318 Electronic Communications in Probability
for 0 ≤ s t ≤ T . Here, the stochastic integrals are defined as Wiener-Itô integrals when i 6= j, while,
when i = j, they are simply given by Z
t s
d B
i u
Z
u s
d B
i v
= 1
2
B
i t
− B
i s
2
.
3 The process B
2
is almost surely an element of C
2 γ
2
[0, 1]; R
d ×d
, and satisfies the algebraic rela- tion
B
2 st
− B
2 su
− B
2 ut
= B
u
− B
s
⊗ B
t
− B
u
, for all 0
≤ s ≤ u ≤ t ≤ 1. These algebraic and analytic properties of the fBm path allow to invoke the rough path machinery
see [9, 10, 13] in order to solve equation 1:
Theorem 2.2. Let B be a Liouville fBm with Hurst parameter 1 3 H 12, and σ : R
n
→ R
n ×d
be a C
2
function, which is bounded together with its derivatives. Then
1 Equation 1 admits a unique solution y ∈ C
γ 1
R
n
for any 13 γ H, with the additional structure of weakly controlled process introduced in [10].
2 The mapping a, B, B
2
7→ y is continuous from R
n
× C
γ 1
R
d
× C
2 γ
2
R
d ×d
to C
γ 1
R
n
. One of the nice aspects of rough paths theory is precisely the second point in Theorem 2.2, which
allows to reduce immediately our weak convergence result for equation 1, namely Theorem 1.2, to the following result on the approximation of B, B
2
:
Theorem 2.3. Recall that the random variables η
i k
satisfy Hypothesis 1.1, and let X
ǫ
be defined by 3. For any
ǫ 0, let X
2, ǫ
= X
2, ǫ
st
i, j
s,t ≥0; i, j=1,...,d
be the natural Lévy’s area associated to X
ǫ
, given by
X
2, ǫ
st
i, j = Z
t s
X
j, ǫ
u
− X
j, ǫ
s
d X
i, ǫ
u
, 7
where the integral is understood in the usual Lebesgue-Stieltjes sense. Then, as ǫ → 0,
X
ǫ
, X
2, ǫ
Law
−→ B, B
2
, 8
where B
2
denotes the Lévy area defined in Proposition 2.1, and where the convergence in law holds in the spaces
C
µ 1
R
d
× C
2 µ
2
R
d ×d
, for any µ H. The remainder of our work is thus devoted to the proof of Theorem 2.3.
As usual in the context of weak convergence of stochastic processes, we divide the proof into weak convergence for finite-dimensional distributions and a tightness type result. Furthermore,
the tightness result in our case is easily deduced from the analogous result in [3]:
Proposition 2.4. The sequence X
ǫ
, X
2,
ǫ ǫ0
defined at Theorem 2.3 is tight in C
µ 1
R
d
×C
2 µ
2
R
d ×d
. Proof. The proof follows exactly the steps of [3, Proposition 4.3], the only difference being that
our Lemma 3.1 has to be applied here in order to get the equivalent of inequality 28 in [3]. Details are left to the reader.
Weak approximation of fractional SDEs 319
With these preliminaries in hand, we can now turn to the finite dimensional distribution f.d.d. in the sequel convergence, which can be stated as:
Proposition 2.5. Under the assumption 1.1, let X
ǫ
, X
2, ǫ
be the approximation process defined by 3 and 7. Then
f.d.d. − lim
ǫ→0
X
ǫ
, X
2, ǫ
= B, B
2
, 9
where f.d.d. − lim stands for the convergence in law of the finite dimensional distributions. Otherwise
stated, for any k ≥ 1 and any family {s
i
, t
i
; i ≤ k, 0 ≤ s
i
t
i
≤ T }, we have L − lim
ǫ→0
X
ǫ t
1
, X
2, ǫ
s
1
t
1
, . . . , X
ǫ t
k
, X
2, ǫ
s
k
t
k
= B
t
1
, B
2 s
1
t
1
, . . . , B
t
k
, B
2 s
k
t
k
. 10
Proof. The structure of the proof follows again closely the steps of [3, Proposition 5.1], except that other kind of estimates will be needed in order to handle the Donsker case.
