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Vol. 8 (2003) Paper no. 14, pages 1–31.
Journal URL
http://www.math.washington.edu/˜ejpecp/
Itô Formula and Local Time for the Fractional Brownian Sheet
Ciprian A. Tudor1 and
Frederi G. Viens2
1 Laboratoire de Probabilités, Université de Paris 6
4, place Jussieu, 75252 Paris, France
Email: tudor@ccr.jussieu.fr
and
2 Dept.
Statistics and Dept. Mathematics, Purdue University
150 N. University St., West Lafayette, IN 47907-2067, USA
Email: viens@purdue.edu
Abstract. Using the techniques of the stochastic calculus of variations for Gaussian processes, we derive an Itô formula for the fractional Brownian sheet with Hurst parameters
bigger than 1/2. As an application, we give a stochastic integral representation for the local
time of the fractional Brownian sheet.
Key words and phrases: fractional Brownian sheet, Itô formula, local time, Tanaka
formula, Malliavin calculus.
AMS subject classification (2000): primary 60H07; seconday 60G18, 60G15, 60J55.
Submitted to EJP on February 18, 2003. Final version accepted on August 20, 2003.
1
1
Introduction
Let (Btα )t∈[0,1] be the fractional Brownian (fBm) motion with Hurst parameter α ∈ (0, 1).
When α = 1/2, B 1/2 is the standard Brownian motion. When α 6= 1/2, this process is not a
semimartingale and in this case the powerful theory and tools of the classical Itô stochastic
calculus cannot be applied to it. However, its special properties such as self-similarity,
stationarity of increments, and long range dependence make this process a good candidate
as a model for different phenomena. Therefore, a development of the stochastic calculus
with respect to the fBm was needed. We refer to [1], [6], [9] and [11] for some approaches
of this stochastic calculus.
The aim of this paper is to develop a stochastic calculus for the two-parameter
fBm introduced in [3], also known as the fractional Brownian sheet. As the one-parameter
fBm, the fractional Brownian sheet is not a semimartingale. But the Malliavin calculus for
Gaussian processes can be adapted to it, Skorohod stochastic integrals can be defined, and
it will allow us to derive an Itô formula when the Hurst parameters are bigger than 1/2
which extends the Itô formula for the two parameters martingales (see [10], [16] and [13]
for a complete exposition of this topic). As an application we study the representation of
the local time of the fractional Brownian sheet in terms of Skorohod stochastic integrals.
Our paper is organized as follows: Section 2 contains the basic notions of the Malliavin calculus for Gaussian processes and the definition of the fractional Brownian sheet. In
Section 3 we give an analogue of the Doob-Meyer decomposition for the square of a martingale, and an Itô formula for the fractional Brownian sheet. The interpretation of a specific
new term appearing in the Itô formula is given in Section 4. Finally, as an application of
this formula, Section 5 contains the study of the local time of the fractional Brownian sheet
using stochastic integrals and an occupation time formula.
2
2.1
Preliminaries
The Malliavin calculus for Gaussian processes
Let T = [0, 1]. Consider (Bt )t∈T a centered Gaussian process, with covariance E(Bt Bs ) =
R(t, s).
We consider the canonical Hilbert space H associated with the process B, defined
as the closure of the linear space generated by the function {1[0,t] , t ∈ T } with respect to
the scalar product h1[0,t] , 1[0,s] i = R(t, s). Then the mapping 1[0,t] → Bt gives an isometry
between H and the first chaos generated by {Bt , t ∈ T } and B(φ) denotes the image of a
element φ ∈ H.
We can develop a stochastic calculus of variations, or a Malliavin calculus, with
respect to B. For a smooth H-valued functional F = f (B(ϕ1 ), · · · , B(ϕn )) with n ≥ 1 ,
f ∈ Cb∞ (Rn ) and ϕ1 , · · · , ϕn ∈ H we put
DB (F ) =
n
X
∂f
(B(ϕ1 ), · · · , B(ϕn ))ϕj
∂xj
j=1
2
and DB F will be closable from Lp (Ω) into Lp (Ω; H). Therefore, we can extend D B to the
closure of smooth functionals on the Sobolev space D k,p defined by the norm
kF kpk,p = E|F |p +
k
X
j=1
kDB,j F kpLp (Ω;H⊗j ) .
Similarly, for a given Hilbert space V we can define Sobolev spaces of V -valued random
variables D k,p (V ) (see [17]).
Consider the adjoint δ B of DB . Then its domain is the class of u ∈ L2 (Ω; H) such
that
E|hDB F, uiH | ≤ CkF k2
and δ B (u) is the element of L2 (Ω) given by
E(δ B (u)F ) = EhD B F, uiH
for every F smooth. It is well-known that D 1,2 (H) is included in the domain of δ B .
Note that Eδ B (u) = 0 and the variance of the divergence operator δ B is given by
Eδ B (u)2 = Ekuk2H + EhD B u, (DB u)∗ iH⊗H
(1)
where (D B u)∗ is the adjoint of D B u in the Hilbert space H ⊗ H. This last identity implies
that
Eδ B (u)2 ≤ Ekuk2H + EkD B uk2H⊗H .
(2)
We will need the property
F δ B (u) = δ B (F u) + hD B F, uiH
(3)
if F ∈ D1,2 and u ∈ Dom(δ B ) such that F u ∈ Dom(δ B ).
2.2
Representation of fBm
Let now B = B α be the fractional Brownian motion with parameter α ∈ (0, 1). This process
is centered Gaussian, B0 = 0 and
´
1 ³ 2α
(4)
s + t2α − |t − s|2α .
R(t, s) =
2
We know that B admits a representation as Wiener integral of the form
Z t
Bt =
K(t, s)dWs ,
(5)
0
where W = {Wt , t ∈ T } is a Wiener process, and K(t, s) is the kernel
µ ¶
t
α− 21
α− 12
K(t, s) = dα (t − s)
+s
F1
,
s
3
(6)
dα being a constant and
F1 (z) = dα
µ
1
−α
2
¶Z
z−1
0
This kernel satisfies the condition (see [15]):
³
´
1
3
θα− 2 1 − (θ + 1)α− 2 dθ.
3
∂K
1 s 1
(t, s) = dα (α − )( ) 2 −α (t − s)α− 2 .
∂t
2 t
(7)
(8)
α
α
We will denote by Hα its canonical Hilbert space and by D B and δ B the corresponding
Malliavin derivative and Skorohod integral with respect to B α .
Fix now α ∈ ( 21 , 1) and define the operator K ? from the set of step functions on T
(denoted by E) to L2 (T ) by
Z 1
∂K α
(r, s)dr.
(K ? f )(s) =
f (r)
∂r
s
It was proved in [1] that the space Hα of the fBm can be represented as the closure of E
with respect to the norm kf kHα = kK ? f kL2 (T ) and K ? is an isometry between Hα and a
closed subspace of L2 (T ). This fact implies the following relation between the Skorohod
integration with respect to B α and W , the Wiener process:
α
δ B (u) = δ W (K ? u)
α
(9)
α
for any Hα -valued random variable u in Dom(δ B ). We will call δ B (u) the Skorohod integral of the process u with respect to B. Since α > 1/2, we have the following representation
of the scalar product in Hα :
Z 1Z 1
hf, giHα = α (2α − 1)
f (a) g (b) |a − b|2α−2 dadb.
(10)
0
0
This formula holds as long as the integral above is finite when f and g are replaced by |f |
and |g|.
2.3
Fractional Brownian sheet
α,β
Consider (Ws,t
)s,t∈[0,1] a fractional Brownian sheet with parameters α, β ∈ (0, 1). This
process is Gaussian, it starts from 0 and its covariance is
³
´
α,β
α,β
E Wt,s
Wu,v
= Rα (t, u)Rβ (s, v)
´
¢ 1 ³ 2β
1 ¡ 2α
s + v 2β − |s − v|2β .
t + u2α − |t − u|2α
2
2
This process self similar and with stationary increments and it admits a continuous version.
We refer to [3] for the basic properties of W α,β . The fractional Brownian sheet with Hurst
parameters α, β ∈ (0, 1) can be defined as (see [4])
Z tZ s
α,β
Ws,t
=
K α (t, u)K β (s, v)dWu,v
=
0
0
4
where (Wu,v )u,v∈[0,1] is the Brownian sheet.
The canonical Hilbert Space Hα,β of the fractional Brownian sheet is the closure
of linear space generated by the indicator functions on [0, 1]2 with respect to the scalar
product
h1[0,t]×[0,s] , 1[0,u]×[0,v] iHα,β = Rα (t, u)Rβ (s, v).
Fix α, β > 21 . Notice that in this case, by tensorization of (10), we have for every f, g ∈ H α,β
Z 1Z 1Z 1Z 1
f (a, b)g(m, n) |a − m|2α−2 |b − n|2β−n dadbdmdn
hf, giHα,β = c(α)c(β)
0
0
0
0
∗,2
and c(α) = α(2α − 1). Define the operator Kα,β
on the space of step function on [0, 1]2 to
L2 ([0, 1]2 ) given by
Z 1Z 1
´
³
∂K α
∂K β 0
∗,2
f (r, r0 )
(r, t)
(r , s)drdr 0 .
