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Stochastics An International Journal of Probabilityand Stochastic Processes: formerly Stochastics andStochastics ReportsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gssr20
Discrete approximation of stochastic integrals withrespect to fractional Brownian motion of Hurst indexH>1/2Francesca Biagini a , Massimo Campanino b & Serena Fuschini ba Department of Mathematics , LMU , Theresienstr. 39, D-80333, Munich, Germanyb Department of Mathematics , University of Bologna , Piazza di Porta S. Donato, 5, I-40127,Bologna, ItalyPublished online: 09 Oct 2008.
To cite this article: Francesca Biagini , Massimo Campanino & Serena Fuschini (2008) Discrete approximation of stochasticintegrals with respect to fractional Brownian motion of Hurst index H>1/2, Stochastics An International Journal of Probabilityand Stochastic Processes: formerly Stochastics and Stochastics Reports, 80:5, 407-426, DOI: 10.1080/17442500701594672
To link to this article: http://dx.doi.org/10.1080/17442500701594672
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Discrete approximation of stochastic integrals with respect tofractional Brownian motion of Hurst index H > 1/2
Francesca Biaginia*, Massimo Campaninob1 and Serena Fuschinib2
aDepartment of Mathematics, LMU, Theresienstr. 39, D-80333 Munich, Germany; bDepartment ofMathematics, University of Bologna, Piazza di Porta S. Donato, 5, I-40127 Bologna, Italy
(Received 7 November 2005; final version received 25 July 2007 )
In this paper we provide a discrete approximation for the stochastic integral withrespect to the fractional Brownian motion of Hurst index H . 1/2 defined in terms ofthe divergence operator. To determine the suitable class of integrands for which theapproximation holds, we also investigate the relations among the spaces of Malliavindifferentiable processes in the fractional and standard case.
Keywords: Fractional Brownian motion; Malliavin derivative; Divergence; Stochasticintegral
AMS 2000 Subject Classifications: Primary 60G15; 60G18; 60HXX; 60H07; 60H40
1. Introduction
Fractional Brownian motion (fBm) of Hurst index H [ (0,1) has been widely used in
applications to model a number of phenomena, e.g. in biology, meteorology, physics and
finance. A natural question is how to define a stochastic integral with respect to fBm and
how to provide a natural discretization of it. The problem of defining a stochastic integral
with respect to the fBm has been extensively studied in literature. Several different
integrals has been introduced by exploiting different properties of the fBm: as Wiener
integrals (exploiting the Gaussianity) in Refs. [12,13,24], as pathwise integrals (using
fractional calculus and forward, backward and symmetric integrals) in Refs.
[2,3,8,9,11,20,23,26–28], as divergence (using fractional Malliavin calculus) in Refs.
[3,4,10,24], as stochastic integral with respect to the standard Brownian motion in Refs.
[6,16], as stochastic integral with respect to the fractional white noise in Refs. [14,17]. For
a complete survey on this subject, an extensive literature and relations among different
definitions of integrals we refer to Biagini et al. and Nualart [7,24].
For applications, in particular for applications to finance, it is relevant to give an
interpretation to a definition of stochastic integral. As it is the case of Ito and Skorohod
integrals, this can be achieved by giving a procedure to obtain the integral as the limit of
discrete approximations that are meaningful in the application field.
In this paper we address the problem of finding a discrete approximation of the
stochastic integral defined as divergence. A discrete realization of the fBm for Hurst index
H . 1/2 is provided in different ways in Refs. [13,19]. This can be extended to find a
discretization of stochastic integral of deterministic functions integrable with respect to
ISSN 1744-2508 print/ISSN 1744-2516 online
q 2008 Taylor & Francis
DOI: 10.1080/17442500701594672
http://www.informaworld.com
*Corresponding author. Email: [email protected]
Stochastics: An International Journal of Probability and Stochastics Processes
Vol. 80, No. 5, October 2008, 407–426
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the fBm. In Ref. [5] they study the convergence of Riemann–Stieltjes integrals with
respect to the fBm for Hurst index H . 1/2 when the integrands are almost surely Holder-
continuous of order l . 1 2 H. Other discrete approximations of stochastic integrals with
respect to the fBm can be found in Refs. [3,8,14,22].
