getdoc7e85. 263KB Jun 04 2011 12:04:33 AM

J
ni
o
r
t
Elec

c

o

u

a
rn l

o

f
P

r

ob
abil
ity

Vol. 8 (2003) Paper no. 18, pages 1–26.
Journal URL
http://www.math.washington.edu/˜ejpecp/

Approximation at first and second order of m-order integrals
of the fractional Brownian motion and of certain semimartingales
Mihai GRADINARU and Ivan NOURDIN
´ Cartan, Universit´e Henri Poincar´e, B.P. 239
Institut de Math´ematiques Elie
54506 Vandœuvre-l`es-Nancy Cedex, France
Mihai.Gradinaru@iecn.u-nancy.fr and Ivan.Nourdin@iecn.u-nancy.fr
Abstract: Let X be the fractional Brownian motion of any Hurst index H ∈ (0, 1) (resp.
a semimartingale) and set α = H (resp. α = 12 ). If Y is a continuous process and if m is a
positive integer, we study the existence of the limit, as ε → 0, of the approximations

¾
½Z t µ
¶
Xs+ε − Xs m
Ys
ds, t ≥ 0
Iε (Y, X) :=
εα
0
of m-order integral of Y with respect to X. For these two choices of X, we prove that the
limits are almost sure, uniformly on each compact interval, and are in terms of the m-th
moment of the Gaussian standard random variable. In particular, if m is an odd integer, the
1
limit equals to zero. In this case, the convergence in distribution, as ε → 0, of ε− 2 Iε (1, X)
is studied. We prove that the limit is a Brownian motion when X is the fractional Brownian
motion of index H ∈ (0, 21 ], and it is in term of a two dimensional standard Brownian motion
when X is a semimartingale.
Key words and phrases: m-order integrals; (fractional) Brownian motion; limit theorems;
stochastic integrals.
AMS subject classification (2000): Primary: 60H05; Secondary: 60F15; 60F05; 60J65;

60G15
Submitted to EJP on April 22, 2003. Final version accepted on October 14, 2003.

1

Introduction

In this paper we investigate the accurate convergence of some approximations of m-order
integrals which appear when one performs stochastic calculus with respect to processes which
are not semimartingales, for instance the fractional Brownian motion. We explain below our
main motivation of this study.

1.1

Preliminaries

Recall that the fractional Brownian motion with Hurst index 0 < H < 1 is a continuH
H
ous centered Gaussian process B = {Bt : t ≥ 0} with covariance function given by
H

H
H
Cov(Bs , Bt ) = 12 (s2H + t2H − |s − t|2H ). It is well known that B is a semi-martingale
H
if and only if H = 21 (see [16], pp. 97-98). Moreover, if H > 21 , B is a zero quadratic variation process. Hence, for H ≥ 21 , a Stratonovich type formula involving symmetric stochastic
integrals holds (see, for instance [17] or [5]):
Z t
H
H
H
f ′ (Bs )d◦ Bs .
(1.1)
f (Bt ) = f (0) +
0

On the other hand, if 0 < H < 12 , some serious difficulties appear. It is a quite technical
computation to prove that (1.1) is still valid when H > 61 (see [8] or [3]). In fact, for H > 61 ,
in [8] was proved, on the one hand,
Z t
Z t

Z t
1
1
H
H
H
H
H
(3)
◦(3) H
′
◦ H
f (Bs )d Bs +
f (5) (Bs )d◦(5) Bs
f (Bt ) = f (0) +
f (Bs )d Bs −
12 0
120 0
0
and on the other hand,

Z

0

t

f

(3)

H

(Bs )d

◦(3)

H

Bs =

Z

0

t

H

H

f (5) (Bs )d◦(5) Bs = 0,

where, for X, Y continuous processes and m ≥ 1, the m-order symmetric integral is given by
Z t
Z
1 t
(Xs+ε − Xs )m
◦(m)
Ys d
Xs := lim prob

(Ys + Ys+ε )
ds,
(1.2)
ε→0
2 0
ε
0

and the m-forward integral is given by
Z t
Z t
(Xs+ε − Xs )m
−(m)
Ys
Ys d
Xs := lim prob
ds,
(1.3)
ε→0
ε

0
0
Rt
Rt
Rt
Rt
(if m = 1, we write 0 Ys d◦ Xs (resp. 0 Ys d− Xs ) instead of 0 Ys d◦(1) Xs (resp. 0 Ys d−(1) Xs ) ).
In [8] one studies, for the fractional Brownian motion, the existence of the m-order symmetric integrals. According to evenness of m, it is proved that
Rt
H
H
• if mH = 2nH > 1 then integral 0 f (Bs )d◦(2n) Bs exists and vanishes;
Rt
H
H
• if mH = (2n − 1)H > 12 then integral 0 f (Bs )d◦(2n−1) Bs exists and vanishes.
Rt
Rt
H
H

H
H
Moreover it is also emphasized that integrals 0 f (Bs )d◦(2n) Bs (resp. 0 f (Bs )d◦(2n−1) Bs )
do not exist in general if 2nH ≤ 1 (resp. (2n − 1)H ≤ 12 ). This last statement is in fact a
direct consequence of results of the present paper (as it is mentioned in the proof of Theorem
4.1 in [8]). An important consequence is that H = 16 is a barrier of validity for the formula
(1.1) (see [1], [3], [4], [7] and [8]).
2