(11)
Kα,β f (s, t) =
∂r
∂r0
s
t
We have
∗,2
∗,2
giL2 ([0,1]2 ) = hf, giHα,β .
f, Kα,β
hKα,β
Indeed,
(12)
∗,2
∗,2
hKα,β
f, Kα,β
giL2 ([0,1]2 )
¶
Z 1 Z 1 µZ 1 Z 1
∂K β
∂K α
=
(a, u)
(b, v)dadb
f (a, b)
∂a
∂b
0
0
u
v
µZ 1 Z 1
¶
∂K β
∂K α
×
(m, u)
(n, v)dmdn dudv
g(m, n)
∂m
∂n
u
v
¶
µZ a∧m Z b∧n
Z 1Z 1Z 1Z 1
∂K β
∂K α
(m, u)
(n, v)dmdndudv dadbdmdn
=
f (a, b)g(m, n)
∂m
∂n
0
0
0
0
0
0
Z 1Z 1Z 1Z 1
∂ 2 Rα
∂ 2 Rβ
f (a, b)g(m, n)
=
(a, m)
(b, n)
∂a∂m
∂b∂n
0
0
0
0
Z 1Z 1Z 1Z 1
f (a, b)g(m, n) |a − m|2α−2 |b − n|2β−n dadbdmdn
= c(α)c(β)
0
0
0
0
= hf, giHα,β .
One can develop a Malliavin stochastic calculus with respect to the fractional Brownian sheet following the method of Section 2.1, using rectangles in [0, 1]2 instead of intervals
α,β
α,β
its associated Skorohod integral and Malliavin derivaand DW
in [0, 1]. Denote by δ W
tive respectively. A consequence of the identity (12) is that the following expression of the
α,β
α,β
Skorohod integral δ W
holds: For f ∈ Dom(δ W ) ⊂ L2 (Ω, Hα,β ),
Z 1Z 1
Z 1Z 1
∗,2
α,β
f (u, v)dWu,v
=
(Kα,β
f )(u, v)dWu,v
0
0
0
0
¶
Z 1 Z 1 µZ 1 Z 1
∂K α
∂K β
f (a, b)
=
(a, u)
(b, v)dadb dWu,v .
∂a
∂b
v
u
0
0
5
All properties of the anticipating Skorohod integration with respect to Gaussian processes
will hold in this case.
α,β
α,β
. These processes are real
and t → Ws,t
Consider now the processes s → Ws,t
β
α
α
fractional Brownian motions with the same law as t W and s W β respectively and with
covariances
´
¢
1¡
1 ³ 2β
2α
2α
2β
R1 (s1 , s2 ) = t2β s2α
and R2 (t1 , t2 ) = s2α
t1 + t2β
−
|t
−
t
|
1
2
1 + s2 − |s1 − s2 |
2
2
2
(13)
respectively. Since these processes are fBm’s themselves, we have the Malliavin calculus
with respect to them. We denote by δ1t the Skorohod integral with respect to the fractional
α,β
Brownian motion s → Ws,t
and by δ2s the Skorohod integral with respect to the fractional
α,β
Brownian motion t → Ws,t
. We will only use the integration by parts formulas
³
´
α,β t
α,β
α,β
f (Ws,t
)δ1 (u) = δ1t f (Ws,t
(14)
)u + t2β f 0 (Ws,t
)hu, 1[0,s] iHα
and
3
³
´
α,β s
α,β
α,β
f (Ws,t
)δ2 (u) = δ2s f (Ws,t
)u + s2α f 0 (Ws,t
)hu, 1[0,t] iHβ .
(15)
The Itô formula for the fractional Brownian sheet
We will start this section with three technical Lemmas that will be used below, and that
rely on the following exponential quadratic growth assumption on f and/or its derivatives:
(H) We say that a function g on R satisfies Condition (H) if, for |x| large enough, we have
|g (x)| ≤ M exp ax2 where a is a constant that depends only on α and/or β, and M
is an arbitrary constant.
Lemma 1 Let s, t ∈ [0, 1] and π 1 : 0 = s0 < s1 < . . . < sn = s and π 2 : 0 = t0 < t1 <
. . . < tm = t be two partitions of the intervals [0, s] and [0, t] respectively. Assume f , f 0 ,
and f 00 satisfy Condition (H). Then we have the following convergences in L 2 (Ω) when the
partition meshes both tend to zero:
1 If (W α )t∈[0,1] is a fBm with parameter α ∈ ( 21 , 1), then for any f ∈ Cb2 (R),
Z t
m−1
´ m−1
X ³
X Z tj+1
α 2α
α
f (Wuα )u2α dWuα .
f
(W
)t
dW
→
δ f (Wtαj )t2α
1
(·)
=
[tj ,tj+1 ]
tj j
u
j
j=0
j=0
0
tj
(16)
α,β
2 If (Ws,t
)s,t∈[0,1] is a fractional Brownian sheet with parameters α, β ∈ ( 12 , 1), then for
any f ∈ Cb2 (R),
n−1
´ n−1
X m−1
X ³
X m−1
X Z si+1 Z tj+1
α,β
α,β
f (Wsα,β
)dWu,v
(17)
δ 1[si ,si+1 ]×[tj ,tj+1 ] (·)f (Wsi ,tj ) =
i ,tj
i=0 j=0
Z tZ s
→
0
0
i=0 j=0
α,β
α,β
f (Wu,v
)dWu,v
.
si
tj
(18)
6
Proof: We use the notation ∆j = [tj , tj+1 ]. Concerning point 1, we prove first the
convergence in L2 (Ω; Hα ) of the sum
m−1
X³
f (Wtαj )t2α
j 1∆j (u)
j=0
´
(19)
to the process
f (Wuα )u2α 1[0,t] (u) =
m−1
X
f (Wuα )u2α 1∆j (u).
(20)
j=0
The L2 -norm of the difference D between the two terms can be estimated as follows
¯
¯2
¯m−1
¯
³
´
X
¯
¯
2α
α
α 2α
¯
D = E¯
f (Wtj )tj − f (W· ) (·)
1∆j ¯¯
¯ j=0
¯ α
H
¯
¯2
¯m−1
¯
¯
¯ X ¯¯
¯
¯
¯
≤ 2E ¯¯
¯f (Wtαj ) − f (W·α )¯ t2α
j 1 ∆j ¯
¯ j=0
¯ α
H
¯2
¯
¯
¯m−1
¯
¯X
¯
|f (W·α )| | (·)2α − t2α
+ 2E ¯¯
j |1∆j ¯
¯ α
¯ j=0
H
We can write that k
P
aj k2Hα ≤
P
|aj |Hα |ak |Hα . Also we have the inequality
¯
¯
¯
¯ 0
¯¯
¯
α¯
α ¯
α ¯¯ α
α
¯
−
W
W
≤
sup
)
−
f
(W
)
f
(W
)
f
(W
¯ tj
¯
u ¯.
u ¯
x
tj
j
j,k
x∈[0,1]
This follows from the mean value theorem and the ¯fact¯ that W is almost-surely continuous.
Since here we will have u ∈ ∆j , so that |u − tj | ≤ ¯π 2 ¯, it also follows that
¯
¯
¯
¯
¯
¯
sup
|Waα − Wbα | .
¯f (Wtαj ) − f (Wuα )¯ ≤ sup ¯f 0 (Wxα )¯
a,b:|a−b|≤|π 2 |
x∈[0,1]
Therefore, using Hölder’s inequality,
# "
"
¯
¯ 0
4
D ≤ 2E sup ¯f (Wxα )¯ E
x∈[0,1]
+ 2E
"
sup |f
x∈[0,1]
(Wxα )|2
#
sup
a,b:|a−b|≤|π 2 |
|π 2 |4α
|Waα − Wbα |4
#
X
j,k
h1∆j , 1∆k iHα
X
h1∆j , 1∆k iHα .
j,k
Since W is Gaussian
and almost-surely continuous,
the general Dudley-Fernique theory
h
i
4
α
α
guarantees that E supa,b:|a−b|≤r E |Wa − Wb | converges to 0 as r tends to 0. To guarantee
that D goes to zero as |π 2 | → 0, we only need to assume that condition (H) holds for f and
7
f 0 with a < 8−1 V ar(Z)−1 where Z = supx∈[0,1] |Wxα |. Indeed, since Z is the supremum of a
£
¤
centered continuous Gaussian process, a classical result of Fernique implies that E exp aZ 2
is finite for a < V ar(Z)−1 2−1 . Using the inequality (2), the proof of point 1 will be finished
if we show that the derivative of (19) converges to the derivatives of (20). It holds that
Ds f (Wtαj ) = f 0 (Wtαj )1[0,tj ] (s) and Ds f (Wuα ) = f 0 (Wuα )1[0,u] (s)
and
¯
¯
¯m−1
¯
³
´
X
¯
¯
α 2α
α 2α
¯
Ds f (Wtj )tj − Ds f (Wu )u
1∆j (u)¯¯
¯
¯ j=0
¯
¯
¯
¯
¯m−1
´
¯
¯X ³ 0
α
0
α
2α
¯
f (Wtj ) − f (Wu ) 1[0,tj ] (s)tj 1∆j (u)¯¯
≤ E¯
¯
¯ j=0
¯
¯
¯m−1
¯
¯X 0
¯
2α
¯
+ E ¯¯
f (Wuα )(t2α
−
u
)1
(s)1
(u)
∆j
[0,tj ]
j
¯
¯ j=0
¯
¯
¯
¯
¯m−1
¯
¯X 0
α
2α
f (Wu )u 1[tj ,u] (s)1∆j (u)¯¯ .