Here we focus on the case H . 1/2 and consider the stochastic integral with respect to
B H defined in terms of the divergence operator d H relative to the fBm.We prove a discrete
approximation for this kind of integral by means of the resolutions of the Fock space
associated to B H and by using its expression in terms of the divergence operator d of the
standard Brownian motion.
In order to find a suitable class of stochastic integrands for which our results hold, we
also study the relations among the spaces of Malliavin differentiable processes with
respect to B and to B H with Hurst index H . 1/2.
2. Basic definitions and notation
We recall in this section the basic definition and the main properties concerning fBm and
the relative Malliavin calculus.
Definition 2.1. The fBm B H of Hurst index H [ (0,1) is a centered Gaussian process
{BHt }t$0 with covariance given by
RHðs; tÞ U1
2ðs2H þ t 2H 2 jt2 sj
2HÞ: ð1Þ
As well-known, for H ¼ 1/2 we obtain the standard Brownian motion (Bm). By
Kolmogorov’s theorem we have that B H admits a continuous modification for every
H [ (0,1). In the sequel we assume to work always with the continuous modification of
B H, that we denote again with B H.
In particular we will focus on the case H . 1/2. In this case, the covariance (1) can be
written as
RHðs; tÞ ¼ aH
ðt0
ðs0
fðr; uÞdr du ð2Þ
where f is given by
fðr; uÞ ¼ jr 2 uj2H22
ð3Þ
and aH ¼ H(2H–1). By Nualart [24], we obtain that RH(s,t) can be expressed in terms of
the deterministic kernel
KHðt; sÞ ¼ cHs122 H
ðts
ðu2 sÞH232 uH21
2 du ð4Þ
where cH ¼ Hð2H 2 1Þ=bð22 2H;H 2 ð1=2ÞÞ1=2, t . s, in the following way
RHðt; sÞ ¼
ðt^s0
KHðt; uÞKHðs; uÞdu:
As in the standard Brownian motion case, we wish to associate to B H a Gaussian
Hilbert space in the following way. We denote E the space of step functions defined on
F. Biagini et al.408
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[0,T ] and let H be the Hilbert space given by the completion of E with respect to the
following inner product
x½0;t�; x½0;s�� �
H¼ RHðt; sÞ:
The function x½0;t� 7! BHt can be extended to an isometry between H and the Gaussian
Hilbert space generated by the fBm. We denote this isometry by B H.
We recall that the following relation holds
L2ð½0; T�Þ # H ð5Þ
since
k fkH # kHk fkL 2ð½0;T�Þ ð6Þ
for every f [ L 2([0,T ]), where kH ¼ T 2H21/(H 2 1/2). For the proof we refer to Alos and
Nualart [4,24].
We introduce now the operator K*H .
Definition 2.2. Let K*H be the linear operator defined on E
K*H : E ! L2ð½0; T�Þ
such that
K*Hw
� �ðsÞ U
ðTs
wðrÞ›KH
›rðr; sÞdr: ð7Þ
We note that
K*Hðx½0;t�ÞðsÞ ¼ KHðt; sÞx½0;t�ðsÞ:
Proposition 2.3. For every f, c [ E, we have
K*Hw;K
*Hc
� �L 2ð½0;T�Þ
¼ kw;clH: ð8Þ
Proof. For the proof we refer to Alos and Nualart [4,23]. A
Hence, we can extend the operator K*H to an isometry between the Hilbert spaceH and
L 2([0,T ]) (see Ref. [24] for further details). We define the process
Bt ¼ BH K*H
� �21x½0;t�
� �t [ ½0; T�: ð9Þ
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We obtain that Bt is a Gaussian process, whose covariance is given by
kBt;BslL 2ð½0;T�Þ ¼ BH K*H
� �21x½0;t�
� �;BH K*
H
� �21x½0;s�
� �D EL 2ð½0;T�Þ
¼ K*H
� �21ðx½0;t�Þ; K*
H
� �21ðx½0;s�Þ
D EH
¼ s ^ t
Hence, Bt (or better its continuous modification) is a standard Brownian motion.