1.2

First order approximation: almost sure convergence

In the definitions (1.2) and (1.3), limits are in probability. One can ask a natural question: is it
possible to have almost sure convergence? For instance, in [10], the almost sure convergence of
a generalized quadratic variation of a Gaussian process is proved using a discrete observation
of one sample path; in particular, their result applies to the fractional Brownian motion.
Here, we prove (see Theorems 2.1 and 2.2 below for precise statements) that, as ε → 0,
Z t
Xs+ε − Xs

Ys f (
)ds
(1.4)
εα
0
converge almost surely, uniformly on each compact interval, to an explicit limit when f
belongs to a sufficiently large class of functions (including the case of polynomial functions,
H
for instance f (x) = xm ), Y is any continuous process, X = B is the fractional Brownian
motion with H ∈ (0, 1) (resp. X = Z a semimartingale) and α = H (resp. α = 12 ). Let us
remark that the case when X is a semimartingale is a non Gaussian situation unlike was the
case in [10] or other papers (at our knowledge).
If m = 2n is an even integer the previous result suffices to study the existence of 2n-order
integrals for the fractional Brownian motion with all 0 < H < 1. Indeed, by choosing
f (x) = x2n , we can write the following equivalent, as ε ↓ 0:
Z t
Z
Z t
1 t
H
H 2n
2nH−1 (2n)!
Ys ds 6= 0.
Ys (Bs+ε − Bs ) ∼ ε
Ys ds, if
ε 0
2n n! 0
0
On the other hand, if m = 2n − 1 is an odd integer, we need to refine our analysis (especialy
for Hurst index 0 < H ≤ 12 ) because, in this case, we do not have an almost sure non-zero
equivalent.

1.3

Second order approximation: convergence in distribution

Set Y ≡ 1 and f (x) = xm in (1.4), with m ≥ 3 an odd integer. For the two same choices of
H
X (that is X = B with α = H or X a semimartingale with α = 12 ), we have, for all T > 0,
¶
Z tµ
Xs+ε − Xs m
a.s., ∀t ∈ [0, T ], lim
ds = 0.
ε→0 0
εα
After correct renormalization, is it possible to obtain the convergence in distribution of
our approximation? We prove (see Theorems 2.4 and 2.5 below for precise statements) that
the family of processes
¶
¾
½
Z tµ
Xs+ε − Xs m
1
√
ds : t ≥ 0
(1.5)
εα
ε 0
converges in distribution, as ε → 0, to an explicit limit:
• If X = B H is the fractional Brownian motion with H ∈ (0, 21 ], we obtain a Brownian
motion and our approach is different to those given by [6] or [19];
• If X is a semimartingale, we express the limiting process in terms of a two-dimensional
standard Brownian motion. This also give an example of a non-Gaussian situation when
the convergence in distribution is studied.
3

We can see that
!m
)
)
(Z t
(Z Ã H
H
³ H
´
t
ε
Bs+ε − Bs
H m
ds, ε > 0
Bs+1 − Bs
ds, ε > 0 equals in law to
εH
0
0
H

by using the self-similarity of the fractional Brownian motion (that is, for all c > 0, B ct equals
H
H
in law - as a process - to c Bt ).
In [6], the authors study the convergence in distribution (but only for finite dimensional
P[t/ε]
H
H
marginals) of the discrete version of our problem, that is of the sum
n=1 f (Bn+1 − Bn )
with f a real function. On the other hand, in [19], the Hermite rank of f is used to discuss
R t/ε
H
H
the existence of the limit in distribution of 0 f (Bs+1 − Bs )ds for H > 12 (recall that, in
the present paper, we assume that H is smaller than 12 ).

2

Statement of results

2.1

Almost sure convergence

In the following, we shall denote by N a standard Gaussian random variable independent of
H
all processes which will appear, by B the fractional Brownian motion with Hurst index H
1

and by B = B 2 the Brownian motion.
Theorem 2.1 Assume that H ∈ (0, 1). Let f : R → R be a function satisfying for all x, y ∈ R
|f (x) − f (y)| ≤ L|x − y|a (1 + x2 + y 2 )b , (L > 0, 0 < a ≤ 1, b > 0),

(2.1)

and {Yt : t ≥ 0} be a continuous stochastic process. Then, as ε → 0,
Z

H

t

Ys f (
0

H

Bs+ε − Bs
)ds → E [f (N )]
εH

Z

t

Ys ds,

(2.2)

0

almost surely, uniformly in t on each compact interval.
The following result contains a similar statement for continuous martingales:
Theorem 2.2 Let f : R → R be a polynomial function. Assume that {Yt : t ≥ 0} is
a continuous process and that {Jt : t ≥ 0} is an adapted locally H¨
older
R t continuous paths
process. Let {Zt : t ≥ 0} be a continuous martingale given by Zt = Z0 + 0 Js dBs . Then, as
ε → 0,
Z t
Z t
Zs+ε − Zs
√
Ys E [f (N Js )|Fs ] ds
(2.3)
Ys f (
)ds →
ε
0
0
almost surely, uniformly in t on each compact interval. Here, Ft = σ(Js , s ≤ t).
Remarks:
1. For instance, if f (x) = xm then the right hand side of (2.3) equals to
Rt
m
m
E [N ] 0 Ys Js ds.
Rt
Rt
2. A similar result holds for continuous semimartingales of type Zt = Z0 + 0 Js dBs + 0 Ks ds
Rt
because the finite variation part 0 Ks ds does not have any contribution to the limit.
✷
4