To be more specific, it should be observed that the first series of simplifications in the proof of [3, Proposition 5.1] can be repeated here. They allow to pass from a convergence of double iterated
integrals to the convergence of some Wiener type integrals with respect to X
ǫ
. Namely, for i = 1, 2 and 0
≤ u t ≤ 1, set Y
i
u, t = Z
t u
B
i v
− B
i u
v − u
H −
3 2
d v, and for 0
≤ u t ≤ 1 and u
1
, . . . , u
6
in a neighborhood of 0 in R
6
, set also Z
u
= u
1
+ u
2
B
2 u
+ u
3
Y
2
u, t + u
4
Z
t u
v − u
H −
1 2
dW
2 v
+u
5
Z
t u
d w Z
w u
w − v
H −
3 2
w − u
H −
1 2
− v − u
1 2
dW
2 v
+u
6
Z
t u
d w Z
u
w − v
H −
3 2
w − u
H −
1 2
dW
2 v
. Consider the analogous processes Y
i, ǫ
, Z
ǫ
defined by the same formulae, except that they are based on the approximations
θ
i, ǫ
of white noise. We still need to recall a little more notation from [3]: for f
∈ L
2
[0, 1] and t ∈ [0, 1], we set Φ
ǫ
f = E e
i R
t
f u θ
ǫ,1
udu
, φ
ǫ f
= Z
1
Z
1
f
2
x f
2
yI
{|x− y|ǫ
2
}
d x d y, 11
and Φ f = E
e
i R
t
f u dW
1 u
= e
1 2
R
t
f
2
u du
. Then it is shown in [3, Proposition 5.1] that one is reduced to prove that lim
ǫ→0
v
a ǫ
= 0, where v
a ǫ
is given by v
a ǫ
= E Φ
ǫ
Z
ǫ
e
iw R
t
θ
ǫ,2
udu
− E ΦZ
ǫ
e
iw R
t
θ
ǫ,2
udu
, for an arbitrary real parameter w in a neighborhood of 0. Furthermore, bounding e
iw R
t
θ
ǫ,2
udu
triv- ially by 1 and conditioning, it is easily shown that v
a ǫ
is controlled by the difference E[ |Φ
ǫ
Z
ǫ
−
320 Electronic Communications in Probability
ΦZ
ǫ
|], for which Lemma 3.3 provides the bound |Φ
ǫ
Z
ǫ
− ΦZ
ǫ
| ≤ E
4 r
1 5
w
3
φ
ǫ Z
ǫ 1
2
kZ
ǫ
k
L
2
k
3 η
exp4w
2
k
2 η
kZ
ǫ
k
2 L
2
+ w
2
ǫ
2 α
kZ
ǫ
k
α
kZ
ǫ
k
L
2
expw
2
kZ
ǫ
k
2 L
2
+ 8
p 5
w
4
φ
ǫ Z
ǫ 1
2
kZ
ǫ
k
2 L
2
k
η 2
exp4w
2
k
2 η
kZ
ǫ
k
2 L
2
+ 1
2 w
4
φ
ǫ Z
ǫ
expw
2
kZ
ǫ
k
2 L
2
, for any
α ∈ 0, 1. In order to reach our aim, it is thus sufficient to check the following inequalities: sup
ǫ
E
kZ
ǫ
k
2 α
≤ M,
lim
ǫ→0
E
φ
ǫ Z
ǫ
2
= 0
and for w w
, where w is a small enough constant,
sup
ǫ
E e
w
2
kZ
ǫ
k
2 L2
≤ M. However, these relations can be deduced, as 39, 40 and 41 in [3], from Lemma 3.1 it should
be noticed however that a one-parameter version of [2, Lemma 5.1] is needed for the adaptation of the latter result to our Donsker setting. The proof is thus finished once the lemmas below are
proven.
3 Moments estimates in the Donsker setting
In order to deal with our technical estimates, let us first introduce a new notation: set ρ
1
= 1 − 5
1 2
2 and ρ
2
= 1 + 5
1 2
2. Then the moments of any integral of a deterministic kernel f with respect to
θ
i, ǫ
can be bounded as follows:
Lemma 3.1. Let m ∈ N, f ∈ L
2
[0, 1], i ∈ {1, 2} and ǫ 0. Recall that the random variable η is assumed to be almost surely bounded by a constant k
η
. Then we have E
Z
1
f r θ
i, ǫ
rd r
2m
12
≤ 2m
2
m
m k f k
2m L
2
+ 2m
5
1 2
m − 2 k
2m η
ρ
2m −1
2
− ρ
2m −1
1
φ
ǫ f
1 2
k f k
2m −2
L
2
, and
E
Z
1
f r θ
i, ǫ
rd r
2m+1
≤
2m + 1 5
1 2
m − 1 k
2m+1 η
ρ
2m 2
− ρ
2m 1
φ
ǫ f
1 2
k f k
2m −1
L
2
, 13
where φ
ǫ f
is the quantity defined at 11. Proof. We focus first on inequality 12 and divide this proof into several steps.
