step processes of the form
u=
m−1
X
Fj 1[tj ,tj+1 ] ,
j=0
8
where Fj ∈ S, and 0 = t1 < ... < tm = T . It is not difficult to check that
ST is dense in D1,2 (|H|).
The following proposition gives sufficient conditions for the existence
of the symmetric integral, and provides a representation of the divergence
operator as a stochastic integral.
Proposition 3 Let u = {ut , t ∈ [0, T ]} be a stochastic process in the space
D1,2 (|H|). Suppose also that a.s.
Z
T
0
Z
0
T
|Ds ut | |t − s|2H−2 dsdt < ∞.
Then the symmetric integral exists and we have
Z T
Z TZ T
Ds ut |t − s|2H−2 dsdt.
ut dBt = δ(u) + αH
0
0
(16)
(17)
0
Proof. The proof of this proposition will be decomposed into several
steps.
Step 1. Let us define, for all u ∈ D1,2 (|H|), the aproximating process uε
given by
Z
utε = (2ε)−1
t+ε
us ds.
t−ε
In this first step we will see that
kuε k2D1,2 (|H|) ≤ dH kuk2D1,2 (|H|) ,
(18)
for some positive constant dH . In fact, we have that
Z TZ T
2
|uεr | |usε | |r − s|2H−2 dsdr
= αH
kuε k|H|
0
0
Z Z Z Z
αH T T ε ε
≤
|ur+y | |us+z | |r − s|2H−2 dydzdsdr
4ε2 0 0 −ε −ε
Z Z Z
αH T T 2ε
≤
|ur | |us | |r − s + x|2H−2 dxdsdr.
ε 0 0 −2ε
2
There exists a constant dH = max(24−2H , 2H−1
) such that
1
ε
Z
2ε
−2ε
|r − s + x|2H−2 dx ≤ dH |r − s|2H−2 .
9
(19)
Indeed, we can assume that r > s, and if |r − s| > 4ε, then
1
ε
Z
2ε
−2ε
|r − s + x|2H−2 dx ≤ 4|r − s − 2ε|2H−2
≤ 24−2H |r − s|2H−2
and for |t − s| ≤ 4ε
1
ε
Z
2ε
2H−2
|r − s + x|
−2ε
dx ≤
=
≤
Hence,
2
ε
Z
4ε
x2H−2 dx
0
2
(4ε)2H−1
2H − 1
2
|t − s|2H−2 .
2H − 1
kuε k2|H| ≤ eH kuk2|H| .
In a similar way we can show that
kDuε k2|H|⊗|H| ≤ eH kDuk2|H|⊗|H| ,
for some constant eH > 0, and now the proof of (18) is complete.
Step 2. Let us see that, for all ε > 0,
Z
−1
(2ε)
0
T
us (Bs+ε − Bs−ε )ds = δ(uε ) + Aε u,
where
Aε u = (2ε)−1
Z
0
{un }
T
(20)
‹
Š
Dus , 1[s−ε,s+ε] H⊗H ds.
In fact, take a sequence
⊂ ST such that un → u in the norm of the
1,2
space D (|H|) as n tends to infinity. By formula (15) it follows that
−1
(2ε)
Z
T
uns (Bs+ε
0
−1
− Bs−ε )ds = (2ε)
Z
T
0
δ(uns 1[s−ε,s+ε] )ds + Aε un ,
which implies
(2ε)−1
Z
0
T
usn (Bs+ε − Bs−ε )ds = δ(un,ε ) + Aε un ,
10
By Step 1 it follows that un,ε converges in D1,2 (|H|) to the process uε as n