+ E ¯¯
¯
¯ j=0
We seek convergence of these three terms as functions of (s, u) in the space (H α )⊗2 . The
convergence of the first two terms to zero can be done using the bounds presented above
in this proof, using Condition (H) on f 0 and f 00 . Finally, using the definition of the scalar
product in (Hα )⊗2 , including the fact that we can use the inequality 1[tj ,u] (s) ≤ 1∆j (s), we
obtain
¯2
¯
¯
¯m−1
¯
¯X 0
2α
α
¯
f (W· ) (·) 1[tj ,·] (?)1∆j (·)¯¯
E¯
¯ α ⊗2
¯ j=0
(H )
#
"
¯
¯2 X
®
­
h1∆ (·) , 1∆ (·)iHα 1∆ (?) , 1∆ (?) α
≤ E sup ¯f 0 (Wxα )¯
j
x∈[0,1]
≤E
"
k
j
k
H
j,k
2
#
X
¯ 0
¯
°
°
2
°1 ∆ ° 2 α  .
sup ¯f (Wxα )¯ 
j H
x∈[0,1]
j
h
i
Condition (H) on f 0 guarantees that E supx∈[0,1] |f 0 (Wxα )|2 is finite, and the other factor
¯ ¯P
¯ ¯
P
converges to 0 with ¯π 2 ¯ since j k1∆j k2Hα ≤ ¯π 2 ¯ j k1∆j kHα converges to zero.
Concerning point 2, we need first to prove that
n−1
X m−1
X
i=0 j=0
´
³
α,β
)
−
f
(W
)
1∆i ×∆j (u, v) f (Wsα,β
u,v
i ,tj
8
(21)
¡
¢
goes to zero in L2 Ω; Hα,β . We have, as before,
¯2
¯
¯n−1
³
´¯¯
X
¯X m−1
α,β ¯
1∆i ×∆j (u, v) f (Wsα,β
) − f (Wu,v
) ¯
E ¯¯
i ,tj
¯ α,β
¯ i=0 j=0
H
#
"
"
¯ ³
´¯4
¯
α,β ¯
sup
≤ E1/2
sup ¯f 0 Wx,y
¯ E1/2
|a1 −b1 |≤|π 1 |,|a2 −b2 |≤|π 2 |
x,y∈[0,1]
·
X
i,i0 ,j,j 0
h1∆i , 1∆i0 iHα h1∆j , 1∆j 0 iHβ
¯2
¯
¯ α,β
α,β ¯
¯Wa1 ,a2 − Wb1 ,b2 ¯
#
¯ ¯
¯ ¯
which, assuming condition (H) on f 0 , converges to zero as ¯π 1 ¯ and ¯π 2 ¯ tend to zero,
due to the almost-sure continuity of W α,β as a centered Gaussian process on [0, 1]2 . The
convergence of the Malliavin derivatives of (21) is no more difficult than the other proofs
above; we leave it to the reader. The proof of the lemma is completed.
¤
The next lemma is trivial.
Lemma 2 Let s, t ∈ [0, 1], π 1 , π 2 as in Lemma 1 and A, B two finite variation processes.
Then, for every f ∈ C(R) and g ∈ C(R2 ) the following convergences hold in L1 (Ω).
m−1
X
j=0
and
n−1
X m−1
X
i=0 j=0
¡
¢
f (tj ) Btj+1 − Btj →
Z
t
f (u)dBu
(22)
0
¡
¢¡
¢
g(si , tj ) Asi+1 − Asi Btj+1 − Btj →
Z
0
sZ t
g(u, v)dAu dBv
(23)
0
The next lemma is useful for dealing with certain specific double Skorohod integrals
that we will encounter.
Lemma 3 For s, t ∈ [0, 1] and for fixed
n and m, let the partitions π 1 , π 2 be again
¢
¡ 1 integers
2 . Also let
as in Lemma 1. Denote π = πn,m = πn , πm
aπ(u1 ,v1 );(u2 ,v2 ) =
n−1
X m−1
X
1[si ,si+1 ]×[0,tj ] ((u1 , v1 ))1[0,si ]×[tj ,tj+1 ] ((u2 , v2 ))
(24)
i=0 j=0
¡
¢⊗2
for every u = (u1 , u2 ), v = (v1 , v2 ) ∈ [0, 1]2 . Then the sequence (aπ )π converges in Hα,β
as the mesh of π tends to 0 (when n and m both tend to infinity) to the function a defined
for every (u1 , v1 ) , (u2 , v2 ) ∈ [0, 1]2 by:
a ((u1 , v1 ) ; (u2 , v2 )) = 1[0,s] (u1 ) 1[0,t] (v2 ) 1[0,u1 ] (u2 ) 1[0,v2 ] (v1 ) .
9
Proof: We denote s0 = t0 = 0 and ∆i = ∆ni = [si ; si+1 ], ∆j = ∆m
j = [tj ; tj+1 ], as
well as Di = Din = [0, si ] and Dj = Djm = [0, tj ]. Although the names of these intervals
are ambiguous, the names of their indices allow us to identify which variables they refer
to, which lightens the notation. To further lighten notation, we allow D j (t) to denote the
indicator function 1Dj (t), and similarly for the other intervals. We only need to show that
the doubly indexed sequence of functions
[an,m ((u1 , v1 ) , (u2 , v2 )) − a ((u1 , v1 ) , (u2 , v2 ))]
£
¡¡
¢ ¡
¢¢
¡¡
¢ ¡
¢¢¤
· an,m u01 , v10 , u02 , v20 − a u01 , v10 , u02 , v20


n−1
X
X m−1
Dj (v1 ) ∆i (u1 ) Di (u2 ) ∆j (v2 ) − a ((u1 , v1 ) , (u2 , v2 ))
=
i=0 j=0

·
n−1
X
X m−1
i=0 j=0

¡ 0¢ ¡ 0¢ ¡ 0¢
¡ 0¢
¡¡ 0 0 ¢ ¡ 0 0 ¢¢
D j v1 ∆ i u 1 D i u 2 ∆ j v2 − a u 1 , v1 , u 2 , v 2 
(25)
¢2
¡
defined on [0, 1]2 × [0, 1]2 converges to 0 as m and n tend to infinity in the space L1 (µ)
where µ is the measure defined by
¡
¢
µ du1 du2 dv1 dv2 du01 du02 dv10 dv20
¯
¯2α−2 ¯
¯
¯
¯
¯
¯
¯u2 − u02 ¯2α−2 ¯v1 − v10 ¯2β−2 ¯v2 − v20 ¯2β−2 du1 du2 dv1 dv2 du01 du02 dv10 dv20 .
= ¯u1 − u01 ¯
We first note that, by virtue of the partitions, for every pair of points (u1 , v1 ) , (u2 , v2 ) ∈
[0, 1]2 , except for the countably many points in the partitions π 1 and π 2 , we have convergence of an,m ((u1 , v1 ) ; (u2 , v2 )) to a ((u1 , v1 ) ; (u2 , v2 )). In other words, an,m converges
to a Lebesgue-almost everywhere in [0, 1]2 × [0, 1]2 . Thus the quantity in (25) converges
¡
¢2
in [0, 1]2 × [0, 1]2 almost everywhere to 0 with respect to the Lebesgue measure, and
also with respect to the measure µ since µ is Lebesgue-absolutely continuous. Since the
sequence in (25) is bounded by 1, and since 1 is integrable with respect to µ, by dominated
convergence, the lemma is proved.
¤
We will also need the basic lemma for the convergence of the Skorohod integral.
¡
¢
Lemma 4 Let un be a sequence of elements in Dom(δ) which converges to u in L2 Ω; Hα,β .
Suppose that δ(un ) converges in L2 (Ω) to some square integrable random variable G. Then
u belongs to the domain of δ and δ(u) = G. This result also holds for sequences in H α , or
in Hβ , and their respective Skorohod integrals.
3.1
α,β 2
A decomposition for (Ws,t
)
Let, from now on, s, t ∈ [0, 1] and π 1 : 0 = s0 < s1 < . . . < sn = s and π 2 : 0 = t0 < t1 <
. . . < tm = t be two partitions of the intervals [0, s] and [0, t] respectively. Without loss of
generality, we can choose the dyadic partitions.
10
Proposition 1 Let
( 21 , 1). Then it holds
³
³
α,β
Ws,t
α,β
Ws,t
´
s,t∈[0,1]
´2
=2
a fractional Brownian sheet with parameters α, β ∈
Z tZ
0
s
0
α,β
α,β
Wu,v
dWu,v
+ 2M̃s,t + s2α t2β
(26)
where, with a and aπ as in Lemma 3, and with δ (2) the double Skorohod integral with respect
to W α,β on [0, 1]2 × [0, 1]2 ,


n−1
X
X m−1
M̃s,t := lim δ (2) 
1[si ,si+1 ]×[0,tj ] (·)1[0,si ]×[tj ,tj+1] (?)
(27)
|π|→0
= lim δ
i=0 j=0
(2)
|π|→0
π
(a ) = δ (2) (a) .
Although this result is contained in the Itô formula that we prove independently in
¡
¢2
the next section, we give a detailed self-contained proof of this decomposition for W α,β as
a dydactic tool that enables us to introduce some of the essential ingredients and techniques
used in proving Itô’s formula, including the random field M̃ . This section helps us make
the next section more concise and easier to read.