Weak approximation of fractional SDEs 321
Step 1: Identification of some key iterated integrals. Notice that E
Z
1
f r θ
i, ǫ
rd r
2m
≤
Z
[0,1]
2m
| f r
1
| · · · | f r
2m
||Eθ
i, ǫ
r
1
· · · θ
i, ǫ
r
2m
|d r
1
· · · d r
2m
. Transforming the symmetric integral on [0, 1]
2m
into an integral on the simplex, and using expres- sion 4 for
θ
ǫ
, we can write the latter expression as: 2m
ǫ
2m n
ǫ
X
k
1
, . . . , k
2m
= 1 k
1
≥ · · · ≥ k
2m
Z
k
1
ǫ
2
k
1
−1ǫ
2
· · · Z
k
2m
ǫ
2
k
2m
−1ǫ
2
| f r
1
| · · · | f r
2m
| × |Eη
i k
1
· · · η
i k
2m
|I
{r
1
≥r
2
≥···≥r
2m
}
d r
1
· · · d r
2m
, 14
where n ǫ = [
1 ǫ
2
] + 1 and where we understand that f x = 0 whenever x 1. Let us study now the quantities E
η
i k
1
· · · η
i k
l
. If there exists l such that k
l
6= k
j
for all j 6= l
then E η
i k
1
· · · η
i k
l
= 0. On the other hand, when k
2l −1
= k
2l
k
2l+1
for any l, we clearly have E
η
i k
1
· · · η
i k
l
= 1. Finally, in the general case, for all l ∈ N, |Eη
i k
1
· · · η
i k
l
| ≤ k
l η
. Sepa- rating the cases in this way for
|Eη
i k
1
· · · η
i k
2m
|, we end up with a decomposition of the form E[
R
1
f r θ
i, ǫ
rd r
2m
] = T
1 m
+ T
2 m
, where T
1 m
= 2m
ǫ
2m n
ǫ
X
k
1
, . . . , k
m
= 1 k
1
· · · k
m
Z
k
1
ǫ
2
k
1
−1ǫ
2
Z
k
1
ǫ
2
k
1
−1ǫ
2
· · · Z
k
m
ǫ
2
k
m
−1ǫ
2
Z
k
m
ǫ
2
k
m
−1ǫ
2
| f r
1
| · · · | f r
2m
| × I
{r
1
≥r
2
≥···≥r
2m
}
d r
1
· · · d r
2m
, and where the term T
2 m
is defined by: T
2 m
= 2m k
2m η
ǫ
2m
X
n
1
, . . . , n
s
≥ 2; s ∈ {1, ..., m − 1} n
1
+ · · · + n
s
= 2m
U
n
1
,...,n
s
, 15
with U
n
1
,...,n
s
=
n ǫ
X
k
1
, . . . , k
s
= 1 k
1
· · · k
s
Z
D
k1···ks
| f r
1
| · · · | f r
2m
|I
{r
1
≥r
2
≥···≥r
2m
}
d r
1
· · · d r
2m
, 16
and where we have set D
k
1
···k
s
= Q
s j=1
[k
j
− 1ǫ
2
, k
j
ǫ
2
]
n
j
. Let us observe at this point that we have split our sum into T
1 m
and T
2 m
because T
1 m
represents the dominant contribution to our moment estimate. This is simply due to the fact that T
1 m
is obtained by assuming some pairwise equalities among the random variables
η
i k
, while T
2 m
is based
322 Electronic Communications in Probability
on a higher number of constraints. In any case, both expressions will be analyzed through the introduction of some iterated integrals of the form
K
ν
k; v, w = 1
ǫ
ν
Z
[k−1ǫ
2
,k ǫ
2
]
ν
ν
Y
j=1
| f r
j
| I
{w≥r
1
···r
ν
≥v}
d r
1
· · · d r
ν
, defined for
ν, k ≥ 1 and 0 ≤ v w ≤ 1. Step 2: Analysis of the integrals K
ν
. Those iterated integrals are treated in a slightly different way according to the parity of
ν. Indeed, for ν = 2n, thanks to the elementary inequality 2a b ≤ a
2
+ b
2
, we obtain a bound of the form:
n ǫ
X
k=1
K
2n
k; v, w 17
≤
n ǫ
X
k=1
1 ǫ
2n
Z
[k−1ǫ
2
,k ǫ
2
]
2n
n
Y
i=1
f
2
x
2i −1
+ f
2
x
2i
2
I
{w≥x
1
≥···≥x
2n
≥v}
d x
1
· · · d x
2n
≤
n ǫ
X