tends to infinity. As a consequence, δ(un,ε ) will converge in L2 (Ω) to δ(uε )
as n tends to infinity. On the other hand,
ŒZ T
Œ2
Œ
Œ
n
Œ
(us − us ) (Bs+ε − Bs−ε )dsŒŒ
Œ
0
! ’Z
“2
T
2
n
sup |Br − Bs |
≤
|us − us | ds
0
|r−s|≤2ε
≤ T
2−2H
2
sup |Br − Bs |
|r−s|≤2ε
!Z
T
0
Z
T
0
|uns − us | |unr − ur | |r − s|2H−2 ds
which converges to zero in probability as n tends to infinity. In a similar
way we can easily check that Aε un converges in probability to Aε u, which
completes the proof of (20).
Step 3. Let us prove that δ(uε )converges in L2 (Ω) to δ(u) as ε → 0. In
fact, take a sequence {un } ⊂ ST such that un → u in the norm of the space
D1,2 (|H|) as n tends to infinity. Then, for all ε > 0, n ≥ 1 we can write
E |δ(uε ) − δ(u)|2
o
n
≤ 3 E |δ(uε ) − δ(un,ε )|2 + E |δ(un,ε ) − δ(un )|2 + E |δ(un ) − δ(u)|2 .
By Step 1 we can easily deduce that for all δ > 0 there exists an integer nδ
such that for all n ≥ nδ ,
o
n
E |δ(uε ) − δ(u)|2 ≤ 3 E |δ(un,ε ) − δ(un )|2 + δ .
Letting now ε → 0 it follows that
lim E |δ(uε ) − δ(u)|2 ≤ 3δ,
ε→0
which implies the desired convergence.
Step 4. It remains to check the convergence in probability of Aε u to
Z TZ T
αH
Ds ut |t − s|2H−2 dsdt
0
0
We have
ε
−1
A u = (2ε)
= (2ε)−1
= (2ε)−1
Z
Z
T
0
T
0
Z
0
Š
‹
Dus , 1[s−ε,s+ε] H⊗H ds
’Z s+ε
“
Z T
2H−2
Dt us
|t − r|
dr dtds
0
T
Z
0
s−ε
ε
T
Dt us
’Z
−ε
11
“
|t − s + σ|2H−2 dσ dtds.
By Step 1 we know that, for some positive constant dH ,
Z ε
−1
(2ε)
|t − s + σ|2H−2 dσ ≤ dH |t − s|2H−2 .
−ε
Applying the dominated
theorem we deduce that the term Aε u
R T Rconvergence
T
converges a.s. to αH 0 0 Ds ut |t − s|2H−2 dsdt, as we wanted to prove.
RT
Remark 1 By a similar arguments one could show that the integral 0 ut dBt
also coincides with the forward and backward integrals, under the assumptions of Proposition 3.
Remark 2 A sufficient condition for (16) is
Z
0
T
’Z
s
T
“ p1
|Ds ut | dt
ds < ∞,
p
1
.
for some p > 2H−1
Remark 3 Let F be a function of class C 2 (R) such that
ˆ
‰
2
max |F (x)|, |F 0 (x)|, |F 00 (x)| ≤ ceλx ,
where c and λ are positive constant such that λ < 2T12H . Then (see [3, The1,2
(|H|) and the following
orem 2]) the process F 0 (Bt ) belongs to the space DB
version of Itô’s formula holds
Z t
€ 0