Moreover, for every w [ H we have
BHðwÞ ¼
ðT0
K*Hw
� �ðtÞdBt
and in particular
BHt ¼ BHðx½0;t�Þ ¼
ðT0
KHðt; sÞdBs; ð10Þ
where the kernel KH is defined in (4). As a consequence of (9) and (10), we immediately
have that B H and B generate the same filtration.
We denote by SH the set of regular cylindric random variables S of the form
S ¼ f ðWHðh1Þ; . . . ;WHðhnÞÞ; ð11Þ
where hi [ H, n $ 1, and f belongs to the set C1b ðRn;RÞ of bounded continuous functionswith bounded continuous partial derivatives of every order.
Definition 2.4. Let H [ [1/2,1) and S [ SH :
(1) The Malliavin derivative of S with respect to B H is defined as the H—valued
random variable
DHS UXni¼1
›f
›xiðWHðh1Þ; . . . ;W
HðhnÞÞhi: ð12Þ
(2) The space SH is endowed with the norm D1;2H such that
kSk2
D1;2H
U kSk2L 2ðVÞ þ kDHSk
2L 2ðV;HÞ:
We denote by D1;2H the closure of the space SH with respect to the norm k·kD1;2
Hand
extend by density the derivative operator to the space D1;2H .
We introduce now the divergence operator. Let H [ [1/2,1).
Definition 2.5.
(1) The domain domd H of the divergence operator is defined as the set of the random
variables F [ L2ðV;HÞ such that
jEðkDHS;FlHÞj # cFkSkL 2ðVÞ ;S [ SH ; ð13Þ
for some constant cF depending on F.
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(2) The divergence operator d H: domd H ! L 2(V) is defined by the duality relation
kDHG;FlL 2ðV;HÞ ¼ kG; dHFlL 2ðVÞ ð14Þ
for every G [ D1;2H .
Hence, the divergence operator is the adjoint of the derivative operator.
From now on, to simplify the notation in the case H ¼ 1/2, we omit to write the index
H and denote
L1;2 U D1;21=2ðL
2ð½0; T�ÞÞ:
We conclude this section by recalling a result that will play a key role in the sequel.
Proposition 2.6. We have that F [ domd H iff K*HF [ domd and
dHðFÞ ¼ d K*HF
� �: ð15Þ
Proof. For the proof we refer to Alos and Nualart [3,24]. A
According to the approach of Decreusefond and Duncan [12,15], we define the
stochastic integral with respect to B H for H . 1/2 as follows.
Definition 2.7. Let B H be a fBm of Hurst index H . 1/2 and F a stochastic process
belonging to domd H. We define stochastic integral of F with respect to B H the divergence
d H(F) of F introduced in Definition 2.5.
3. Relations between the standard case and the case H > 1/2
Before proceeding to prove our main results concerning the discretization of the stochastic
integral as divergence, we need to clarify some relations between the standard and the
fractional case.
3.1 Chaos expansions
By the Chaos Expansion Theorem, for every F [ L 2(V,L 2([0,T ])) we have
;t [ ½0; T�; Ft ¼Xnq¼0
Iqðf q;tÞ in L2ðVÞ ð16Þ
where f q;tðt1; . . . ; tqÞ U f qðt1; . . . ; tq; tÞ [ L2ð½0; T�qþ1Þ are symmetric with respect to the
first q variables (t1, . . . ,tq) and Iq denotes the q-dimensional iterated Wiener integral with
respect to (t1, . . . ,tq)
Iqð f q;tÞ U
ðT0
ðtq0
. . .