k=1
1 ǫ
2n
Z
[k−1ǫ
2
,k ǫ
2
]
2n
f
2
x
1
· · · f
2
x
n
I
{w≥x
1
≥···≥x
n
≥v}
d x
1
· · · d x
2n
=
n ǫ
X
k=1
Z
[k−1ǫ
2
,k ǫ
2
]
n
f
2
x
1
· · · f
2
x
n
I
{w≥x
1
≥···≥x
n
≥v}
d x
1
· · · d x
n
≤ Z
[0,1]
n
f
2
x
1
· · · f
2
x
n
I
{x
1
−x
n
ǫ
2
}
I
{w≥x
1
≥···≥x
n
≥v}
d x
1
· · · d x
n
. The case
ν = 2n + 1 can be treated along the same lines, except for the fact that one has to cope with some expressions of the form
n ǫ
X
k=1
K
3
k; v, w ≤
n ǫ
X
k=1
1 ǫ
Z
[k−1ǫ
2
,k ǫ
2
]
2
| f x
1
| f
2
x
2
I
{w≥x
1
≥x
2
≥v}
d x
1
d x
2
≤ 1
ǫ Z
[0,1]
2
| f x
1
| f
2
x
2
I
{x
1
−x
2
ǫ
2
}
I
{w≥x
1
≥x
2
≥v}
d x
1
d x
2
. 18
Combining 18 and 17 we can state the following general formula: let ν ≥ 1, and define a
couple ν
∗
, ˆ ν as: i ν
∗
= ν2, ˆ ν = 0 if ν is even, ii ν
∗
= ν + 12, ˆ ν = 1 if ν is odd. With this
notation in hand, we have:
n ǫ
X
k=1
K
ν
k; v, w ≤ 1
ǫ
ˆ ν
Z
[0,1]
ν∗
| f x
1
|
2 −ˆ
ν
f
2
x
2
· · · f
2
x
ν
∗
I
{x
1
−x
ν∗
ǫ
2
}
× I
{w≥x
1
≥···≥x
ν∗
≥v}
d x
1
· · · d x
ν
∗
. 19
Step 3: Bound on T
1 m
. It is readily checked that T
1 m
can be decomposed into blocks of the form
Weak approximation of fractional SDEs 323
K
2
k; w, v, for which one can apply 19. This yields T
1 m
≤ 2m
2
m n
ǫ
X
k
1
,...,k
m
=1
Z
k
1
ǫ
2
k
1
−1ǫ
2
· · · Z
k
m
ǫ
2
k
m
−1ǫ
2
f
2
r
1
· · · f
2
r
m
I
{r
1
≥r
2
≥···≥r
m
}
d r
1
· · · d r
m
≤ 2m
2
m
m k f k
2m L
2
. Step 4: Bound on U
n
1
,...,n
s
. Recall that U
n
1
,...,n
s
is defined by 16. We introduce now a recursion procedure in order to control this term. Namely, integrating with respect to the last n
s
variables, one obtains that
1 ǫ
2m
U
n
1
,...,n
s
= 1
ǫ
2m −n
s
n ǫ
X
k
1
, . . . , k
s −1
= 1 k
1
· · · k
s −1
Z
D
k1···ks−1
2m −n
s
Y
l=1
| f r
l
| I
{r
1
≥r
2
≥···≥r
2m −ns
}
× K
n
s
k
s
; 0; r
2m −n
s
d r
1
· · · d r
2m −n
s
Plugging our bound 19 on K
n
s
into this expression, we get 1
ǫ
2m
U
n
1
,...,n
s
≤ 1
ǫ
2m −n
s
+ˆ n
s
n ǫ
X
k
1
, . . . , k
s −1
= 1 k
1
· · · k
s −1
Z
D
k1···ks−1
Z
[0,1]
n∗ s
2m −n
s
Y
l=1
| f r
l
| | f y
1
|
2 −ˆn
s
×
n
∗ s
Y
j=2
f
2
y
j
I
{ y
1
− y
n∗ s
ǫ
2
}
I
{r
1
≥r
2
≥···≥r
2m −ns
≥ y
1
≥···≥ y
n∗ s
}
d r
1
· · · d r
2m −n
s
d y
1
· · · d y
n
∗ s
. We can now proceed, and integrate with respect to the variables r
l
for 2m −n
s
−n
s −1
l ≤ 2m−n
s
. In the end, since
P n
s
= 2m, the remaining singularity in ǫ is of the form Q ǫ
−ˆn
s
. However, each of the singularity
ǫ
−ˆn
s
comes with an integral that compensates the singularity ǫ
−ˆn
s
recall that ˆ
n
s
≤ 1. Hence, iterating the integrations with respect to the variables r, we end up with a bound of the form
1 ǫ
2m
U
n
1
,...,n
s
≤ 1
m − 2 k f k
2m −2
L
2
φ
ǫ f
≤ 1
m − 2 k f k
2m −2
L
2
φ
ǫ f
1 2
. 20
Notice that when one of the terms n
1
, n
2
, . . . , n
s