F (Bt ) = F (0) + δ F (B· )1[0,t] + H
F 00 (Bs )s2H−1 ds.
0
Taking into account that
Z tZ
r
€

Ds F 0 (Br ) (r − s)2H−2 dsdr
0
0
Z t
Z r
F 00 (Br )
(r − s)2H−2 dsdr
=
0
0
Z t
1
F 00 (Br )r2H−1 dr
=
2H − 1 0
we obtain
F (Bt ) = F (0) +
Z
0
12
t
F 0 (Bs )dBs .
5
Estimates for the stochastic integral
Suppose that u = {ut , t ∈ [0, T ]} is a stochastic process in the space D1,2 (|H|)
such that condition (16) holds. Then, for any t ∈ [0, T ] the process u1[0,t]
also belongs to D1,2 (|H|) and satisfies (16). Hence, by Proposition 3 we can
Rt
RT
define the indefinite integral 0 us dBs = 0 us 1[0,t] (s)dBs and the following
decomposition holds
Z
t
0
us dBs = δ(u1[0,t] ) + αH
Z tZ
0
0
T
Dr us |s − r|2H−2 drds.
The second summand in this expression is a process with absolutely continuous paths that can be studied by means of usual methods. Therefore,
in
Rt
p
order to deduce L estimates and to study continuity properties of 0 us dBs
we can reduce our analysis to the process δ(u1[0,t] ). In this section we will
establish Lp maximal
€ this divergence process. We will make
R t estimates for
use of the notation 0 us δBs = δ u1[0,t] .
1,p
For any p > 1 we denote by L1,p
H the set of processes u in D (|H|) such
that
kukp 1,p =: E kukpL1/H ([0,T ]) + E kDukpL1/H ([0,T ]2 ) < ∞.
LH
From (11) and Lemma 1 we obtain
kukD1,p (|H|) ≤ bH kukL1,p .
H
(21)
1,2
and, as a consequence, the space LH
is included in the domain of the divergence operator δ.
By Meyer’s inequalities (see for example [18]), if p > 1, a process u ∈
D1,p (|H|) belongs to the domain of the divergence in Lp (Ω), and we have
‘

p
p
.
E |δ (u)|p ≤ CH,p kEuk|H|
+ E kDuk|H|⊗|H|
1,p
we can write
As a consequence, if u belongs to LH
‘

E |δ (u)|p ≤ CH,p kEukpL1/H ([0,T ]) + E kDukpL1/H ([0,T ]2 )
(22)
≤ CH,p kukp 1,p .
LH
The following
theorem will give us a maximal Lp -estimate for the indefinite
Rt
integral 0 us δBs .
13
Theorem 4 Let p > 1/H. Let u = {ut , t ∈ [0, T ]} be a stochastic process
1
in L1,p
H−ε for some 0 < ε < H − p . Then the following inequality holds
E
"’Z
ŒZ t
Œp !
Œ
Œ
sup ŒŒ us δBs ŒŒ
≤ C
0
t∈[0,T ]
“p(H−ε)
1
|Eus | H−ε ds
T
0

Z
+E 
T
0
’Z
T
1
|Ds ur | H dr
0
p(H−ε) 

ds
,
H
“ H−ε
where the constant C depends on H, ε and T .
Proof. Set α = 1 − p1 − ε. Then 1 − H < α < 1 − p1 . Using the equality
Rt
cα = r (t − θ)−α (θ − r)α−1 dθ we can write
Z
t
us δBs = cα
0
Z
0
t
us
’Z
t
(t − r)
s
α−1
−α
(r − s)
“
dr δBs .
Using Fubini’s stochastic theorem (see for example [18]) we have that
“
’Z r
Z t
Z t
α−1
−α
δBs dr.
us (r − s)
(t − r)
us δBs = cα
0
0
0
Hölder’s inequality and condition α < 1 −
ŒZ t
Œp
Z
Œ
Œ
Œ us δBs Œ ≤ cα,p
Œ
Œ
0
0
Œ
Œ
Œ
t ŒZ r
0
0≤t≤T
T
0
Using now inequality (22) we obtain
(Z
Œp !
ŒZ t
Œ
Œ
≤ Cα,p
E sup ŒŒ us δBs ŒŒ
0
T
0
+E
Z
T
0
yields
us (r − s)
0
from where it follows that
ŒZ t
Œp !
Z
Œ
Œ
Œ
Œ
E sup Œ us δBs Œ
≤ cα,p E
0≤t≤T
1
p
ŒZ
Œ
Œ
Œ
’Z
0
r
0
us (r − s)
r
(r − s)
Z
0
= : Cα,p (I1 + I2 ) .
14
Œp
Œ
δBs ŒŒ dr,
r
0
’Z
α−1
α−1
H
T
(r − s)
α−1
Œ
Œp
δBs ŒŒ dr.
|Eus |
α−1
H
1
H
“pH
dr
ds
|Dθ us |
1
H
dθds
“pH
dr
)
For the term I1 we apply the Hardy-Littlewood inequality (12) with q = pH
H
and we obtain
and p = H−ε
’Z
I1 ≤ Cα,p,H
T
0
|Eur |
1
H−ε
dr
“p(H−ε)
.
A similar argument can be used in order to estimate the term I2 and we
obtain
p(H−ε)