ðt20
f q;tðt1; . . . ; tqÞdBt1 . . . dBtq21
� �dBtq
(for further details see Refs. [18,23]).
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We introduce the function
K*H : L2ðV;HÞ! L2ðV; L2ð½0; T�ÞÞ ð17Þ
that associates to F [ L2ðV;HÞ the random variable K*HF [ L2ðV; L2ð½0; T�ÞÞ defined as
ðK*HFÞðvÞ U K*
HðFðvÞÞ:
Proposition 3.1. The following properties hold
(i) K*H : L2ðV;HÞ! L2ðV; L2ð½0; T�ÞÞ is an isometry;
(ii) L2ðV; L2ð½0; T�ÞÞ # L2ðV;HÞ and
kFkL 2ðV;HÞ # kHkFkL 2ðV;L 2ð½0;T�ÞÞ:
Proof. The proof follows by (5), (6) and (8). A
Proposition 3.2. Let F belong to L 2(V,L 2([0,T])) with chaos expansion given by (16).
Then
K*HF ¼
X1q¼0
Iq K*H;·ðf q;·Þ
� �in L2ðV; L2ð½0; T�ÞÞ: ð18Þ
Moreover, for almost every t [ [0,T]
K*HF
� �t¼X1q¼0
Iq K*H;·f q;·
� �t
� �in L2ðVÞ; ð19Þ
where by (7) we have
K*H;·f q;·
� �t¼
ðTt
f qðt1; . . . ; tq; rÞ›KH
›rðr; tÞdr:
Proof. We start with some preliminary remarks.
(A) If we put
ðSQÞt UXQq¼0
Iqðf q;tÞ; Q [ N;
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then by the Chaos Expansion Theorem [23] we obtain the following convergences
limQ!1
ðSQÞt 2 Ft
L 2ðVÞ
¼ 0 ;t [ ½0; T� ð20Þ
limQ!1
SQ 2 F
L 2ðV;L 2ð½0;T�ÞÞ¼ 0: ð21Þ
(B) Consider F * ¼ K*HF. Then F * admits chaos expansion given by
F * ¼X1q¼0
Iq f *q;·
� �in L2ðV; L2ð½0; T�ÞÞ;
where f *q;· [ L2ð½0;T�qþ1Þ are symmetric function with respect to the variables
t1, . . . ,tq, for every q [ N: Hence if we set
S*Q
� �tUXQq¼0
Iq f *q;t
� �; Q [ N;
the following convergences hold
limQ!1
S*Q
� �t2F*
t
L 2ðVÞ
¼ 0 ð22Þ
limQ!1
S*Q 2 F *
L 2ðV;L 2ð½0;T�ÞÞ¼ 0: ð23Þ
(C) By the stochastic version of the Fubini–Tonelli theorem [25], we can exchange the
order of integration between the multiple integral Iq and the integral operator K*H
K*H;·ðIqð f q;·ÞÞ
h it¼ Iq K*
H;·ð f q;·Þ� �
ðtÞ; ð24Þ
for every t [ [0,T].