Proof of the proposition: First write Taylor’s formula
α,β 2
(Ws,t
)
=2
n−1
X
i=0
´
³
¯
α,β
α,β
W
−
W
Wsα,β
s
,t
s
,t
,t
i+1
i
i
¯
where Wsα,β
is an arbitrary point located between Wsα,β
and Wsα,β
. Observe first that the
i ,t
i ,t
i+1 ,t
2
1
L (Ω)-limit as |π | → 0 of the difference
n−1
X³
i=0
¯
Wsα,β
− Wsα,β
i ,t
i ,t
´³
Wsα,β
− Wsα,β
i+1 ,t
i ,t
´
is equal to 0. Indeed, by Hölder inequalities,
¯n−1
¯
¯X ³
´³
´¯2
¯
¯
¯
Wsα,β
− Wsα,β
Wsα,β
− Wsα,β
E¯
¯
i ,t
i ,t
i+1 ,t
i ,t
¯
¯
i=0
n−1
X
≤E
≤
≤
i,l=0
sup
¯³
¯¯
¯
´³
´
¯ α,β
α,β
α,β
α,β
α,β ¯
α,β ¯ ¯ α,β
Wsα,β
−
W
¯Wsi+1 ,t − Wsi ,t ¯ ¯Wsl+1 ,t − Wsl ,t ¯ Wsi+1 ,t − Wsi ,t
,t
s
,t
l+1
l
|a−b|≤|π 1 |
sup
|a−b|≤|π 1 |
µ ¯
¯2 ¶ 12
¯2 ¶ 12 n−1
X µ ¯¯ α,β
¯ α,β
α,β
α,β
α,β ¯
α,β ¯
E ¯(Wsi+1 ,t − Wsi ,t )(Wsl+1 ,t − Wsl ,t )¯
E ¯Wa,t − Wb,t ¯
i,l=0
Ã
!
µ ¯
µ ¯
¯2 ¶ 12 n−1
¯4 ¶ 41 2
X
¯
¯ α,β
¯
¯
α,β
E ¯Wa,t − Wb,t
≤ |π 1 |4α−2
− Wsα,β
E ¯Wsα,β
¯
¯
i+1 ,t
i ,t
i=0
11
and this goes to 0 when α > 21 . Therefore it suffices to study the limit of the sum
T =2
n−1
X
i=0
Using that g(t) = g(0) +
T =2
n−1
Xh
X m−1
i=0 j=0
=2
n−1
X m−1
X
i=0 j=0
i=0 j=0
=2
n−1
X m−1
X
i=0 j=0
=I +J
j=0
(g(tj+1 ) − g(tj )) we will obtain
³
´
³
´i
α,β
α,β
α,β
α,β
α,β
Wsα,β
W
−
W
−
W
W
−
W
si+1 ,tj+1
si ,tj+1
si ,tj
si+1 ,tj
si ,tj
i ,tj+1
´
³
α,β
α,β
α,β
α,β
+
W
−
W
−
W
W
Wsα,β
si ,tj
si+1 ,tj
si ,tj+1
si+1 ,tj+1
i ,tj+1
n−1
X³
X m−1
+2
Pm−1
³
´
α,β
α,β
Wsα,β
W
−
W
si+1 ,t
si ,t .
i ,t
Wsα,β
− Wsα,β
i ,tj+1
i ,tj
´³
¢
¡
Wsα,β
δ 1∆i ×∆j
i ,tj+1
Wsα,β
− Wsα,β
i+1 ,tj
i ,tj
+2
n−1
X m−1
X³
i=0 j=0
´
− Wsα,β
Wsα,β
i ,tj+1
i ,tj
´³
Wsα,β
− Wsα,β
i ,tj
i+1 ,tj
´
where we denoted by ∆i = [si , si+1 ] and ∆j = [tj , tj+1 ]. Now, by property (3), the summand
I becomes
I=2
n−1
X m−1
X
i=0 j=0
=2
n−1
X m−1
X
i=0 j=0
n−1
³
´
X m−1
X
α,β
δ Wsi ,tj+1 1∆i ×∆j + 2
h1[0,si ]×[0,tj+1 ] , 1∆i ×∆j iHα,β
i=0 j=0
³
´
δ Wsα,β
+2
1
∆
×∆
,t
i
j
i j+1
n−1
X m−1
X
i=0 j=0
h1[0,si ] , 1∆i iHα h1[0,tj+1 ] , 1∆j iHβ .
By Lemma 1 point 2, and Lemma 2, we have the following convergences in L2 (Ω)
n−1
X m−1
X
i=0 j=0
Z tZ
³
´
δ Wsα,β
1
→
i ,tj+1 ∆i ×∆j
0
s
0
α,β
α,β
Wu,v
dWu,v
(28)
and
n−1
X
X m−1
i=0 j=0
=
h1[0,si ] , 1∆i iHα h1[0,tj+1 ] , 1∆j iHβ
´
¢ X ³ 2β
1
1 X ¡ 2α
2β
2α
tj+1 − t2β
→ s2α t2β .
si+1 − s2α
i − (si+1 − si )
j − (tj+1 − tj )
4
4
i
j
12
(29)
The term J admits the following decomposition
J =2
n−1
X
X m−1
δ
i=0 j=0
= 2M (π) + 2
h³
n−1
´
i
X
X m−1
α,β
Wsα,β
−
W
1
+
2
h1[0,si ] , 1∆i iHα h1[0,tj ] , 1∆j iHβ
∆i ×[0,tj ]
si ,tj
i ,tj+1
i=0 j=0
n−1
X
X m−1
i=0 j=0
h1[0,si ] , 1∆i iHα h1[0,tj ] , 1∆j iHβ
where
M (π) =
n−1
X
X m−1
i=0 j=0

= δ (2) 
δ
h³
´
i
α,β
Wsα,β
−
W
1
∆
×[0,t
]
,t
s
,t
i
j
i j+1
i j
n−1
X
X m−1
i=0 j=0

1∆i ×[0,tj ] (·)1[0,si ]×∆j (?) = δ (2) (aπ )
(30)
where δ (2) denotes the double Skorohod integral and the process aπ is defined in Lemma 3.
As before, we can show that
n−1
X
X m−1
i=0 j=0
h1[0,si ] , 1∆i iHα h1[0,tj ] , 1∆j iHβ →|π|→0
1 2α 2β
s t in L2 .
4
(31)
Lemma 4 can be used in combination with Lemma 3 to assert immediately that δ (2) (aπ )
converges in L2 (Ω) to δ (2) (a); indeed the convergence of the deterministic integrands in
¡ α,β ¢⊗2
H
is in Lemma 3, and since the corresponding Skorohod integrals are equal to
α,β 2
(Ws,t ) minus other terms that were already proved to converge in L2 (Ω), they converge
in L2 (Ω). Note that Lemma 4 is not stated for the convergence of the double Skorohod
integral, but the simple one; however it can be still applied because a double integral can
be written as a simple one in which the integrand is convergent in L2 (Ω; Hα,β )). Combining
this convergence and (28), (29), (31), we find the result of the proposition.
¤
3.2
Itô formula
α,β
In the sequel of this article we will make use of the notation du Wu,v
to denote the differential
α,β
α,β
α,β
. As before, dWu,v
of the f Bm given by u 7→ Wu,v for fixed v, and similarly for dv Wu,v
denote the differential of the fractional Brownian sheet with respect to both parameters.
α,β
Theorem 1 Let f ∈ C 4 (R) and (Ws,t
)s,t∈[0,1] a fractional Brownian sheet with α, β ∈
13
( 12 , 1). Also assume that f, f 0 , f 00 , f 000 and f (iv) satisfy Condition (H). Then it holds
Z tZ s
α,β
α,β
α,β
f (Ws,t
) = f (0) +
f 0 (Wu,v
)dWu,v
0
0
Z tZ s
Z tZ s
α,β
00
α,β 2α−1 2β−1
f 00 (Wu,v
)dM̃u,v
f (Wu,v )u
v
dudv +
+ 2αβ
0
0
0
0
Z tZ s
Z tZ s
α,β 2α 2β−1
α,β
000
α,β 2α−1 2β
α,β
f 000 (Wu,v
)u v
du Wu,v
dv
f (Wu,v )u
v dudv Wu,v + β
+α
0
0
0
0
Z tZ s
α,β 4α−1 4β−1
f iv (Wu,v
)u
v
dudv
(32)
+ αβ
0
0
where, by definition,
Z tZ
0
s
0

α,β
f 00 (Wu,v
)dM̃u,v = lim δ (2) 
|π|→0
n−1
X
X m−1
i=0 j=0
³
´

f 00 Wsα,β
1[0,si ]×[tj ,tj+1 ] (·)1[si ,si+1 ]×[0,tj ] (?)
i ,tj
where the last limit exists in L2 (Ω). More specifically, we have the following formula, which
can be taken as a definition of the integral with respect to M̃ which would otherwise only be
a formal notation:
Z Z
t
0
s
0
α,β
f 00 (Wu,v
)dM̃u,v := δ (2) (N )
where for every (u1 , v1 ) , (u2 , v2 ) ∈ [0, 1]2 the function N is defined by:
³
´
N ((u1 , v1 ) ; (u2 , v2 )) = f 00 Wuα,β
1[0,s] (u1 ) 1[0,t] (v2 ) 1[0,u1 ] (u2 ) 1[0,v2 ] (v1 ) .
1 ,v2
Proof: Let s, t ∈ [0, 1] and write Taylor’s formula with fixed t. It holds that
f
³
α,β
Ws,t
´
= f (0) +
n−1
X
i=0
+
n−1
X
i=0
´
´³
³
α,β
α,β
W
−
W
f 0 Wsα,β
si+1 ,t
si ,t
i ,t
´2
³
´³
¯
α,β
α,β
f 00 Wsα,β
W
−
W
si+1 ,t
si ,t
i ,t
14
(33)
(34)
¯
where Wsα,β
is a point on the segment from Wsα,β
to Wsα,β
. First, we can prove that the
i ,t
i+1 ,t
i ,t
2
last term goes to zero in L (Ω). Indeed, by Hölder’s inequality,
¯
¯
¯X ³
´³
´2 ¯2
¯α,β
¯
¯
α,β
α,β
00
f Wsi ,t
E¯
Wsi+1 ,t − Wsi ,t ¯
¯
¯
i
≤E
n−1
X
i,l=0
≤ E1/2
≤ CK
"
¯ ³
´ ³
´¯ ³
´2 ³
´2
¯α,β
¯α,β ¯
¯ 00
α,β
α,β
α,β
Wsα,β
−
W
¯f Wsi ,t f 00 Wsl ,t ¯ Wsi+1 ,t − Wsi ,t
sl ,t
l+1 ,t
#
·¯
·¯
¯8 ¸
¯8 ¸
¯ ³
´¯4 X
¯ 00
¯
α,β ¯
α,β ¯
1/4 ¯ α,β
1/4 ¯ α,β
α,β
E
sup ¯f Wx
¯Wsl+1 ,t − Wsl ,t ¯
¯Wsi+1 ,t − Wsi ,t ¯ E
¯
x∈[0,1]2
X
i,l
|si+1 − si |
i,l
2α
|sl+1 − sl |2α
where K is a constant obtained by using Condition (H) on f 00 , C is a universal constant;
this all tends clearly to zero since 2α > 1. It remains to study the limit of the term
T =
n−1
X
i=0
´
´³
³
α,β
α,β
.