H
“ H−ε
Z T ’Z T
1
,
I2 ≤ Cα,p,H E 
dr
|Ds ur | H ds
0
0
which completes the proof.
Remark 5 The terms I1 and I2 in the proof of Theorem 4 can also be
estimated by Hölder’s inequality, taking into account that α > 1 − H. With
this approach we obtain the following maximal inequality, for any p > H1
and any process u in L1,p
1/H
"Z
Œp
ŒZ t
Œ
Œ
E sup ŒŒ us δBs ŒŒ ≤ C
t∈[0,T ]
0
T
p
|Eus | ds + E
0
Z
T
0
’Z
T
0
|Ds ur |
1
H
“pH #
ds
dr ,
where the constant C > 0 depends on p, H and T .
6
Continuity of the integral process
In this section we will use the preceeding estimates for the divergence opin order to study the continuity of the integral processes of the form
Rerator
t
0 us δBs .
Theorem 5 Let u = {ut , t ∈ [0, T ]} be a stochastic process in the space
L1,p
H , where pH > 1 and assume that
Z
T
0
p
|Eur | dr +
Z
T
0
Then the integral process Xt =
E
’Z
T
0
nR
t
|Dθ ur |
1
H
dθ
“pH
dr < ∞.
o
δB
u
,
t
∈
T
]
has an a.s. continuous
[0,
s
s
0
modification. Moreover, for all γ < H −
Cγ a.s. finite such that
1
p
there exists a random constant
|Xt − Xs | ≤ Cγ |t − s|γ .
15
Proof. Using the estimate (22) we can write
E |Xt − Xs |p
(’Z
“pH
’Z t ’Z
t
1
|Eur | H dr
≤ CH,p
+E
s
≤ CH,p (t − s)
pH−1
(Z
p
|Eur | dr +
s
|Dθ ur |
0
s
t
T
Z
s
t
E
’Z
1
H
dθ dr
T
0
“
|Dθ ur |
1
H
dθ
“pH )
“pH
dr
)
,
for some constant CH,p > 0. So, there exist a non-negative function A :
RT
[0, T ] → R+ such that 0 Ar dr < ∞ and
Z t
pH−1
p
Ar dr.
E |Xt − Xs | ≤ (t − s)
s
By Fubini’s theorem we obtain that, for all α ∈ (2, pH + 1)
Z TZ T
|Xt − Xs |p
E
dsdt
|t − s|α
0
0
“
’Z t
Z TZ t
pH−1−α
Ar dr dsdt
(t − s)
≤ 2
s
0
0
“
’Z r Z T
Z T
pH−1−α
(t − s)
dtds dr
= 2
Ar
0
=
0
r
Z
2
(pH − α) (pH − α + 1)
Then the random variable
Γ :=
h
i
Ar (T − r)pH−α+1 − T pH−α+1 + rpH−α+1 dr.
T
0
Z
0
T
Z
0
T
|Xt − Xs |p
dsdt
|t − s|α
is a.s. finite. Now, by Garsia-Rodemish-Ramsey lemma (see [9]) we deduce
that for γ := α−2
p , there exist a random constant Cγ a.s. finite such that
|Xt − Xs | ≤ Cγ |t − s|γ .
Taking into account that the condition α < pH +1 is equivalent to γ < H − p1
the proof is complete.
1,p
Remark 6. The above theorem proves
o u ∈ ∩p>1 L H ,
nR that, for a process
t
the indefinite integral process Xt = 0 us δBs , t ∈ [0, T ] is γ-Hölder continuous for all γ < H. .
Remark
R t 7.
R s If we assume also that hypothesis (16) holds, the integral
process 0 0 Dr us (s − r)2H−2 drds is continuous a.s. in the variable t, which
Rt
gives us the continuity of the Stratonovich integral process 0 us dBs .
16
7
Itô’s formula
The purpose of this section is to prove a change-of-variable formula for
the indefinite divergence integral. We recall first the local property of the
divergence operator.
Lemma 6 Let u be an element of D1,2 (H). If u = 0 a.s. on a set A ∈ F ,