By (8) and Proposition 3.1 we obtain that
K*HSQ 2K*
HF 2
L 2ðV;L 2ð½0;T�ÞÞ¼ K*
HðSQ 2 FÞ 2
L 2ðV;L 2ð½0;T�ÞÞ# kH SQ 2 F
2L 2ðV;L 2ð½0;T�ÞÞ
:
Hence by (21) we have
limQ!1
K*HSQ 2K*
HF
L 2ðV;L 2ð½0;T�ÞÞ¼ 0: ð25Þ
Now it remains only to prove that
limQ!1
K*HSQ
� �t2 K*
HF� �
t
L 2ðVÞ
¼ 0; for a:e: t [ ½0; T�: ð26Þ
By (23) and (25) we get immediately
limQ!1
S*Q 2K*HSQ
2L 2ðV;L 2ð½0;T�ÞÞ
¼ 0: ð27Þ
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Moreover, we know
S*Q 2K*HSQ
2L 2ðV;L 2ð½0;T�ÞÞ
¼XQq¼0
Iq f *q;· 2 K*Hf q;·
� �2
L 2ðV;L 2ð½0;T�ÞÞ
: ð28Þ
and that the following relation holds between the norms:
Iqð f q;tÞ 2
L 2ðV;L 2ð½0;T�ÞÞ¼ q! ð f q;tÞ
2L 2ð½0;T�qþ1Þ
; ð29Þ
if (fq,t) is symmetric with respect to the first q variables t1, . . . ,tq, for every q [ N: Sincealso f *q;· 2 K*
H;·f q;· are symmetric with respect to (t1, . . . ,tq), for every q [ N, we get
XQq¼0
Iq f *q;· 2 K*H;·f q;·
� �2
L 2ðV;L 2ð½0;T�ÞÞ
¼XQq¼0
q! f *q;· 2 K*H;·f q;·
2L 2ð½0;T�qþ1Þ
:
and by (27)
X1q¼0
q! f *q;· 2 K*H;·f q;·
2L 2ð½0;T�qþ1Þ
¼ 0:
Finally
f *q;· 2 K*H;·f q;·
2L 2ð½0;T�qþ1Þ
¼
ðT0
ð½0;T�q
ðf *q;· 2 K*H;·f q;·Þ
2ðt1; . . . ; tq; tÞdt1 . . . dtq
� �dt ¼ 0
for every q [ N. We conclude that for almost every t [ [0,T]
f *q;t 2 K*H;·f q;·
� �t¼ 0 in L2ð½0; T�qÞ
for every q [ N: Finally, for almost every t [ [0,T] we have that ðS*QÞt ¼ ðK*HSQÞt, for
every Q [ [0,T], and then
limQ!1
K*HSQ
� �t2 K*
HF� �
t
L 2ðVÞ
¼ limQ!1
S*Q
� �t2 K*
HF� �
t
L 2ðVÞ
:
By (22) we obtain the result. A
3.2 Relation between D1;2H ðHÞ and L1;2
We can generalize the Definition 2.5 to the case of H—valued random variables. In this
case the space SHðHÞ is defined as the set of random variables U with values in H of the
form
U ¼Xnj¼1
Fjvj; ð30Þ
where Fj [ D1;2H , vj [ H, ;j [ {1, . . . ,n}. If U [ SHðHÞ, the Malliavin derivative D HU
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of U is defined as the element in L2ðV;H^HÞ given by
DHU UXnj¼1
DHFj^vj; ð31Þ
where ^ denotes the Hilbert–tensor product between the Hilbert spaces H. We refer to
Aubin [1] for the definition and the properties of the tensor product.
The derivative operator
DH : SHðHÞ! L2ðV;H^HÞ
induces on SHðHÞ the norm
Uk k2
D1;2HðHÞ
U Uk k2L 2ðV;HÞ þ DHU
2L 2ðV;H^HÞ
:
We extend Definition 2.5 as follows.
Definition 3.3. The space D1;2H ðHÞ is defined as the closure of SHðHÞ with respect to the
norm k·kD1;2HðHÞ
. The derivative operator D H can be extended to D1;2H ðHÞ.
We note that the space D1;2H ðHÞ is an Hilbert space with respect to the inner product
kF;GlD1;2HðHÞ
U kF;GlL 2ðV;HÞ þ kDHHF;D
HHGlL 2ðV;H^HÞ; ð32Þ
for every F;G [ D1;2H ðHÞ:
In particular, we have
Proposition 3.4.