If we apply Theorem 2.1 with f (x) = x, we obtain, almost surely on each compact interval,
Z t
1
H
H
Ys (Bs+ε − Bs )ds = 0.
lim
ε→0 εH 0
In the following result, we prove that, by replacing ε−H by ε−1 , we obtain a non-zero limit
H
for integrands of the form Ys = g(Bs ):
Corollary 2.3
1) Assume that g ∈ C2 (R) and that H belongs to [ 12 , 1). Then
Z

t
0

H

H

B
− Bs
g(Bs ) s+ε
ds
ε
H

converges,
surely, on each compact interval. Consequently, the forward
R t as ε H→ 0, almost
H
integral 0 g(Bs )d− Bs can be defined path to path.

2) Assume that g ∈ C3 (R) and that H belongs to [ 13 , 1). Then
1
2

Z

0

t

H

H

H

(g(Bs+ε ) + g(Bs ))

H

Bs+ε − Bs
ds
ε

converges, as ε R→ 0, almost surely, on each compact interval. Consequently, the symH
H
t
metric integral 0 g(Bs )d◦ Bs can be defined path to path.

3) Let {Zt : t ≥ 0} be a continuous martingale as in Theorem 2.2. Assume that g ∈ C2 (R).
Then
Z t
Zs+ε − Zs
ds
g(Zs )
ε
0
converges,
as ε → 0, almost surely, on each compact interval, to the classical Itˆ
o integral
Rt
g(Z
)dZ
.
s
s
0

Remarks: 1. In [2], it is proved, for g regular enough, that
Z t
Z t
H
H
H
H
◦ H
g(Bs )δBs + TrDg(B )t ,
g(Bs )d Bs =

(2.4)

0

0

Rt
H
H
H
H
where 0 g(Bs )δBs denotes the usual divergence integral with respect to B and TrDg(B )t
is defined as the limit in probability, as ε → 0, of
Z t
1
H
g ′ (Bs ) [R(s, (s + ε) ∧ t) − R(s, (s − ε) ∨ 0)] ds
2ε 0
H

H

with R(s, t) = Cov(Bs , Bt ). For any H ∈ (0, 1), it is a simple computation to see that the
Rt
H
previous limit exists almost surely, on each compact interval and equals to H 0 g ′ (Bs )s2H−1 ds.
Consequently, by using (2.4) and the part 2 of Corollary 2.3, we see that the divergence inteRt
H
H
gral 0 g(Bs )δBs can be defined path-wise.
H
2. In [20], it was introduced a path-wise stochastic integral with respect to B when the
integrator has γ-H¨older continuous paths with γ > 1 − H. When the integrator is of the form
H
g(Bt ), the condition on γ implies that H > 21 (see p. 354). Hence, the first two parts of
Corollary 2.3 could be viewed as improvements of the results in [20].
✷
5

2.2

Convergence in distribution

Let m be an odd integer. It is well known that the monomial xm may be expanded in terms
of the Hermite polynomials:
m

x

=

m
X

k

ak,m Hk (x), with Hk (x) = (−1) e

x2
2

k=1

dk
dxk

µ
¶
2
− x2
, k = 0, 1, 2, . . .
e

(2.5)

Note that the sum begin with k = 1 since m is odd (for instance x = H1 (x), x3 = 3H1 (x) +
H3 (x) and so on).
Theorem 2.4 Let m ≥ 3 be an odd integer and assume that H belongs to (0, 12 ]. Then, as
ε → 0,
!m
Z tà H
H
Bs+ε − Bs
1
√
L
{√
ds : t ≥ 0} −→ { cm,H βt : t ≥ 0}.
(2.6)
H
ε 0
ε
Here {βt : t ≥ 0} denotes a one-dimensional standard Brownian motion starting from 0 and
cm,H is given by
cm,H := 2

Z
m
X
a2k,m
k=1

k!

∞h

(x + 1)

0

2H

+ |x − 1|

2H

− 2x

2H

ik

dx,

where the coefficients ak,m are given by (2.5).
Remark: Let us note that if 0 < H < 21 ,
(x + 1)

2H

+ |x − 1|

and if H = 12 ,
(x + 1)

2H

2H

− 2x

2H

+ |x − 1|

∼ H(2H − 1)x
2H

Hence cm,H < ∞ if and only if 0 < H ≤ 21 .

− 2x

2H

−2(1−H)

, as x → ∞,

= 0, as x ≥ 1.
✷

Finally, let us state the result concerning martingales:
Theorem 2.5 Let m ≥ 3 be an odd integer and assume that Rσ is an element of C2 (R; R).
t
Let {Zt : t ≥ 0} be a continuous martingale given by Zt = Z0 + 0 σ(Bs )dBs . Then, as ε → 0,
1
{√
ε

Z tµ

(1)

0

Zs+ε − Zs
√
ε

¶m

L

ds : t ≥ 0} −→ {

Z

(2)

t

0

σ(βs(1) )m d(κ1 βs(1) + κ2 βs(2) ) : t ≥ 0}.

(2.7)

Here {(βt , βt ) : t ≥ 0} denotes a two-dimensional standard Brownian motion starting from
(0,0) and κi , i = 1, 2 are some constants.