then δ(u) = 0 a.s. on A.
Given a set L of H-valued random variables we will denote by Lloc
the set of H-valued random variables u such that there exists a sequence
{(Ωn , un )}, n ≥ 1} ⊂ F × L with the following properties:
i) Ωn ↑ Ω a.s.
ii) u = un , a.e. on [0, T ] × Ωn .
We then say that {(Ωn , un )} localizes u in L. If u ∈ D1,2
loc (H) by Lemma
6 we can define without ambiguity δ(u) by setting
δ(u)|Ωn = δ(un )|Ωn
for each n ≥ 1, where {(Ωn , un )} is a localizing sequence for u in L.
The following proposition asserts that the divergence processes have zero
quadratic variation.
Proposition 7 Assume that u = {ut , t ∈ [0, T ]} is a process in the space
D1,2
loc (|H|). Then,
“2
n−1
X ’Z ti+1
us δBs → 0
i=0
ti
in probability, as |π| → 0, where π denotes a partition π = {0 = t0 < ... < tn = T }
of the interval [0, T ]. Moreover, the convergence is in L1 (Ω) if u ∈ D1,2 (|H|).
17
Proof. It suffices to assume that u ∈ D1,2 (|H|). By the isometry
property of the divergence operator we have
E
n−1
X ’Z ti+1
i=0
= αH E
us δBs
ti
n−1
X Z ti+1
Z
“2
ti+1
ur us |r − s|2H−2 drds
ti
i=0 ti
Z
n−1
X ti+1 Z ti+1
+α2H E
i=0 ti
2H−2
× |r − η|
ti
Z
[0,T ]2
Dθ ur Dη us
|θ − s|2H−2 dθdηdrds,
and this expression converges to zero as |π| tends to zero because u belongs
to D1,2 (|H|).
Now we are in a position to prove the main result of this section.
Theorem 8 Let F be a function of class C 2 (R). Assume that u = {ut , t ∈
[0, T ]} is a process in the space D2,2
loc (|H|) such that the indefinite integral
Rt
Xt = 0 us δBs is a.s. continuous. 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
(23)
0
’Z T
“ “
’Z s
Z t
2H−2
00
F (Xs ) us
+ αH
|s − σ|
Dσ uθ δBθ dσ ds
0
0
0
“
’Z s
Z t
2H−2
00
dθ ds.
+ αH
uθ (s − θ)
F (Xs )us
0
0
Proof. uppose that (Ωn , un ) is a localizing sequence for un in D2,2 (|H|).
For each positive integer k let ψ k be a smooth function such that 0 ≤ ψ k ≤ 1,
ψ k (x) = 0 if |x| ≥ k + 1, and ψ k (x) = 1 if |x| < k + 1. Define
2
un,k
).
= unt ψ k (kun k|H|
t
Set Xtn,k =
Rt
un,k
s δBs and consider the family of sets
)
(
n
o
Gn,k = Ωn ∩ sup |Xt | ≤ k ∩ kun k2|H| ≤ k .
0
t∈[0,T ]
18
Define also F k = F ψ k . Then it suffices to show the result for the process
un,k and the function F k . In this way we can assume that u ∈ D2,2 (|H|),
kuk|H| is uniformly bounded and the functions F , F 0 and F 00 are bounded.
Moreover we can assume that the process X has a continuous version.
Set ti = it
n , i = 0, ..., n. Applying Taylor expansion up to the second
order we obtain
•
n−1
X”
2
 1 00 €  €
€
0
F (Xt ) − F (0) =
,
F (Xti ) Xti+1 − Xti + F X̄ti Xti+1 − Xti
2
i=0
where X̄ti denotes a random intermediate point between Xti and Xti+1 .
Applying now equality (15) it follows that
Z ti+1
Z ti+1
‹
Š €