D1;2H ðHÞ # domdH and kdHFkL 2ðVÞ # kFkD1;2
HðHÞ
:
Proof. For the proof and further details, see Ref. [18]. A
Consider now the operator ðK*HÞ
^2 : L2ðV;H^HÞ! L2ðV; L2ð½0; T�2ÞÞ; defined on
F [ L2ðV;H^HÞ as follows
K*H
� �^2ðvÞ U K*
H
� �^2ðFðvÞÞ:
Proposition 3.5. The operator K*H has the following properties:
(i) Let F [ L2ðV;HÞ. Then
F [ D1;2H ðHÞ , K*
HF [ L1;2 ð33Þ
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and
K*H
� �^2ðDHFÞ ¼ D K*
HF� �
: ð34Þ
(ii) K*H : D
1;2H ðHÞ! L1;2 is an isometry with respect to the inner product defined in (32),
i.e.
K*HF;K
*HG
� �L1;2
¼ kF;GlD1;2HðHÞ
for every F;G [ D1;2H ðHÞ:
Proof.
(i) Let F [ D1;2H ðHÞ , L2ðV;HÞ: Then exists a sequence (Un) in SHðHÞ; such that
Un ! F in L2ðV;HÞ
DHUn ! DHF in L2ðV;H^HÞ
(ð35Þ
for n ! 1. We recall that by (30) the term Un is given by
Un ¼Xknj¼1
Fnj h
nj n [ N
where Fnj [ D1;2 and hj [ H, ;j [ {1, . . . ,n}.
We define U*n U K*
HUn; n [ N: By Proposition 3.1 and (35) we have
limn!1
kUn 2 FkL 2ðV;HÞ ¼ 0:
That implies
U*n !K*
HF in L2ðV; L2ð½0; T�ÞÞ ð36Þ
for n ! 1.
In Ref. [24] is proved that D1;2H ¼ D1;2 and in particular
K*HðD
HFÞ ¼ DF; ;F [ D1;2H ¼ D1;2: ð37Þ
By (37) we obtain the following relation
K*H
� �^2DHUn
� �¼Xknj¼1
DFnj^K*
Hhnj ¼ D U*
n
� �:
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Then
D U*n
� �2 K*
H
� �^2ðDHFÞ
2L 2ðV;L 2½0;T�2Þ
¼ K*H
� �^2DHUn
� �2 K*
H
� �^2DHF� � 2
L 2ðV;L 2ð½0;T�2ÞÞ
¼ DHUn 2 DHF 2
L 2ðV;H^HÞ:
The last term goes to zero by (35) and we have
D U*n
� �! K*
H
� �^2ðDHFÞ in L2ðV; L2ð½0; T�2ÞÞ ð38Þ
for n ! 1.
By (36) and (38) we obtain that (34) holds. If we now assume K*HF [ L1;2, we can
proceed in the same way using ðK*HÞ
21 instead of K*H : Hence we have proved also (33).
(ii) This result follows immediately by (32), and from that the fact that K*H is an
isometry. A
We can prove now the following result, that is essential to characterize the space of
integrands F, whose stochastic integral d H(F) admits the discretization provided in
Section 4.
Proposition 3.6. The following inclusion holds
L1;2 # D1;2H ðHÞ ð39Þ
and
kFkD1;2HðHÞ
# kHkFkL1;2 ;F [ L1;2: ð40Þ
where kH ¼ T 2H21=ðH 2 ð1=2ÞÞ as in (6).
Proof. Let F [ L1;2: We recall that it is equivalent to the condition
X1q¼0
qðq!Þ f q;·Þ 2
L 2ð½0;T�qþ1Þ, 1 ð41Þ
(see Ref. [23]). By Proposition 3.2 and 3.5 we have that F belongs to D1;2H ðHÞ iff
X1q¼0
qðq!Þ K*H;·f q;·
2L 2ð½0;T�qþ1Þ
, 1:
By (6) we have
K*H;·f q;·
2L 2ð½0;T�qþ1Þ
¼
ð½0;T�q
f qðt1; . . . ; tqÞ 2
Hdt1 . . . dtq # kHkf qk2L 2ð½0;T�qþ1Þ
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and then
X1q¼0
qðq!Þ K*H;·f q;·
� � 2L 2ð½0;T�qþ1Þ
# kHX1q¼0
qðq!Þ f q 2
L 2ð½0;T�qþ1Þ: ð42Þ
Since F [ L1;2, the seriesP1
q¼0 qðq!Þkf qk2L 2ð½0;T�qþ1Þ is convergent in L 2(V,L 2([0,T])).