W
−
W
f 0 Wsα,β
s
,t
s
,t
,t
i+1
i
i
Writing the fact that g(t) = g(0) +
T =
n−1
Xh
X m−1
i=0 j=0
=
n−1
X m−1
X
i=0 j=0
+
j=0
(g(tj+1 ) − g(tj )) we get
³
´³
´i
³
´³
´
α,β
α,β
α,β
α,β
α,β
0
W
W
−
W
W
−
W
−
f
f 0 Wsα,β
si ,tj
si+1 ,tj
si ,tj
si+1 ,tj+1
si ,tj+1
i ,tj+1
³
´³
´
α,β
α,β
α,β
α,β
f 0 Wsα,β
W
−
W
−
W
+
W
,t
s
,t
s
,t
s
,t
s
,t
i j+1
i+1 j+1
i j+1
i+1 j
i j
n−1
X m−1
X³
i=0 j=0
Pm−1
´
³
´
´´ ³
³
α,β
α,β
α,β
0
f 0 Wsα,β
−
f
W
W
−
W
,t
s
,t
s
,t
s
,t
i j+1
i+1 j+1
i j+1
i j
= A + B.
3.2.1
The estimation of the term A
Note that
³
´
¢
¡
α,β
α,β
α,β
(·)
Wsα,β
−
W
−
W
+
W
=
δ
1
∆
×∆
,t
s
,t
s
,t
s
,t
i
j
i+1 j+1
i j+1
i+1 j
i j
15
and by the properties of the Skorohod integral we obtain
A=
n−1
X
X m−1
i=0 j=0
+
n−1
X m−1
X
i=0 j=0
h ³
´
i
δ f 0 Wsα,β
1
(·)
∆
×∆
i
j
i ,tj+1
³
´
h1[0,si ]×[0,tj+1 ] , 1∆i ×∆j iHα,β
f 00 Wsα,β
,t
i j+1
= A1 + A2 .
The convergence of A1 follows from Lemma 1 point 2, assuming condition (H) for f 0 , f 00
and f 000 , yielding
n−1
X
X m−1
h
δ f
i=0 j=0
0
³
Wsα,β
i ,tj+1
´
i
1∆i ×∆j (·) →L2 (Ω)
Z tZ
0
s
0
α,β
α,β
f 0 (Wu,v
)dWu,v
.
(35)
Concerning the term A2 , we can write
A2 =
n−1
X m−1
X
i=0 j=0
=
n−1
X m−1
X
i=0 j=0
³
´
f 00 Wsα,β
h1[0,si ] , 1∆i iHα h1[0,tj+1 ] , 1∆j iHβ
i ,tj+1
´
³
´1¡
¢ ³ 2β
2β
2β
2α
2α
2α
t
−
t
−
(t
−
t
)
s
−
s
−
(s
−
s
)
f 00 Wsα,β
j+1
j
i+1
i
i
j+1
j
i ,tj+1
4 i+1
and as above this converges, due to Lemma 2, to
Z t ³
Z t ³
´
´
α,β
α,β
f 00 Wu,v
f 00 Wu,v
u2α−1 v 2β−1 dudv
dRuα dRvβ = αβ
(36)
0
0
where Ruα = Rα (u, u) = u2α and Rvβ = Rβ (v, v) = v 2β . due to Lemma 2.
3.2.2
Since
The estimation of the term B
³
´
³
´
´³
´
³
¯α,β
α,β
α,β
α,β
0
00
f 0 Wsα,β
−
f
W
=
f
W
W
−
W
si ,tj
si ,tj
si ,tj+1
si ,tj
i ,tj+1
¯
where Wsα,β
is a point located between Wsα,β
and Wsα,β
, the term B becomes
i ,tj
i ,tj
i ,tj+1
B=
X
i,j
´³
´³
´
³
¯
α,β
α,β
α,β
α,β
f 00 Wsα,β
W
−
W
W
−
W
si ,tj+1
si ,tj
si+1 ,tj
si ,tj
i ,tj
and using the argument as above, assuming Condition (H) for f 000 , we can prove that the
limit of B is the same as the limit of the term
´³
´
X ³ α,β ´ ³ α,β
α,β
α,β
B0 =
f 00 Wsi ,tj Wsi ,tj+1 − Wsα,β
W
−
W
si+1 ,tj
si ,tj .
i ,tj
i,j
16
It holds that
B0 =
X
i,j
+
X
i,j
´i
³
´ h
³
α,β
α,β
−
W
W
δ
1
f 00 Wsα,β
[0,si ]×∆j
si ,tj
si+1 ,tj
i ,tj
f
00
³
Wsα,β
i ,tj
´
h1[0,si ] , 1∆i iHα h1[0,tj+1 ] , 1∆j iHβ = B1 + B2 .
The second term is similar with the term A2 studied before and we have its convergence to
Z t ³
Z t ³
´
´
α,β
α,β
dRuα dRvβ = αβ
u2α−1 v 2β−1 dudv.
(37)
f 00 Wu,v
f 00 Wu,v
0
0
For the term B1 we can write
´i
³
X ³ α,β ´ s h
α,β
−
W
B1 =
f 00 Wsi ,tj δ2i 1∆j (·) Wsα,β
s
,t
,t
i j
i+1 j
i,j
=
X
i,j
´i
´
³
h ³
α,β
α,β
−
W
1
(·)
W
δ2si f 00 Wsα,β
∆
si ,tj
si+1 ,tj
j
i ,tj
´
³
´³
³
´
α,β
α,β
2β
2β
2α 1
2β
f 000 Wsα,β
s
W
−
W
t
−
t
−
(t
−
t
)
j+1
j
i
si+1 ,tj
si ,tj
j
i ,tj
2 j+1
i,j
´i
³
X s h ³ α,β ´ tj
α,β
−
W
=
δ2i f 00 Wsi ,tj δ1 (1∆i (?)) 1∆j (·) Wsα,β
si ,tj
i+1 ,tj
+
X
+
X
i,j
i,j
³
´
³
´
tj
2β
2β
2α 1
2β
f 000 Wsα,β
δ
(1
(?))
s
−
(t
−
t
)
−
t
t
j+1
j
∆
i
i
1
j
i ,tj
2 j+1
and by applying the integration by parts several times for the Skorohod integral with respect
to the corresponding fractional Brownian motion, we obtain
i
X s tj h ³ α,β ´
B1 =
δ2i δ1 f 00 Wsi ,tj 1∆j (·)1∆i (?)
i,j
h ³
´
i1¡
¢
2α
2α 2β
s2α
tj
δ2si f 000 Wsα,β
1
(·)
∆
i+1 − si − (si+1 − si )
j
i ,tj
2
i,j
³
´¸
X tj · ³ α,β ´
2β
2β
000
2α 1
2β
+
δ1 f
Wsi ,tj 1∆i (?)si
t
− tj − (tj+1 − tj )
2 j+1
i,j
´ ¡
³
X
¢
2β
2β
2β 1
2α
2α 2β
2α 1
−
(t
−
t
)
−
t
s2α
tj
t
+
f (iv) (Wsα,β
)s
j+1
j
i+1 − si − (si+1 − si )
i
j
j+1
i ,tj
2
2
+
X
i,j
= B11 + B12 + B13 + B14 .
For the summand B12 we have
i¡
¢
1 X si h 000 ³ α,β ´ 2β
2α
B12 =
δ2 f
Wsi ,tj tj 1∆j (·) s2α
i+1 − si
2
i,j
i
h ³
´
X
1
2β
+
(·)
(si+1 − si )2α .
δ2si f 000 Wsα,β
t
1
∆
,t
j
j
i j
2
i,j
17
By Lemma 1 point 1, assuming condition (H) for f 000 and f (iv) , it holds that, for every si
m−1
X
j=0
δ2si
h
f
000
³
Wsα,β
i ,tj
´
t2β
j 1∆j (·)
i
→L2 (Ω)
Z
t
0
³
´
f 000 Wsα,β
v 2β dv Wsα,β
i ,v
i ,v
and this fact
together with Lemma 2 implies that the first part of B12 goes to the integral
R s R t 000 ³ α,β ´ 2α−1
α,β
α 0 0f
Wu,v u
duv 2β dv Wu,v
. Using Condition (H) on f 000 , since 2α > 1 it is easy
to observe that the last part of B12 converges to zero and therefore we have the convergence
Z sZ t
³
´
α,β
α,β
f 000 Wu,v
B12 → α
u2α−1 duv 2β dv Wu,v
.