F 0 (Xti ) us δBs + D F 0 (Xti ) , u1[ti ,ti+1 ] H .
us δBs =
F 0 (Xti )
ti
ti
Observe that by our assumptions, F 0 (Xti ) u belongs to D1,2 (|H|), and all
the terms in the above equality are square integrable. On the other hand,
Š € 0
‹

‹
Š
D F (Xti ) , u1[ti ,ti+1 ] H = F 00 (Xti ) u1[0,ti ] , u1[ti ,ti+1 ] H

œZ ti
00
.
Duθ δBθ , u1[ti ,ti+1 ]
+F (Xti )
0
H
Now the proof will be decomposed into several steps.
Step 1. The term
n−1
X
i=0
2
€ €
F 00 X̄ti Xti+1 − Xti
converges to zero in L1 (Ω) as n tends to infinity by Proposition 7.
Step 2. The term
n−1
X
i=0
= αH
Š
F 00 (Xti ) u1[0,ti ] , u1[ti ,ti+1 ]
n−1
X Z ti+1
i=0
ti
00
F (Xti ) ur
converges in L1 (Ω) to
’Z
Z t
00
αH
F (Xs )us
0
s
0
’Z
0
‹
H
ti
2H−2
us (r − s)
2H−2
uθ (s − θ)
19
“
dθ ds
“
ds dr
as n tends to infinity by the dominated convergence theorem and the continuity of F (Xt ).
Step 3. Let us see that
n−1
X
00
F (Xti )
i=0
œZ
0
ti
Duθ δBθ , u1[ti ,ti+1 ]

H
converges in L1 (Ω) to
αH
Z
t
00
F (Xs ) us
0
’Z
T
’Z
2H−2
|s − σ|
0
“
s
“
Dσ uθ δBθ dσ ds.
0
Notice that
œZ
ti
Duθ δBθ , u1[ti ,ti+1 ]
0
= αH
Z
Z
T
0
ti+1
ti
’Z
ti
0

H
“
Dσ uθ δBθ us |s − σ|2H−2 dsdσ.
Then we can write
n−1
X
00
F (Xti )
i=0
Z
−αH
= αH
t
−αH
:
Duθ δBθ , u1[ti ,ti+1 ]
0
F (Xs ) us
0
n−1
X Z ti+1
’Z
ti
00
i=0 ti
T
×
œZ
‚
2H−2
|s − σ|
ti
T
2H−2
|s − σ|
0
’Z
s
“
s
“
“
Dσ uθ δBθ dσ ds
“
Dσ uθ δBθ dσ ds
0
00
H
’Z
0
ƒ
F 00 (Xti ) − F 00 (Xs ) us
0
n−1
X Z ti+1
i=0
’Z

F (Xti ) us
’Z
T
0
= T1 + T2 .
20
|s − σ|
2H−2
’Z
s
ti
“
“
Dσ uθ δBθ dσ ds
Let us first consider the term T2 . We have the following estimate
Œ
“ Œ
’Z s
n−1
X Z ti+1 ŒZ T
Œ
2H−2
Œ
Πds
αH E
|s
−
σ|
u
u
δB
dσ
D
s
σ
θ
θ
Œ
Œ
n−1
XZ
0
ti
i=0
ŒZ
Œ
Œ
Œ
ti+1
T
ti
Z
s∧σ
∂K
(s, r)
∂s
0
0
i=0 ti
Œ
’Z s
“
Œ
∂K
Œ
×
(σ, r)drdσ Œ ds
Dσ uθ δBθ
∂σ
ti
“
Z t ’Z t
∂K
≤
kus k2
(s, r)ds
∂s
0
r
Z T ’Z s