of the decomposition of 2m is even the last bound is easily obtained. On the other hand, we will illustrate with an example how the bound can be
obtained when all the terms are odd. Let us consider the case m = n
1
= n
2
= 3. Following our procedure we obtain that
1 ǫ
6
U
3,3
≤ 1
ǫ
2
Z
[0,1]
4
f
2
y
2
· | f y
1
| · f
2
y
4
· | f y
3
| ×I
{ y
1
− y
2
ǫ
2
}
I
{ y
3
− y
4
ǫ
2
}
I
{ y
3
≥ y
4
≥ y
1
≥ y
2
}
d y
1
· · · d y
4
≤ 1
ǫ
2
Z
[0,1]
4
f
2
y
2
f
2
y
4
f
2
y
1
+ f
2
y
3
2
×I
{ y
1
− y
2
ǫ
2
}
I
{ y
3
− y
4
ǫ
2
}
I
{ y
3
≥ y
4
≥ y
1
≥ y
2
}
d y
1
· · · d y
4
≤ φ
ǫ f
1 ǫ
2
Z
[0,1]
2
f
2
y
4
I
{0≤ y
3
− y
4
ǫ
2
}
d y
3
d y
4
= k f k
2 L
2
φ
ǫ f
.
324 Electronic Communications in Probability
Step 5: Bound on T
2 m
. Owing to inequality 20, our bound on T
2 m
can be reduced now to an estimate of the number of terms in the sum over n
1
, . . . , n
s
in formula 15. This boils down to the following question: given a natural number n, how can we write it as a sum of natural numbers
larger than one? This is arguably a classical problem, and in order to recall its answer, let us take a simple example:
for n = 6, the possible decompositions can be written as {6; 2 + 2 + 2; 2 + 4; 4 + 2; 3 + 3}. Further-
more, notice that the decompositions of 6 can be obtained by adding +2 to the decompositions of 4 or adding 1 to the last number of the decompositions of 5. Extrapolating to a general integer n,
it is easily seen that the number of decompositions can be expressed as u
n −1
, where u
n n
≥1
stands for the Fibonacci sequence. We have thus found a number of decompositions of the form
N
n
= 5
−12
ρ
n −1
2
− ρ
n −1
1
,
where the quantities ρ
1
, ρ
2
appear in formula 12. Moreover, the number of terms in T
2
is given by N
2m
− 1, the −1 part corresponding to the term T
1
. Putting together this expression with 20 and the result of Step 3, our claim 12 is now easily
obtained. Step 6: Proof of 13. The proof of 13 follows the same arguments as for 12. We briefly sketch
the main difference between these two proofs, lying in the analysis of the term U
n
1
,...,n
s
. Indeed, since we are now dealing with an odd power 2m + 1, the equivalent of 20 is an upper bound of
the form 1
ǫ Z
1
Z
1
| f y
1
| f
2
y
2
I
{| y
1
− y
2
|ǫ
2
}
d y
1
d y
2
Z
[0,1]
m −1
m −1
Y
j=1
f
2
x
j
I
{x
1
≥···≥x
m −1
}
d x
1
· · · d x
m −1
. 21
Furthermore, applying Hölder’s inequality twice, we obtain 1
ǫ Z
1
Z
1
| f y
1
| f
2
y
2
I
{| y
1
− y
2
|ǫ
2
}
d y
1
d y
2
= 1
ǫ Z
1
f
2
y
2
Z
1 ∧ y
2
+ǫ
2
∨ y
2
−ǫ
2
| f y
1
|d y
1
d y
2
≤ Z
1
Z
1
f
2
y
1
f
2
y
2
I
{| y
1
− y
2
|ǫ
2
}
d y
1
d y
2
1 2
k f k
L
2
= φ
ǫ f
1 2
k f k
L
2
, and thus we can bound 21 by
1 m−1
φ
ǫ f
1 2
k f k
2m −1
L
2
, which ends the proof. Our next technical lemma compares the moments of a Wiener type integral with respect to
θ
ǫ
and with respect to the white noise.