(38)
0
In the same way
Z
B13 → β
0
0
sZ t
0
³
´
α,β
α,β
u2α v 2β−1 du Wu,v
dv.
f 000 Wu,v
(39)
Lemma 2 also gives that B1,4 converges to
1
4
Z
0
sZ t
f
(iv)
0
α,β 2α 2β
(Wu,v
)u v dRα (u)dRβ (v)
= αβ
Z
0
sZ t
0
α,β 4α−1 4β−1
f (iv) (Wu,v
)u
v
dudv. (40)
Now, note that
B11 =
X
i,j
h ³
´
i
δ (2) f 00 Wsα,β
1
(·)1
(?)
[0,s
]×∆
∆
×[0,t
]
i
j
i
j
i ,tj
where δ (2) denotes again the double Skorohod integral, and specifically,
h ³
´
i Z si Z tj+1 Z si+1 Z tj
α,β
(2)
00
δ
f Wsi ,tj 1[0,si ]×∆j (·)1∆i ×[0,tj ] (?) =
f 00 (Wsα,β
)dWuα,β
dWuα,β
.
1 ,v1
2 ,v2
i ,tj
0
tj
si
0
We now show
Lemma 5 Using the same partitions as in Lemma 3, define the process
X ³ α,β ´
π
N(u
=
f 00 Wsi ,tj 1∆i ×[0,tj ] ((u1 , v1 ))1[0,si ]×∆j ((u2 , v2 )).
1 ,v1 );(u2 ,v2 )
(41)
i,j
³ ¡
¢⊗2 ´
to
Then assuming condition (H) for f 00 the sequence (N π )π converges in L2 Ω; Hα,β
the function N defined in the statement of Theorem 1.
Proof: The proof uses Condition (H) combined with the proof of Lemma 3. We
¡
¢⊗2
only need to establish
convergence
in Hα,β
for fixed randomness ω ∈ Ω. Indeed, the
´
³ the
¡ α,β ¢⊗2
2
follows by dominated convergence under condition (H)
convergence in L Ω; H
18
¯
¯¢−1
¡
for f 00 , as long as a < 2−1 V ar max[0,1]2 ¯W α,β ¯
for instance. Now, using the almosteverywhere convergence established in proving Lemma 3, and noticing that we can rewrite
N π as


X ³ α,β ´
π
N(u
=
f 00 Wsi ,tj 1∆i (u1 )1∆j (v2 )
1 ,v1 );(u2 ,v2 )
i,j


X
·
1∆i ×[0,tj ] ((u1 , v1 ))1[0,si ]×∆j ((u2 , v2 )) ,
i,j
we only need to show that for almost every ω ∈ Ω, for fixed u1 , v2 , the quantity
´
X ³ α,β
00
f Wsi ,tj (ω) 1∆i (u1 )1∆j (v2 )
i,j
³
´
α,β is almost-surely continuous on [0, 1]2 , this conconverges to f 00 Wuα,β
1 ,v2 (ω) . Since W
vergence is trivial. The lemma is proved.
¤
To guarantee that Lemma 4 can be invoked to conclude that the term


n−1
´
³
X
X m−1
1[0,si ]×∆j (·)1∆i ×[0,tj ] (?)
(42)
f 00 Wsα,β
δ (2) (N π ) := δ (2) 
i ,tj
i=0 j=0
is convergent in L2 (Ω)
integral, since we already have the convergence of
³ to¡ a Skorohod
´
¢
⊗2
from Lemma 5, we only need to guarantee that the
the integrands in L2 Ω; Hα,β
Skorohod integrals in (42) converge to a random variable in L2 (Ω). However this is trivial
since the term in (42) is the difference of all the other terms that have been estimated in
α,β
this proof, and have been proved to converge in L2 (Ω) to f (Ws,t
) minus the sum of all
terms on the right hand side of (32) except for the one denoted by
Z tZ
0
s
0
³
´
α,β
f 00 Wu,v
dM̃u,v .
Therefore, the Skorohod integral in (42) converges to the Skorohod integral δ (2) (N ), and
(32) is established, so Theorem 1 is proved.
¤
4
Interpretation of the integral with respect to M̃
As we mentioned before in the statement of the Itô formula (Theorem 1), the notation M̃
is defined using a double Skorohod integral. However, the rationale for using a specific
19
differential notation for M̃ is the following. We can write
Z
Z
M̃ (s, t) =
dWuα,β
dWuα,β
1
(u1 ) 1[0,t] (v2 ) 1[0,u1 ] (u2 ) 1[0,v2 ] (v1 )
1 ,v1
2 ,v2 [0,s]
=
=
=
Z
u1 ,v1
s
u2 ,v2
t
Z
Z
u1
u1 =0 v2 =0 u2 =0
Z s Z t µ Z v2
u1 =0 v2 =0
Z s Z t
u1 =0
v2 =0
Z
v1 =0
v2
dWuα,β
dWuα,β
1 ,v1
2 ,v2
v1 =0
¶ µZ u1
¶
α,β
α,β
dWu1 ,v1
dWu2 ,v2
(43)
(44)
u2 =0
du1 Wuα,β
· dv2 Wuα,β
1 ,v2
1 ,v2
(45)
which shows that dM̃ (s, t) should be interpreted as
α,β
α,β
ds Ws,t
· dt Ws,t
.
(46)
To make the string of equalities above rigorous, the Fubini-type property used to go from
(43) to (44) must be justified, and the double stochastic integral in (45) must be properly
defined. This can be done without needing to invoke a new Skorohod integration theory,
by simply noting that the integral in (45) can be defined as a random variable in the
second Gaussian chaos of W α,β with an appropriate distribution, and checking that it
equals M̃ (s, t). We omit these details.
Defining stochastic integration with respect to M̃ is also trivial if the integrands
are deterministic. However, in connection with the general Itô formula (Theorem 1), we
are much more interested in random integrands. A proper theory of general Skorohod
integration with respect to M̃ can be given. We will not present this theory here, since it
constitutes only a tangent to the Itô formula that is the subject of this article, and especially
since it is not clear that it brings any more computational information than formulas (33)
and (34) which can be taken as a definition of general Skorohod integration against M̃ .
What we will do instead is to present briefly a somewhat formal calculation that shows
exactly how formula (33) is related to integration against M̃ . The main point in this
calculation is that, although the integrands are non-deterministic, the familiar Malliavinderivative-type corrections that come from pulling random non-time-dependent terms out
of Skorohod integrals do not appear in the final formula. We recall the general formula for
Skorohod integration with respect to an arbitrary Gaussian field, and a fixed L 2 random
variable F :
F δ (u) = δ (uF ) + hu; DF iH
where h·; ·iH is the inner product in the canonical Hilbert space of the underlying Gaussian
field.
We also recall that the stochastic differential of W α,β can be formally understood
as follows, using a standard Brownian sheet W :
Z u Z v
∂K β
∂K α
α,β
(u, a)
(v, b) dWa,b .
dWu,v
= dudv
∂v
a=0 b=0 ∂u
20
Therefore we have
I :=
=
Z
Z
u
u0 =0
u
u0 =0
Z
Z
v
v 0 =0
α,β
f (u, v) dWuα,β
0 ,v dWu,v 0
v
f (u, v)
v 0 =0
u0 Z v
∂K α ¡ 0 ¢ ∂K β
u ,a
(v, b) dWa,b
∂v
a=0 b=0 ∂u
Z u Z v0
¢ ∂K β ¡ 0 0 ¢
∂K α ¡
· dudv 0
u, a0
v , b dWa0 ,b0 .
∂v
a0 =0 b0 =0 ∂u
0
· du dv
Z
Now we pull the constant L2 (Ω)-random variable f (u, v) into the integral with respect to
dWa,b , modulo a correction term, then use Fubini to pull the Riemann integral with respect
to u0 all the way in (which requires no correction), yielding
·Z u Z v Z u
¸
Z v
α ¡
¢ ∂K β
0 ∂K
0
du
·dv
u ,a
(v, b) f (u, v) dWa,b − correction term
I=
∂u
∂v
a=0 b=0 u0 =a
v 0 =0
Z u Z v0
¢ ∂K β ¡ 0 0 ¢
∂K α ¡
0
u, a0
v , b dWa0 ,b0
· dudv
∂v
a0 =0 b0 =0 ∂u
¸
Z v ·Z u Z v
∂K β
α
α
= dudv
(v, b) f (u, v) dWa,b − correction term
(K (u, a) − K (a, a))
∂v
v 0 =0
a=0 b=0
Z u Z v0
¢ ∂K β ¡ 0 0 ¢
∂K α ¡
0
u, a0
v , b dWa0 ,b0
· dv
∂v
a0 =0 b0 =0 ∂u
·Z u Z v
¸
∂K β
α
K (u, a)
= dudv
(v, b) dWa,b
∂v
a=0 b=0
Z u Z v0
Z v
¢ ∂K β ¡ 0 0 ¢
∂K α ¡
dv 0 f (u, v)
u, a0
v , b dWa0 ,b0
·
∂v
a0 =0 b0 =0 ∂u
v 0 =0
Now we perform the same operation on f and the integral with respect to v 0 , yielding
¸
·Z u Z v
∂K β
α
(v, b) dWa,b
I = dudv
K (u, a)
∂v
a=0 b=0
#
"
Z v
Z u Z v0
α ¡
β ¡
¢
¢
∂K
∂K
u, a0
v 0 , b0 f (u, v) dWa0 ,b0 − correction term
·
dv 0
∂u
∂v
0
0
0
v =0
a =0 b =0
·Z u Z v
¸
∂K β
α
K (u, a)
= dudv
(v, b) dWa,b
∂v
·Z u Z va=0 b=0
¸
¢ β ¡ 0¢
∂K α ¡
0
·
u, a K v, b f (u, v) dWa0 ,b0 − correction term
a0 =0 b0 =0 ∂u
21
u
v
¸
∂K β
K (u, a)
(v, b) dWa,b
= dudv
∂v
a=0 b=0
¸
·Z u Z v
¢
¡
¢
∂K α ¡
u, a0 K β v, b0 dWa0 ,b0
· f (u, v)
a0 =0 b0 =0 ∂u
¸
¸· Z u
· Z v
¢
∂K α ¡
∂K β
1/2,β
α,1/2
0
du
u, a da0 Wa0 ,v
(v, b) db Wu,b
= f (u, v) dv
a0 =0 ∂u
b=0 ∂v
·Z
Z
α
α,β
α,β
= f (u, v) dv Wu,v
· du Wu,v
.