“


∂K

D
u
× sup sup 
δB
r)dσ
(σ,
σ θ
θ

 dr.
∂σ
i s∈[ti ,ti+1 ]
r
ti
2
= E
us
Notice that the term
Z t ’Z
0
t
∂K
(s, r)ds
kus k2
∂s
r
“2
dr
is finite because kuk2 belongs to the space H. On the other hand, we can
write
“•
“ ’Z s
”’Z s
Dσ0 uθ δBθ
E
Dσ uθ δBθ
ti
ti
Z
= αH
E(Dσ uθ Dσ0 uθ0 )|θ − θ0 |2H−2 dθdθ0
[ti ,s]2
Z
Z
E(Dη Dσ uθ Dη0 Dσ0 uθ0 )
+α2H
[0,T ]2 [ti ,s]2
0 2H−2 0
|θ − η|2H−2 dθdθ0 dηdη 0 .
×|θ − η |
Hence,
E
Z
T
0
= αH E
’Z
Z
T
r
T
0
Z
’Z
T
0
0 2H−2
×|σ − σ |
“
“2
∂K
(σ, r)dσ dr
∂σ
ti
’Z s
“ ’Z s
“
Dσ uθ δBθ
Dσ0 uθ δBθ
s
Dσ uθ δBθ
ti
ti
0
dσdσ ,
and this converges to zero, uniformly in s and i, because u belongs to the
space D2,2 .
21
In a similar way we can show that
’Z
Z t ŒZ T
Œ
2H−2
Œ
u
|s
E
−
σ|
s
Œ
0
s
0
0
“ Œ
Œ
Dσ uθ δBθ dσ ŒŒ ds < ∞.
(24)
(24) which implies that the term T1 converges to zero in L1 (Ω).
Step 4. Observe that F 0 (Xs ) us belongs to D1,2 (|H|), because kuk|H| is
uniformly bounded together with the function F and its first two derivatives.
The following convergence
n−1
X
n=0
€ 
F 0 X̄ti u1[ti ,ti+1 ] → F 0 (X) u
(25)
holds in the norm of the space L2 (Ω; |H|). In fact,
2


n−1

X 0 € 
E
F X̄ti u1[ti ,ti+1 ] → F 0 (X) u


n=0
= E
n−1
X m−1
X Z ti+1
n=0 m=0 ti
Z
|H|
tj+1
tj
Œ€ 0 € 
 Œ
Œ F X̄t − F 0 (Xs ) us Œ
i
Œ€ € 
 Œ
× Œ F 0 X̄ti − F 0 (Xr ) ur Œ |r − s|2H−2 drds,
which converges to zero as n → ∞, by the dominated convergence theorem.
As a consequence, for any smooth and cylindrical random variable G we
haveLet us see that
!
’ Z t
“
n−1
X Z ti+1 € 
0
0
lim E G
F X̄ti us δBs = E G
F (Xs ) us δBs . (26)
n
0
n=0 ti
On the other hand, by the previous steps, the sequence
converges in L1 (Ω), as n tends to infinity to
Z
t
00
’Z
T
2H−2
F (Xt ) − F (0) − αH
F (Xs ) us
|s − σ|
0
0
’Z s
“
Z t
2H−2
00
− αH
F (Xs )us
uθ (s − θ)
dθ ds.
0
0
This allows us to complete the proof.
22
Pn−1 R ti+1
’Z
n=0 ti
s
€ 
F 0 X̄ti us δBs
“
“
Dσ uθ δBθ dσ ds
0
Remarks: If the process u is adapted, then Itô’s formula can be written as
Z t
F (Xt ) = F (0) +
(27)
F 0 (Xs )us δBs
0
“
“
’Z s ’Z θ
Z t
2H−2
00
F (Xs ) us
Dσ uθ dσ δBθ ds
|s − σ|
+ αH
0
0
0
’Z s
“
Z t
2H−2
00
+ αH
F (Xs )us
uθ (s − θ)
dθ ds.
0
0
On the other hand, 2αH (s − θ)2H−2 1[0,s] (θ) is an approximation of the identity as H tends to 12 . Therefore, taking the limit as H converges to 12 in
equation (27) we recover the usual Itô’s formula for the classical Brownian
motion, and taking the limit in equation (23) we obtain the Itô’s formula
for the Skorohod integral proved by Nualart and Pardoux (see, for instance,
[18]).
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