6

3
3.1

Proofs
Proof of almost sure convergence

The idea to obtain almost sure convergence is firstly, to verify L2 -type convergence and
secondly, to use a Borel-Cantelli type argument and the regularity of paths (see Lemma 3.1
below).
To begin with, let us recall a classical definition: the local H¨
older index γ0 of a continuous paths process {Wt : t ≥ 0} is the supremum of the exponents γ verifying, for any
T > 0:
P ({ω : ∃ L(ω) > 0, ∀s, t ∈ [0, T ], |Wt (ω) − Ws (ω)| ≤ L(ω)|t − s|γ }) = 1.

(3.1)

We can state now the following almost sure convergence criterion which will be used in proving
Theorems 2.1 and 2.2:
Lemma 3.1 Let f : R → R be a function satisfying (2.1), {Wt : t ≥ 0} be a locally H¨
older
continuous paths process with index γ0 and {Vt : t ≥ 0} be a bounded variation continuous
paths process. Set
Z t
Ws+ε − Ws
(f )
)ds, t ≥ 0, ε > 0,
(3.2)
Wε (t) :=
f(
εγ 0
0
and assume that for each t ≥ 0, as ε → 0,
°2
° (f )
°
°
°Wε (t) − Vt ° 2 = O(εα ), with α > 0.

(3.3)

L

(f )

Then, for any t ≥ 0, limε→0 Wε (t) = Vt almost surely, and if f is non-negative, for any
continuous process {Yt : t ≥ 0}, as ε → 0,
Z t
Z t
(f )
Ys dVs
(3.4)
Ys dWε (s) →
0

0

almost surely, uniformly in t on every compact interval.
Proof. We split the proof in several steps.
Step 1. We set, for n ∈ N∗ , εn := n−2/α . For every δ > 0
´
³ (f )
i cst.
1 h (f )
P |Wεn (t) − Vt | > δ ≤ 2 E (Wεn (t) − Vt )2 ≤ 2 εαn .
δ
δ
P α
Since
εn < +∞, we deduce, by applying Borel-Cantelli lemma that, for each t ≥ 0,
(f )
limn→∞ Wεn (t) = V (t) almost surely.
Step 2. Fix ε > 0 and consider n ∈ N∗ such that εn+1 < ε ≤ εn . Let us fix ω ∈ Ω. We
shall denote, for each t ≥ 0,
(f )

(f )

Wε (t)(ω) = Wεn (t)(ω) + ξn (t)(ω) + ζn (t)(ω),
with
ξn (t)(ω) :=

Z t"
0

#
(f )
(f )
(f )
(f )
Ws+εn (ω) − Ws (ω)
Ws+ε (ω) − Ws (ω)
) − f(
) ds,
f(
εγ 0
εγ 0
7

ζn (t)(ω) :=

Z t"
0

#
(f )
(f )
(f )
(f )
Ws+εn (ω) − Ws (ω)
Ws+εn (ω) − Ws (ω)
f(
) − f(
) ds.
εγ 0
εγn0

We prove that ξn (t)(ω), ζn (t)(ω) tend to zero, as n → ∞, hence we shall deduce that, for each
(f )
t ≥ 0, limε→0 Wε (t) = V (t) almost surely.
For notational convention, we will drop the argument ω and the superscript (f ). We can
write, for δ > 0,
"
¶2 µ
¶2 #b
µ
Z
W
−
W
−
W
W
cst. t
s+ε
s
s
s+εn
+
ds
|Ws+ε − Ws+εn |a 1 +
|ξn (t)| ≤ aγ0
ε
εγ 0
εγ 0
0
≤

cst.
(a+2b)γ
εn+1 0

0 −δ)
|εn − εn+1 |a(γ0 −δ) ε2b(γ
t.
n
2a+4b

Since εn − εn+1 = O(n−1−2/α ), we have |ξn (t)| = O(n−aγ0 +δ(a+ α ) ) as n → ∞. Hence,
limn→∞ ξn (t) = 0 by choosing δ small enough. Similarly,
"
µ
µ
¶ Z
µ
¶
¶ #b
1
Wu+ε − Wu 2
Ws+εn − Ws 2
1 a t
a
|ζn (t)| ≤ cst. γ0 − γ0
|Ws+εn − Ws | 1 +
+
ds
ε
εγ 0
εγ 0
εn
0
µ
¶
1
1 a (a+2b)(γ0 −δ)
cst.
≤ 2bγ0
εn
t.
− γ0
γ0
εn+1 εn+1 εn
2a+4b

0
0
−1−2γ0 /α ), we have |ζ (t)| = O(n−a+δ(a+ α ) ) as n → ∞. Again,
− ε−γ
Since ε−γ
n
n
n+1 = O(n
limn→∞ ζn (t) = 0 by choosing δ small enough.
(f )
Step 3. We will show that the exceptional set of the almost sure convergence W ε (t) →
V (t) can be choosed independent of t. Let Ω∗ the set of probability 1, such that for every
(f )
ω ∈ Ω∗ , limε→0 Wε (t)(ω) = Vt (ω), ∀t ∈ Q ∩ R+ . Fix such a ω ∈ Ω∗ , t ∈ R+ and assume that
{sn } and {tn } are rational sequences such that sn ↑ t and tn ↓ t. Clearly,
(f )

(f )

(f )