α,β
α,β
Modulo the definition of Skorohod integration with respect to dv Wu,v
· du Wu,v
, this calculation justifies the formula
Z tZ s
f (u, v)dM̃u,v
0
0
Z
Z
:=
dWuα,β
dWuα,β
f (u1 , v2 ) 1[0,s] (u1 ) 1[0,t] (v2 ) 1[0,u1 ] (u2 ) 1[0,v2 ] (v1 )
1 ,v1
2 ,v2
[0,1]2 [0,1]2
s
=
Z tZ
0
0
α,β
α,β
f (u, v) dv Wu,v
· du Wu,v
,
which can be summarized by
α,β
α,β
dM̃u,v = dv Wu,v
· du Wu,v
as announced in (46).
5
¤
Application to the local time of the fractional Brownian
sheet
As an application of the Itô formula, we will establish in this section the existence and the
stochastic integral representation of the local time of the fractional Brownian sheet.
5.1
Known results and motivation
We start with a short summary of the results on the local times for one and multiparameter
fractional Brownian motion.
motion B H was introduced
• The local time λat of the one parameter fractional Brownian
Rt
in [5] as the density of the occupation measure Γ → 0 1Γ (BsH )ds. The author proved
that λat has a jointly continuous version in the variables a and t. Moreover λat has
Hölder-continuous paths of order δ < 1 − H in time and of order γ < 1−H
2H in the
space variable a provided that H ≥ 31 . As a density of an occupation measure, the
local time is increasing in t and the measure La (dt) is concentrated on the level set
{s : BsH = a}. A chaos expansion of λat is given in [14], along with an L2 estimate.
22
• Another version of the local time of the fractional BrownianR motion was introduced in [8]
t
as the density of the occupation measure mt (Γ) = 2H 0 1Γ (BsH )s2H−1 ds. If H ≥ 31 ,
this local time satisfy the Tanaka formula
Z t
H
|Bt − a| = | − a| +
sign(Bsa )dBsH + Lat
0
where the stochastic integral is defined in the Skorohod sense. This formula was recently extended by [7] to H ∈ (0, 1). The regularity properties of λat can be transferred
to Lat by integrating by parts.
• In [12] the local time of the d-dimensional fractional sheet with
R N -parameters was
studied. This local time can be formally defined as Lat = [0,t] δa (WsH )ds, where
t = (t1 , . . . , tN ) ∈ T N , H = (H1 , . . . , HN ) and δa denotes the Dirac function. The
smoothness of the local time in the Sobolev-Watanabe spaces and its asymptotic behavior were studied.
• Recently, in [18], using the techniques of the Fourier analysis the authors proved the
existence of the local time of the N -parameters d-dimensional fractional Brownian
sheet satisfying the occupation density formula
Z
Z
´
³
f W Ht d t =
f (x)L(x, A)dx
d
A
R
for any Borel set A ⊂ RN and for any measurable function f : Rd → R. The existence
of a jointly continuous version of the local time is proved. Obviously, this local time
coincides with the one used in [12].
³
In this section we define the local time Las,t , a ∈ R of the fractional Brownian sheet
´
α,β
Ws,t
as the density of the occupation measure
s,t∈[0,1]
ms,t (Γ) = αβ
Z tZ
0
s
0
α,β 4α−1 4β−1
1Γ (Wu,v
)u
v
dudv.
(47)
Note that in the case α = β = 21 this local time coincide with the local time introduced in
[13] and [16] for the standard Wiener process with two parameters.
Remark 1 Using the integration by parts we can transfer the joint continuity of the local
time defined in [18] to Las,t . The details are left to the reader.
5.2
Let
Stochastic representation of local time
1
2
e−x /2ε
pε (x) = √
2πε
23
be the heat kernel with variance ε > 0. The following results gives an approximation in L 2
of the local time of the fractional Brownian sheet. It can be deduced from [12], where the
authors used the Wiener chaos decomposition of the local time.
Proposition 2 Let a ∈ R and s, t ∈ [0, 1] and define
La,ε
s,t
= αβ
Z tZ
0
s
0
α,β 4α−1 4β−1
pε (Wu,v
)u
v
dudv.
a
2
Then the random variable La,ε
s,t converges to Ls,t in L (Ω) as ε tends to zero.
The next result gives the stochastic integral representation of the local time.
Theorem 2 For any a ∈ R and s, t ∈ [0, 1], we have
¯³
¯³
´2 1 Z t Z s ¯
´
1 ¯¯ α,β
¯ α,β
¯
¯
α,β
α,β
α,β
− a dWu,v
¯Ws,t − a¯ Ws,t − a −
¯Wu,v − a¯ Wu,v
6
2 0 0
Z tZ s¯
¯
¯ α,β
¯
− 2αβ
¯Wu,v − a¯ u2α−1 v 2β−1 dudv
0
0
Z tZ s¯
¯
¯ α,β
¯
−
¯Wu,v − a¯ dM̃u,v
0
0
Z tZ s
α,β
α,β
sign(Wu,v
− a)u2α−1 v 2β dv Wu,v
du
−α
0
0
Z tZ s
α,β
α,β
sign(Wu,v
− a)u2α v 2β−1 du Wu,v
dv.
−β
Las,t =
0
0
Proof: Let us introduce the functions
gεIV
00
gε (x) =
Z
x
0
(x) = pε (x),
000
gε (y)dy,
000
gε (x) = 2
0
gε (x) =
Clearly, as ε tends to zero, we have
000
Z
x
gε (x) → g (x) = sign(x),
1
0
0
gε (x) → g (x) = x|x|,
2
x
0
pε (y)dy − 1,
gε (y)dy,
0
000
00
Z
00
gε (x) =
x
0
00
gε (x) → g (x) = |x|,
gε (x) → g(x) =
24
Z
1 2
x |x|.
6
0
gε (y)dy.
Let’s write Itô’s formula for the function gε (x − a), where a is a real number. We get
α,β
gε (Ws,t
− a)
Z tZ s
α,β
α,β
gε0 (Wu,v
− a)dWu,v
= gε (0) +
0
0
Z tZ s
Z tZ s
α,β
00
α,β
2α−1 2β−1
gε00 (Wu,v
− a)dM̃u,v
gε (Wu,v − a)u
v
dudv +
+ 2αβ
0
0
0
0
Z tZ s
Z tZ s
000
α,β
2α−1 2β
α,β
α,β
α,β
+α
gε (Wu,v − a)u
v dudv Wu,v + β
gε000 (Wu,v
− a)u2α v 2β−1 dvdu Wu,v
0
0
0
0
Z tZ s
α,β
+ αβ
gεIV (Wu,v
− a)u4α−1 v 4β−1 dudv.
0
0
We will decompose the proof into several steps.
Step 1.
Given the existence of moments of all orders for the supremum of continuous
Gaussian fields, dominated convergence implies the following convergences hold in L 2 (Ω):
1
α,β
α,β
(Ws,t
− a)2 |Ws,t
− a|,
6
Z tZ s¯
¯
¯ α,β
¯
2α−1 2β−1
− a)u
v
dudv → 2αβ
¯Wu,v − a¯ u2α−1 v 2β−1 dudv,
α,β
gε (Ws,t
− a) →
2αβ
Z tZ
0
s
0
α,β
gε00 (Wu,v
and by Proposition 2,
0
αβ
Z tZ
0
s
0
0
α,β 4α−1 4β−1
pε (Wu,v
)u
v
dudv → Las,t .
Step 2. We will show that the integrand of the stochastic integral with respect to the
fractional Brownian sheet is convergent. More precisely, consider the process
α,β
gε0 (Wu,v
)1[0,t]×[0,s] (u, v)
¡
¢
and let us show its convergence in L2 Ω; Hα,β as ε → 0 to
α,β
g 0 (Wu,v
)1[0,t]×[0,s] (u, v)
where g 0 (x) = 21 x|x|. Equivalently, we will show that K ∗,2 (g 0 (W α,β )1[0,t]×[0,s] )(u, v) converges to K ∗,2 (g 0 (W α,β )1[0,t]×[0,s] )(u, v) in L2 ([0, 1]2 × Ω). We have, by Jensen’s inequality
¯2
Z t Z s ¯Z t Z s ³
´ ∂K α
¯
¯
∂K β
α,β
α,β
0
0
¯ dudv
¯
(a,
u)
(b,
v)dadb
E
)
)
−
g
(W
g
(W
ε
a,b
a,b
¯
¯
∂a
∂b
0
0
u v
¶
Z t Z s µZ t Z s ¯
¯2
¯ 0
α,β
α,β ¯
H− 32
0
H− 32
(b − v)
dadb dudv
≤ cE
¯gε (Wa,b ) − g (Wa,b )¯ (a − u)
u v
0
0
Z tZ s¯
¯2
¯ 0
α,β
α,β ¯
≤ cE
¯gε (Wa,b ) − g 0 (Wa,b )¯ dadb
0
0
25
which goes to zero as ε → 0.