Lemma 3.2. Let m ∈ N, f ∈ C
α
[0, 1], i ∈ {1, 2}, ǫ 0 and for m ≥ 1, set J
m
= 1
2m E
Z
1
f r θ
i, ǫ
rd r
2m
− 1
2
m
m Z
[0,1]
m
f
2
s
1
· · · f
2
s
m
ds
1
· · · ds
m
. Then
1 We have J
1
≤ ǫ
2 α
k f k
L
2
k f k
α
.
Weak approximation of fractional SDEs 325
2 For any m 1, the following inequality holds true, where we recall that ρ
1
, ρ
2
have been defined just before Lemma 3.1:
J
m
≤ 1
m − 1 ǫ
2 α
k f k
α
k f k
2m −1
L
2
+ k
2m η
p 5m
− 2
ρ
2m −1
1
− ρ
2m −1
2
φ
ǫ f
1 2
k f k
2m −2
L
2
+ 1
m − 2 k f k
2m −2
L
2
φ
ǫ f
. Proof. We divide again this proof into several steps.
Step 1: Variance estimates. We prove here the first of our assertions: Notice that 1
2 Z
1
f
2
s
1
ds
1
= 1
2 ǫ
2 n
ǫ
X
k=1
Z
k ǫ
2
k−1ǫ
2
Z
k ǫ
2
k−1ǫ
2
f
2
s
1
ds
2
ds
1
. On the other hand
1 2
E
Z
1
f r θ
i, ǫ
rd r
2
=
1 2
ǫ
2 n
ǫ
X
k=1
Z
k ǫ
2
k−1ǫ
2
Z
k ǫ
2
k−1ǫ
2
f r
1
f r
2
d r
2
d r
1
. We thus get
J
1
= 1
2 ǫ
2 n
ǫ
X
k=1
Z
k ǫ
2
k−1ǫ
2
Z
k ǫ
2
k−1ǫ
2
f r
1
f r
2
− f r
1
d r
2
d r
1
= 1
2 ǫ
2
Z
1
Z
1
f r
1
f r
2
− f r
1 +∞
X
k=1
I
[k−1,k
2
r
1
ǫ
2
, r
2
ǫ
2
d r
2
d r
1
, and hence this quantity can be bounded as follows:
J
1
≤ 1
2 ǫ
2
Z
1
Z
1
| f r
1
|| f r
2
− f r
1
|I
{|r
1
−r
2
|ǫ
2
}
d r
2
d r
1
≤ 1
2 ǫ
2
Z
1
| f r
1
|k f k
α
Z
1
|r
2
− r
1
|
α
I
{|r
1
−r
2
|ǫ
2
}
d r
2
d r
1
≤ ǫ
2 α
k f k
L
2
k f k
α
, which is the first claim of our lemma.