Wε (sn )(ω) ≤ Wε (t)(ω) ≤ Wε (tn )(ω).
First, letting ε goes to zero we get
(f )

(f )

Vsn (ω) ≤ lim inf Wε (t)(ω) ≤ lim sup Wε (t)(ω) ≤ Vtn (ω),
ε→0

ε→0

and then, letting n goes to infinity we deduce that for each ω ∈ Ω∗ and each t ∈ R+ ,
(f )
limε→0 Wε (t)(ω) = Vt (ω).
(f )
Step 4. If f is non-negative we can apply Dini’s theorem to obtain that Wε (t) converges
almost surely toward Vt , uniformly on every compact interval.
Step 5. Further, the reasoning is pathwise, hence we fix ω ∈ Ω, we drop the argument
(f )
ω and write small letters instead capital letters. Since wε simply converges toward v, the
(f )
distribution function of the measure dwε converges toward the distribution function of the
(f )
measure dv, hence dwε weakly converges toward dv. Clearly, the measure dv does not charge
points and the function s 7→ ys 1[0,t] (s) is dv-almost everywhere continuous. Consequentely,
Z t
Z ∞
(f )
ys dvs .
lim
ys 1[0,t] (s) dwε (s) =
ε→0 0

0

The proof of the almost sure convergence criterion is done.
8

Proof of Theorem 2.1.
First, let us note that if f satisfies (2.1) then the positive part f+ and the negative part
f− also satisfy (2.1). Hence by linearity, we can assume that f is a non-negative function. We
H
shall apply Lemma 3.1 to W = B , the fractional Brownian motion which is a locally H¨older
continuous paths process with index H (as we can see by applying the classical Kolmogorov
theorem, see [15], p. 25) and to the process Vt = E [f (N )] t. We need to verify (3.3). First
we note that
H
H
2H
Var(Bu+ε − Bu ) = ε
√
and, if u + ε ≤ u + ε < v,
H

H

H

H

Cov(Bu+ε −Bu , Bv+ε −Bv ) = (v−u−ε)

2H

+(v−u+ε)

2H

−2(v−u)

2H

≤

cst.ε2
1+H
,
2−2H ≤ cst.ε
(v − u)

as we can see by using Taylor expansion. Hence, by classical linear regression we obtain, for
√
u + ε < v,
H
H
³
´
Bv+ε − Bv
1+H
2(1+H)
=
O(ε
)N
+
1
+
O(ε
)
Mu,ε ,
u,ε
εH

B

H

−B

(3.5)

H

uniformly with respect to u, where Nu,ε = u+εH u and Mu,ε are two independent standard
ε
Gaussian random variables. We write
" Ã
! #2 
H
H
 Z t

Bs+ε − Bs
f(
E
)
−
E
[f
(N
)]
ds
= T1 (ε) + T2 (ε) + T3 ,
(3.6)
 0

εH

where

#
H
H
H
H
B
−
B
B
−
B
1u 0,
L

then, we have
³ (f )
´
˜ ε(f ) (t) = 0
lim Wε (t) − W

(3.11)

ε→0

almost surely, uniformly on every compact interval.
The proof is straightforward and we leave it to the reader. Using this result, we obtain that
¸
¶
¶
µ
·Z t
Z tµ
Zs+ε − Zs m
Bs+ε − Bs m
√
√
ds −
ds = 0
(3.12)
Jsm
lim
ε→0
ε
ε
0
0
almost surely, uniformly on each compact interval. Combining (3.12) with (3.7), we get (2.3).
H

Proof of Corollary 2.3. To prove the first part, we set Ys = g ′′ (Bs ) and f (x) = x2 in
(2.2). Then, we obtain the existence of
lim

Z

ε→0 0

t

H

H

g ′′ (Bs )

H

(Bs+ε − Bs )2
ds, almost surely, uniformly on each compact interval.
ε

On the other hand, we can write, for a, b ∈ R, a < b,
g(b) = g(a) + g ′ (a)(b − a) +
H

g ′′ (θ)
(b − a)2 ,
2

H

with θa,b ∈ (a, b). Setting a = Bs and b = Bs+ε , integrating in s on [0, t] and dividing by ε
we get:
Z
Z
Z t
1
1 t ′ H
1 t
H
H
H
H
H
H
(g(Bs+ε )−g(Bs ))ds =
g (Bs )(Bs+ε −Bs )ds+
g ′′ (θB H ,B H )(Bs+ε −Bs )2 ds.
s
s+ε
ε 0
ε 0
2ε 0
By a simple change of variable we can transform the left-hand side as
Z
Z
1 t+ε
1 ε
H
H
g(Bs )ds −
g(Bs )ds,
ε t
ε 0
H

which tends, as ε → 0, almost surely uniformly on each compact interval toward g(B t )−g(0).
The last term on the right-hand side of the previous equality converges, almost surely and
uniformly on each compact interval and therefore the term which remains on the right-hand
side is also forced to have a limit. The third part can be proved in a similar way.
H
Let us turn to the second part. By setting Ys = g (3) (Bs ) and f (x) = x3 in (2.2), we obtain
the existence of
Z t
H
H
− Bs )3
H (B
lim
g (3) (Bs ) s+ε
ds, almost surely, uniformly on each compact interval.
ε→0 0
ε
11

On the other hand, by setting Ys ≡ 1 and f (x) = |x|3 in (2.2) we obtain that
lim

Z

t

ε→0 0

H

H

|Bs+ε − Bs |3
ds < +∞, almost surely, ∀t > 0.
ε

Consequently, it suffices to use the following Taylor formula
g(b) = g(a) +

g ′ (a) + g ′ (b)
g (3) (θ)
(b − a) −
(b − a)3
2
12

and the dominated convergence, in order to conclude as previously.