Step 3. Lemma 5 together with the proof of Theorem 1 shows that the limit in L 2 (Ω)
i
X h ³ α,β ´
f 00 Wsi ,tj 1[0,si ]×∆j (·)1∆i ×[0,tj ] (?)
lim δ (2)
|π|→0
i,j
exists for every f satisfying Condition (H). Consequently, for every ε > 0, denote by A ε the
following random variable in L2 (Ω):
i
X h ³ α,β ´
(2)
00
Aε = lim δ
gε Wsi ,tj 1[0,si ]×∆j (·)1∆i ×[0,tj ] (?) .
|π|→0
i,j
Since gε00 converges to g(x) = 12 |x|x, we obtain by Lemma 4 (with the observation made
2
at the end of the proof of Proposition 1) theh convergence
³
´ in L (Ω) of Aε , as iε goes to
P
zero, to the random variable lim|π|→0 δ (2) i,j g Wsα,β
1[0,si ]×∆j (·)1∆i ×[0,tj ] (?) which is
i ,tj
in L2 (Ω).
Step 4. We consider now the term
Z tZ s
α,β
α,β
β
gε000 (Wu,v
− a)u2α v 2β−1 dvdu Wu,v
0
0
and we will prove its convergence to
Z tZ s
α,β
α,β
− a)u2α v 2β−1 dvdu Wu,v
.
sign(Wu,v
β
0
0
An argument from the one-parameter case can be used to conclude this fact. The proof of
this step follows from the next lemma and the dominated convergence theorem.
Lemma
B α a one-dimensional fBm
with α ∈ ( 12 , 1). Then, for every s ∈ T ,
R s 000 α6 Consider
R
s
2α
α
2
α
2α
α
0 gε (Bu − a)u dWu converges in L (Ω) to 0 sign(Bu − a)u dWu .
Proof: Let us write Itô’s formula with hε (t, x) = t2α gε00 (x) (this is a straightforward
application of Theorem 1 of [1]). We get
Z s
Z s
00
α
2α−1
2α 00
α
gε000 (Buα − a)u2α dBuα
gε (Bu − a)u
du +
s gε (Bs − a) = 2α
0
0
Z s
+ 2α
pε (Buα − a)u4α−1 du
0
The following convergences in L2 (Ω) can be proved exactly as in [8]:
s2α gε00 (Bsα − a) → s2α |Bsα − a|
26
2α
Z
s
0
gε00 (Buα − a)u2α−1 du → 2α
2α
Lau
Z
s
0
pε (Buα
− a)u
4α−1
Z
s
0
du →
|Buα − a| u2α−1 du
Z
s
0
Lau u2α du
(here
is the local time of the one-parameter fBm defined in [8] and presented at the
beginning of this section), and
1[0,s] (u)gε000 (Buα − a)u4α−1 → 1[0,s] (u)sign(Buα − a)u2α ,
the last convergence being in L2 (Ω; H). That gives the conclusion using Lemma 4.
Lemma 4 again and Steps 1 to 4 above finish the proof of the theorem.
¤
¤
Remark 2 Using standard arguments, additional versions of the stochastic integral representation of the local time of the fractional Brownian sheet can be obtained by using, instead
0
00
000
0
00
of the functions g, g , g and g appearing in the proof of Theorem 2, the functions j, j , j
000
and j , where
1
1
0
j(x) = [(x − a)+ ]3 , j (x) = [(x − a)+ ]2
6
2
and
00
000
j (x) = (x − a)+ , j (x) = 1(a,∞) (x).
More precisely, we have
Z tZ s ³
´
1 a
α,β
α,β
α,β
Ls,t = j(Ws,t ) −
j 0 Wu,v
− a dWu,v
2
0
0
Z tZ s ³
´
α,β
− 2αβ
j 00 Wu,v
− a u2α−1 v 2β−1 dudv
0
0
Z tZ s ³
´
α,β
−
j 00 Wu,v
− a dM̃u,v
0
0
Z tZ s
³
´
α,β
α,β
−α
j 000 Wu,v
− a u2α−1 v 2β dv Wu,v
du
0
0
Z tZ s
³
´
α,β
α,β
−β
j 000 Wu,v
− a u2α v 2β−1 du Wu,v
dv.
0
0
Negative part functions can be also used to express the local time.
Our last result is an occupation time formula for the local time of the fractional
Brownian sheet, for which the previous remark is useful.
Proposition 3 For any Borel function f : R → R and and for any a ∈ R, s, t ∈ [0, 1], it
holds that
Z tZ s
Z
α,β 4α−1 4β−1
2αβ
f (Wu,v )u
v
dudv =
f (a)Las,t da.
(48)
0
0
R
27
Proof: We use the same idea as in [16]. Let us introduce the function hIV
x (y) =
1(0,x) (y) and its anti-derivatives. Notice that
000
hx (y) =
and
0
hx (y) =
Z
x
0
Z
x
0
1(y>x0 ) dx0 ,
00
hx (y) =
1
[(y − x0 )+ ]2 dx0 ,
2
Z
hx (y) =
x
0
Z
(y − x0 )+ dx0 ,
x
0
1
[(y − x0 )+ ]3 dx0 .
6
On the one hand, we can write Itô’s formula for hx (y) (although the fourth derivative
is not continuous, is it not difficult to see that Itô’s formula still holds, by approximating
hx by smooth functions) and we get
Z tZ
s
α,β 4α−1 4β−1
1(0,x) (Wu,v
)u
v
dudv
Z tZ s
0
α,β
α,β
α,β
) − hx (0) −
hx (Wu,v
= hx (Ws,t
)dWu,v
0
0
Z tZ s
Z tZ s
00
00
α,β
α,β 2α−1 2β−1
hx (Wu,v
hx (Wu,v )u
)dM̃u,v
v
dudv −
− 2αβ
0
0
0
0
Z tZ s
Z tZ s
000
000
α,β 2α−1 2β
α,β
α,β 2α 2β−1
α,β
−α
)u
v dudv Wu,v
−β
hx (Wu,v
)u v
du Wu,v
dv.
hx (Wu,v
αβ
0
0
0
0
0
0
On the third hand, using the integral expression of the local time Lys,t given in Remark 2,
multiplying both sides by 1(0,x) (y), integrating over the real line and changing the order of
integration (see exercise 3.2.8 of [17] for the Fubini anticipating theorem), we will obtain
that
Z
Z tZ s
1
y
α,β 4α−1 4β−1
1(0,x) (Wu,v
)u
v
dudv.
1
(y)Ls,t dy = αβ
2 R (0,x)
0
0
The formula (48) is proved for f (y) = 1(0,x) (y). A monotone class argument is sufficient to
conclude the proof.
¤
5.3
Relation with the variation of M̃
When α = β = 21 the process M̃ is a martingale and in this case (see [16]) the formula (48)
can be written as
Z Z
Z
1 t s
α,β
f (Wu,v )dhM̃ iu,v =
f (a)Las,t da.
(49)
2 0 0
R
This can be seen immediately by formally taking the square of the differential dM̃u,v =
³
´2
du M̃u,v · dv M̃u,v which yields dM̃u,v = uvdudv.
If α or β are not 12 , then M̃ is not a semimartingale. To investigate whether a
formula such as (49) still holds, the results of Section 4 can be invoked. They show that
the quadratic variation of M̃ is obviously not the right quantity to look at, since it is 0.
28
As with one-parameter fBm, there is a specific non-quadratic variation of M̃ that will be
non-zero and non-infinite. One can easily prove that, for small h, k > 0, and p, q > 1
´q i
³
³
´p ³
h³
´´
α,β
α,β
α,β
α,β
− Wu,v
= v βp hαp uαq k βq + o max hαp , k βq . (50)
− Wu,v
Wu,v+k
E Wu+h,v
In order for the increment
³
´p ³
´
α,β
α,β
α,β
α,β
α,β
α,β
dM̃u,v = dv Wu,v
· du Wu,v
= Wu+du,v
− Wu,v
Wu,v+dv
− Wu,v
to yield the differential of a non-trivial bounded variation process, one must choose the
powers p and q above in order to get the product dudv. Therefore we define the (p, q)variation of M̃ by the formula
D
E(p) D
E(q)
D E(p,q)
α,β
α,β
= d W·,v
d M̃
d Wu,·
(51)
u,v
u
v
where for any number r > 1, the r-variation of a one-parameter process X is defined as
usual using partitions π = {t1 , · · · , tn } of [0, t] by
hXi
(r)
n
X
¯
¯
¯Xt − X t ¯r .
(t) = lim
i
i−1
|π|→0
(52)
i=1
D E(α−1 ,β −1 )
exists and is non-zero, and in fact
Formula (50) proves that M̃
u,v
D E(α−1 ,β −1 )
= uα/β v β/α dudv.
d M̃
u,v
D
Despite the somewhat formal nature of the above development, the definition of
E(α−1 ,β −1 )
M̃
can be made rigorous because of the tensor-type nature of M̃ , and the last
u,v
formula can be established rigorously by only referring to the definition of M̃ as a double
Skorohod integral in the statement of Proposition 1 or Theorem 1. By comparing with
formula (48) in Proposition 3, we can state the following, which answers the issue of the
validity
D E of formula (49) negatively, leaving a rigorous proof, and a rigorous definition of
M̃ , to the reader.
Proposition 4 The (p, q)-variation of the fractional Brownian sheet (u, v) 7→
D E(p,q)
,
M̃
u,v
defined in equations (51) and (52), is a non-trivial bounded-variation process on [0, 1] 2 if
and only if p = 1/α and q = 1/β. In that case
D E(α−1 ,β −1 ) Z s Z t
M̃
=
dudvuα/β v β/α .
s,t
0
0
The formula (49) does not hold unless α = β = 1/2.
29
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31