Step 2: decomposition for higher moments: We can follow exactly the computations of Lemma 3.1, Step 1, in order to get
1 2m
E
Z
1
f r θ
i, ǫ
rd r
2m
= ˜
T
1 m
+ ˜ T
2 m
, with ˜
T
j m
=
T
j m
2m
for j = 1, 2. Furthermore, the term ˜ T
2 m
can be bounded as in Lemma 3.1, and we obtain
| ˜ T
2 m
| ≤ k
2m η
p 5m
− 2
ρ
2m −1
2
− ρ
2m −1
1
φ
ǫ f
1 2
k f k
2m −2
L
2
. 22
326 Electronic Communications in Probability
Step 3: Study of ˜ T
1 m
: We analyze ˜ T
1 m
in a slightly different way as in Lemma 3.1. Namely, we first write
˜ T
1 m
= 1
2
m
ǫ
2m n
ǫ
X
k
1
, . . . , k
m
= 1 k
1
· · · k
m
Z
k
1
ǫ
2
k
1
−1ǫ
2
Z
k
1
ǫ
2
k
1
−1ǫ
2
· · · Z
k
m
ǫ
2
k
m
−1ǫ
2
Z
k
m
ǫ
2
k
m
−1ǫ
2
f r
1
· · · f r
2m
× I
{{r
1
,r
2
}≥···≥{r
2m −1
,r
2m
}}
d r
1
· · · d r
2m
, 23
where we have written {a, b} ≥ {c, d} for a ∧ b ≥ c ∨ d. We will now compare this quantity with
another expression of the same type, called ˆ T
1 m
and defined by ˆ
T
1 m
= 1
2
m
m Z
[0,1]
m
f
2
s
1
· · · f
2
s
m
ds
1
· · · ds
m
. Let us thus write ˆ
T
1 m
as ˆ
T
1 m
= 1
2
m
Z
[0,1]
m
m
Y
j=1
f
2
s
j
I
{s
1
≥···≥s
m
}
ds
1
· · · ds
m
= 1
2
m n
ǫ
X
k
1
, . . . , k
m
= 1 k
1
≥ · · · ≥ k
m
Z
k
1
ǫ
2
k
1
−1ǫ
2
· · · Z
k
m
ǫ
2
k
m
−1ǫ
2
m
Y
j=1
f
2
s
j
I
{s
1
≥···≥s
m
}
ds
1
· · · ds
m
= 1
2
m n
ǫ
X
k
1
, . . . , k
m
= 1 k
1
· · · k
m
Z
k
1
ǫ
2
k
1
−1ǫ
2
· · · Z
k
m
ǫ
2
k
m
−1ǫ
2
m
Y
j=1
f
2
s
j
I
{s
1
≥···≥s
m
}
ds
1
· · · ds
m
+ ˜ T
3 m
,
where ˜ T
3 m
represents the part of the sum taken over the indices k
1
, . . . , k
m
such that there exist l satisfying k
l
= k
l+1
. However, this latter term can be bounded as in 20, yielding | ˜
T
3 m
| ≤ m
− 1 2
m
1 m − 2
k f k
2m −2
L
2
φ
ǫ f
≤ 1
2 1
m − 2 k f k
2m −2
L
2
φ
ǫ f
. 24
Step 4: Conclusion. Putting together the decompositions we have obtained so far, we end up with J
m
≤ | ˜ T
2 m
| + | ˜ T
3 m
| + 1
2
m
ǫ
2m n
ǫ
X
k
1
, . . . , k
m
= 1 k
1
· · · k
m
Z
k
1
ǫ
2
k
1
−1ǫ
2
Z
k
1
ǫ
2
k
1
−1ǫ
2
· · · Z
k
m
ǫ
2
k
m
−1ǫ
2
Z
k
m
ǫ
2
k
m
−1ǫ
2
f r
1
· · · f r
2m
− f
2
r
1
f
2
r
3
· · · f
2
r
2m −1
I
{{r
1
,r
2
}≥···≥{r
2m −1
,r
2m
}}
d r
1
· · · d r
2m
≤ | ˜ T
2 m
| + | ˜ T
3 m
| + 1
2
m
m ǫ
2m
Z
[0,1]
2m
f r
1
· · · f r
2m
− f
2
r
1
f
2
r
3
· · · f
2
r
2m −1
× I
{|r
1
−r
2
|ǫ
2
}
· · · I
{|r
2m −1
−r
2m
|ǫ
2
}
d r
1
· · · d r
2m
.