3.2

Proofs of the convergence in distribution

Proof of Theorem 2.4. First, let us explain the main ideas in the simpliest
R tT situation ofmthe
1
−1/2
Brownian motion (H = 2 ). In this case we are studying MT (t) = T
0 (Bs+1 − Bs ) ds
(see Step 1 below) and we write it, thanks to succesive applications of Itˆ
o’s formula and of the
R tT
stochastic version of Fubini theorem, as 0 RT (s)dBs plus a remainder which tends to zero
R tT
in L2 , as T → ∞. Then we can show that limT →∞ 0 RT (s)2 ds = cst.t, hence, by Dubins√
Schwarz theorem, we obtain that MT → cst.β, as T → ∞, in the sense of finite dimensional
time marginals. Finally we prove the tightness. Let us remark that similar technics have been
used in [14], precisely in the proof of Proposition 3 (see also Step 10 below).
For the fractional Brownian motion case (0 < H < 21 ) technical difficulties appear because
Rt
H
the kernel K in its moving average representation (Bt = 0 K(s, t)dBs ) is singular at the
points s = 0 and s = t. Again we split the proof in several steps.
H
Step 1. By the self-similarity of the fractional Brownian motion, that is {Bct : t ≥
L

H

H

0} = {c Bt : t ≥ 0}, for all c > 0, we can see that
!m
Z t³
Z tà H
H
´
ε
Bs+ε − Bs
1
H
H m
L √
{√
ds
:
t
≥
0}
=
{
ε
B
ds : t ≥ 0} = {M 1 (t) : t ≥ 0},
−
B
s+1
s
ε 0
ε
εH
0
where
1
MT (t) := √
T

Z

0

tT

³

H

H

Bs+1 − Bs

´m

ds, t ≥ 0.

(3.13)

Hence, to get (2.6) it suffices to prove that
√
L
{MT (t) : t ≥ 0} −→ { cm,H βt : t ≥ 0}, as T → ∞.

(3.14)

Moreover, this convergence is a consequence of the following two facts:
√
i) as T → ∞, {MT (t) : t ≥ 0} → { cm,H βt : t ≥ 0} in law in sense of finite

(3.15)

dimensional time marginals;
ii) for T ≥ 1, the family of distributions of processes MT is tight.

12

(3.16)

Step 2. Before proceeding with the proof of (3.15), let us show how the constant c m,H
appears. We claim that, for each t ≥ 0,
£
¤
lim E MT (t)2 = cm,H t.
(3.17)
T →∞

H

H

H

H

Set G1 = Bu+1 − Bu , G2 = Bv+1 − Bv and θ(u, v) = Cov(G1 , G2 ). We need to estimate the
m
expectation of the product Gm
1 G2 . Thanks to (2.5), we have
m
E[Gm
1 G2 ]

=

X

ak,m aℓ,m E[Hk (G1 )Hℓ (G2 )] =

m
X
a2k,m
k=1

1≤k,ℓ≤m

k!

θ(u, v)k .

Replacing this in the expression of the second moment of MT (t), we obtain, noting also that
2H
2H
2H
θ(u, v) = 21 (|v − u + 1| + |v − u − 1| − 2|v − u| ),
£
¤
2
E MT (t)2 =
T
=2

Z

0

t

dy

Z
m
X
a2k,m
k=1

k!

ZZ

dudv

0≤u 0. Let us note that, by (3.17), we can choose b > 0 small
enough such that
£
¤
lim E MT (b)2 ≤ ̺.
(3.19)
T →∞

Step 4. Let us recall (see, for instance, [1], p. 122) that the fractional Brownian motion
can be written as
Z 0
Z t
1
1
H− 2
H− 1
H− 2
H
ˇtH = γ
Bt = At + B
[(t
−
s)
(t − s) 2 dBs , t ≥ 0, (3.20)
−
(−s)
]dB
+
γ
s
H
H
−∞

0

where, here and elsewhere we denote by γH the constant Γ(H + 12 )−1 . We can write MT(b) (t) =
ˇ (b) (t) + D (b) (t), where
M
T
T
Z tT ³
´m
H
ˇ (b) (t) := √1
ˇsH
ˇs+1
ds, t ≥ 0.
(3.21)
−
B
B
M
T
T bT
ˇ
Since the process A has absolutely continuous trajectories and using the fact that A and B
are independent as stochastic integrals on disjoint intervals, it is not difficult to prove that,
for each t ≥ 0,
h
i
h
i
ˇ (b) (t))2 = 0.
lim E DT(b) (t)2 = lim E (MT(b) (t) − M
(3.22)
T
T →∞

T →∞

13

Step 5. By (3.21) and (3.20) we can write
¶m
Z s
Z tT µZ s+1
H− 1
H− 1
γH
2
2
(b)
ˇ
(s − u)
ds
(3.23)
dBu −
dBu
(s + 1 − u)
MT (t) = √
T bT
0
0
¶m
Z tT µZ s
Z s+1
1
H− 1
H− 1
H− 2
γH
2
2
[(s + 1 − u)
=√
(s + 1 − u)
ds, t ≥ 0.
− (s − u)
]dBu +
dBu
T bT
0
s
We need to introduce a second technical notation. Let us denote:
Z tT ÃZ s
H− 1
H− 1
γH
(b,c)
ˇ
(3.24)
NT (t) := √
[(s + 1 − u) 2 − (s − u) 2 ]dBu
T bT
(s−c)∨0
+

Z

s+1

(s + 1 − u)

s

1
H− 2

dBu

¶m

ds, t ≥ 0,

where the positive constant c will be fixed and specified by the statement a) below. We shall
prove successively the following statements:
a) for each t ≥ 0, there exists c > 0 large enough, such that
h
i
ˇ (b) (t) − N
ˇ (b,c) (t))2 ≤ cst.̺;
lim sup E (M
T
T

(3.25)

T →∞

b) there exists two families of stochastic processes {RT (t) : t ≥ 0} and {ST (t) : t ≥ 0} such
that, for each t ≥ 0,
Z tT
ˇ (b,c) (t) =
RT (s)dBs + ST (t), with lim E[ST (t)2 ] = 0;
(3.26)
N
T
T →∞

0

c) for each t ≥ 0,
lim Var
T →∞

µZ

0

tT

2

RT (s) ds

¶

= 0.

(3.27)

Step 6. Suppose for a moment that a), b), c) are proved and let us finish the proof of the
convergence in law in sense of finite dimensional time marginals (3.15). First, we can write,
Z

tT
0

2

RT (s) ds − cm,H t = {

Z

tT
0

2

RT (s) ds − E[

Z

0

tT

2

RT (s) ds]} + {E[

µZ

tT
0

RT (s)dBs

¶2

]

£
¤
ˇ (b) (t)2 ]} + {E[M
ˇ (b) (t)2 − M (b) (t)2 ]} + {E[M (b) (t)2 − M (t)2 ]} + {E M (t)2 − cm,H t}.
−E[M
T
T
T
T
T
T

By using (3.27) for the first term, (3.25)-(3.26) for the second term, (3.22) for the third term,
(3.19) for the forth term and (3.17) for the fifth one, we obtain
Z tT
RT (s)2 ds − cm,H t)2 ] ≤ cst.̺,
lim sup E[(
T →∞

0

or equivalently, for each t ≥ 0,
lim sup E[(a(T ) (t) − a(t))2 ] ≤ cst.̺,

(3.28)

T →∞

14

R tT

RT (s)2 ds and a(t) := cm,H t. Second, we fix d ∈ N∗ and 0 ≤ t1 <
P
t2 < . . . < td and we shall denote for any u ∈ Rd and f : R+ → R, u • f := dj=1 uj f (tj ). We
consider the characteristic functions:
h
i
|E [exp(iu • MT )] − E [exp(iu • (β ◦ a))]| ≤ E | exp(iu • (MT − MT(b) )) − 1|
with notations a(T ) (t) :=

0

h
i
h
i
ˇ (b) )) − 1| + E | exp(iu • (M
ˇ (b) − N
ˇ (b,c) )) − 1|
+E | exp(iu • (MT(b) − M
T
T
T

+E [| exp(iu • ST ) − 1|] + |E[exp(iu •

Z

·T

0

RT (s)dBs )] − E[exp(iu • (β ◦ a))]|.

By (3.19), (3.22), (3.25), (3.26) and using the classical inequality |eix − 1| ≤ |x|, we obtain,
for T large enough
|E [exp(iu • MT )] − E [exp(iu • (β ◦ a))]| ≤ cst.̺
+|E[exp(iu •

Z

0

·T

(3.29)

RT (s)dBs )] − E[exp(iu • (β ◦ a))]|.

R ·T
By Dubins-Schwarz theorem, we can write, for each T , 0 RT (s)dBs = β(T ) ◦ a(T ) , with β(T )
a one-dimensional standard Brownian motion starting from 0. Therefore, we have
Z ·T
RT (s)dBs )] − E[exp(iu • (β ◦ a))]|
(3.30)
|E[exp(iu •
0

h
i
≤ 2P(ka(T ) − ak > δ) + E | exp(iu • (β(T ) ◦ a(T ) )) − exp(iu • β(T ) ◦ a))| : ka(T ) − ak ≤ δ
≤

i
2 h
2
E
ka
−
ak
+ kuk E[ sup k(βv1 , . . . , βvd ) − (βw1 , . . . , βwd )k].
(T )
δ2
kv−wk≤δ

Combining (3.29), (3.30) and letting ̺ → 0, (3.15) follows.
Step 7. We verify (3.16), that is, the tightness of the family of distributions of processes
MT . It suffices to verify the classical Kolmogorov criterion (see [15], p. 489):
£
¤
sup E (MT (t) − MT (s))4 ≤ cR |t − s|2 , ∀0 ≤ s, t ≤ R.
(3.31)
T ≥1

Let s, t ∈ [0, R]. Then, by (3.13),
ZZZZ
h H
£
¤
1
H
H
H
H
H
4
E (MT (t) − MT (s)) = 2
E (Bu1 +1 − Bu1 )m (Bu2 +1 − Bu2 )m (Bu3 +1 − Bu3 )m
T
[sT,tT ]4
ZZZZ
i
1
H
H m
m m m
× (Bu4 +1 − Bu4 ) du1 du2 du3 du4 = 2
E [Gm
1 G2 G3 G4 ] du1 du2 du3 du4
T
[sT,tT ]4
H