Weak approximation of fractional SDEs 327
Invoking our estimates 22 and 24 on ˜ T
2 m
and ˜ T
3 m
, our bound on J
m
easily reduced to check that 1
2
m
m ǫ
2m
Z
[0,1]
2m
f r
1
· · · f r
2m
− f
2
r
1
f
2
r
3
· · · f
2
r
2m −1
× I
{|r
1
−r
2
|ǫ
2
}
· · · I
{|r
2m −1
−r
2m
|ǫ
2
}
d r
1
· · · d r
2m
≤ 1
m − 1 ǫ
2 α
k f k
α
k f k
2m −1
L
2
. The latter inequality can now be obtained from the decomposition
f r
1
· · · f r
2m
− f
2
r
1
f
2
r
3
· · · f
2
r
2m −1
= f r
1
f r
2
− f r
1
f r
3
f r
4
· · · f r
2m
+ f
2
r
1
f r
3
f r
4
− f r
3
f r
5
f r
6
· · · f r
2m
+ · · · + f
2
r
1
f
2
r
3
... f
2
r
2m −3
f r
2m −1
f r
2m
− f r
2m −1
, the inequalities
1 2
ǫ
2
Z
1
Z
1
f
2
r
1
I
|r
1
−r
2
|ǫ
2
d r
2
d r
1
≤ k f k
2 L
2
, 1
2 ǫ
2
Z
1
Z
1
f r
1
f r
2
I
|r
1
−r
2
|ǫ
2
d r
2
d r
1
≤ k f k
2 L
2
, and from the estimate we have already obtained for J
1
. This finishes the proof.
Finally, the characteristic function of a Wiener type integral of the form R
1
f r θ
k, ǫ
rd r can be compared to its expected limit
R
1
f rdW
k r
in the following way:
Lemma 3.3. Let f ∈ C
α
[0, 1] for a certain α ∈ 0, 1, k ∈ {1, . . . , d} and ǫ 0. For any u ∈ R, we have:
E e
iu R
1
f r θ
k, ǫ
rd r
− Ee
iu R
1
f rdW
k r
≤ 415
1 2
u
3
φ
ǫ f
1 2
k f k
L
2
k
3 η
exp4u
2
k
2 η
k f k
2 L
2
+ u
2
ǫ
2 α
k f k
α
k f k
L
2
expu
2
k f k
2 L
2
+8 5
−12
u
4
φ
ǫ f
1 2
k f k
2 L
2
k
2 η
exp4u
2
k
2 η
k f k
2 L
2
+ 12u
4
φ
ǫ f
expu
2
k f k
2 L
2
. Proof. Let us control first the imaginary part of the difference. Using lemma 3.1, and invoking the
328 Electronic Communications in Probability
fact that the odd moments of a Gaussian random variable are null, we get Im
E
e
iu R
1
f r θ
k, ǫ
rd r
− Ee
iu R
1
f rdW
k r
≤
∞
X
m=1
|u|
2m+1
2m + 1 E
Z
1
f r θ
k, ǫ
rd r
2m+1
≤
∞
X
m=1
15
1 2
|u|
2m+1
m − 1 k
2m+1 η
ρ
2m 2
− ρ
2m 1
φ
ǫ f
1 2
k f k
2m −1
L
2
≤ 415
1 2
u
3
φ
ǫ f
1 2
k f k
L
2
k
3 η
∞
X
m=1
1 m − 1
4k
2 η
k f k
2 L
2
u
2 m
−1
≤ 415
1 2
u
3
φ
ǫ f
1 2
k f k
L
2
k
3 η
exp4u
2
k
2 η
k f k
2 L
2
. In order to control the real part of the difference, we will use Lemma 3.2. This yields:
Re E
e
iu R
1
f r θ
k, ǫ
rd r
− Ee
iu R
1
f rdW
k r
≤
+∞
X
m=1
u
2m
1
m − 1 ǫ
2 α
k f k
α
k f k
2m −1
L
2
+ u
4
φ
ǫ f
1 2
8 p
5 k f k
2 L
2
k
2 η
+∞
X
m=2
1 m − 2
4k
2 η
u
2
k f k
2 L
2
m −2
+ 1
2 u
4
φ
ǫ f
+∞
X
m=2
1 m − 2
u
2
k f k
2 L
2
m −2
.
The latter quantity can be bounded by u
2
ǫ
2 α
k f k
α
k f k
L
2
expu
2
k f k
2 L
2
+ 8
p 5
u
4
φ
ǫ f
1 2
k f k
2 L
2
k
2 η
exp4u
2
k
2 η
k f k
2 L
2
+ 1
2 u
4
φ
ǫ f
expu
2
k f k
2 L
2
, which ends the proof.
References
[1] E. Alòs, O. Mazet and D. Nualart 2000: Stochastic calculus with respect to fractional Brown- ian motion with Hurst parameter lesser than 12. Stoch. Proc. Appl.
86, 121-139. MR1741199