H

where, as in Step 2, we denoted the standard Gaussian random variables G i = Bui +1 − Bui ,
i = 1, 2, 3, 4. Let us also denote θij = Cov(Gi , Gj ), i, j = 1, . . . , 4. We need to estimate the
m m m
expectation of the product Gm
1 G2 G3 G4 . By using (2.5), we get
X
m m m
ak1 ,m ak2 ,m ak3 ,m ak4 ,m E [Hk1 (G1 )Hk2 (G2 )Hk3 (G3 )Hk4 (G4 )]
E [Gm
1 G2 G3 G4 ] =
k1 ,k2 ,k3 ,k4 ≥1

15

and we need to estimate E[Hk1 (G1 )Hk2 (G2 )Hk3 (G3 )Hk4 (G4 )]. Using the result in [18], p.
210, we can write
E [Hk1 (G1 )Hk2 (G2 )Hk3 (G3 )Hk4 (G4 )]
=

½

k1 !k2 !k3 !k4 !
2q q!

0
P

P

1 θi1 j1

. . . θiq jq ,

(3.32)

if k1 + k2 + k3 + k4 = 2q and 0 ≤ k1 , k2 , k3 , k4 ≤ q
otherwise,

where 1 is the sum over all indices i1 , j1 , . . . , iq , jq ∈ {1, 2, 3, 4} such that i1 6= j1 , . . . , iq 6= jq
and there are k1 indices 1, . . . , k4 indices 4. For instance E [H1 (G1 )H1 (G2 )H1 (G3 )H1 (G4 )] =
1
8 (θ12 θ34 + θ13 θ24 + θ14 θ23 ). Similarly, we can compute E [H3 (G1 )H3 (G2 )H3 (G3 )H3 (G4 )] in
terms of θij and so on. Since Gi have variance 1, we deduce, using the conditions on the
indices appearing in (3.32), that
X
E |Hk1 (G1 )Hk2 (G2 )Hk3 (G3 )Hk4 (G4 )| ≤ cst.
|θij | |θkℓ |
{i,j}6={k,ℓ}

Therefore, to get (3.31), we need to consider the following two type of terms: {i, j}∩{k, ℓ} = ∅,
for instance i = 1, j = 2, k = 3, ℓ = 4, or {i, j} ∩ {k, ℓ} =
6 ∅, for instance i = 1, j = 2, k =
1, ℓ = 3. Clearly, by simple change of variables,
1
T2

=

Ã

ZZZZ

1
2T
Ã

[sT,tT ]4

ZZ

[sT,tT ]2

|θ12 | |θ34 |du1 du2 du3 du4 =

Ã

1
T

ZZ

[sT,tT ]2

|θ12 |du1 du2

!2

¯
¯
2H
2H
2H ¯
¯
|u
−
u
+
1|
+
|u
−
u
−
1|
−
2|u
−
u
|
¯ du1 du2
¯ 2
1
2
1
2
1

!2

!2
Z (t−s)T ¯
Z tT
¯
1
2H
2H
2H ¯
¯
=
≤ λ(t − s)2 ,
du1
¯(x + 1) + |x − 1| − 2x ¯ dx
2T sT
0
R∞
2H
2H
2H
where λ := ( 21 0 |(x + 1) + |x − 1| − 2x |dx)2 , and, similarly,
ZZZZ
1
|θ12 | |θ13 |du1 du2 du3 du4
T2
[sT,tT ]4
ZZZ
¯
¯
t−s
2H
2H
2H ¯
¯
=
¯|u2 − u1 + 1| + |u2 − u1 − 1| − 2|u2 − u1 | ¯
4T
[sT,tT ]3
¯
¯
2H
2H
2H ¯
¯
× ¯|u3 − u1 + 1| + |u3 − u1 − 1| − 2|u3 − u1 | ¯ du1 du2 du3 ≤ λ(t − s)2 .

Hence (3.31) is verified so the family of distributions of processes MT is tight.
The proof of Theorem 2.4 will be finished once we prove statements a)-c) in Step 5.
Step 8. We prove (3.25) and at the same time we precise the choice of the constant c.
For notational convenience we will drop superscripts “(b)” or “(b,c)” during the proof. Using
again (3.23) and (3.24) we can write
ˇ (t) − N
ˇ (t) =
M
T
T

m µ ¶
X
m
k=1

k

PˇT(k) (t),

(3.33)

16

with
γ
PˇT(k) (t) = √H
T

Z

tT

ds
bT

"Z

(s−c)∨0

K(s, u)dBu

0

and where we denoted
(
H− 1
H− 1
(s + 1 − u) 2 − (s − u) 2 ,
K(s, u) :=
H− 1
(s + 1 − u) 2 ,

#k "Z

s+1

(s−c)∨0

K(s, u)dBu

if 0 ≤ u ≤ s

#m−k

.

if s ≤ u ≤ s + 1

(3.34)

We shall prove that the second moment of each term in (3.33) can be made small enough and
then (3.25) will follows.
Step 9. We can write
¸
ZZ
i 2γ 2 ·Z Z
′
′
′
′
(k)
2
H
ˇ
∆(s, s )dsds +
∆(s, s )dsds
E PT (t) =
T
bT