Moreover (42) implies that
X1q¼0
qðq!Þ K*H;·f q;·
2L 2ð½0;T�qþ1Þ
converges in L 2(V,L 2([0,T])).
We prove now (40). We note that
kDFk2L 2ðV;L 2ð½0;T�2ÞÞ ¼
X1q¼0
qðq!Þkf qk2L 2ð½0;T�qþ1Þ
and
D K*HF
� � 2L 2ðV;L 2ð½0;T�2ÞÞ
¼X1q¼0
qðq!Þ K*H;·f q;·
2L 2ð½0;T�qþ1Þ
:
Hence by (42), it follows that
D K*HF
� � 2L 2ðV;L 2ð½0;T�2ÞÞ
# kHkDFk2L 2ðV;L 2ð½0;T�2ÞÞ
Finally, we get
kFk2
D1;2HðHÞ
¼ kFk2L 2ðV;HÞ þ kDHFk
2L 2ðV;H^HÞ
¼ kK*HFk
2L 2ðV;L 2ð½0;T�ÞÞ þ kðK*
HÞ^2ðDHFÞk
2L 2ðV;L 2ð½0;T�2ÞÞ
¼ kK*HFk
2L 2ðV;L 2ð½0;T�ÞÞ þ kDðK*
HFÞk2L 2ðV;L 2ð½0;T�2ÞÞ
# kHkFk2L 2ðV;L 2ð½0;T�ÞÞ þ kHkDFk
2L 2ðV;L 2ð½0;T�2ÞÞ
¼ kHkFk2L1;2 :
A
Remark 3.7. The space L1;2 is strictly included in D1;2H ðHÞ> L2ðV; L2ð½0; T�ÞÞ: We prove
this by showing that there exists an element F [ D1;2H ðHÞ> L2ðV; L2ð½0; T�ÞÞ that does
not belong to L1;2. Let F have the following chaos expansion
F ¼X1q¼0
Iqð f q;·Þ in L2ðV; L2ð½0; T�ÞÞ
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where
f q;tðt1; . . . ; tqÞ ¼1ffiffiffiffiffiffiT q
p1ffiffiffiffiffiffiffiffiffiffiq!qb
p f ðqatÞ a;b . 0: ð43Þ
Suppose that f satisfies the following hypotheses:
(i) f $ 0;
(ii) f [ L 2([0,1));
(iii)Ð10
Ð10f ðuÞf ðvÞfðu; vÞdudv , 1, where f is defined in (3).
Since by (18) and (33) the following must hold
F [ L1;2 ,X1q¼0
qðq!Þk f q;·k2L 2ð½0;T�qþ1Þ , 1;
F [ D1;2H ðHÞ ,
X1q¼0
qðq!Þ K*H;·f q;·
2L 2ð½0;T�qþ1Þ
, 1;
it is sufficient to prove that there exist fq,· of the form (43) such that
X1q¼0
qðq!Þkf q;·k2L 2ð½0;T�qþ1Þ ¼ 1; ð44Þ
X1q¼0
qðq!Þ K*H;·f q;·
2L 2ð½0;T�qþ1Þ
, 1: ð45Þ
By standard integral calculations we get the following inequalities
qðq!Þk f q;·k2L 2ð½0;T�qþ1Þ $ k fk
2L 2ð½0;T�Þ
1
qaþb21ð46Þ
and
qðq!Þk f q;·k2L 2ð½0;T�qþ1Þ # cf
1
q2aHþb21: ð47Þ
where cf ¼Ð10
Ð10f ðuÞf ðvÞfðu; vÞdudv:
Finally (44) and (45) hold if we choose a and b such that
aþ b # 2;
2aH þ b . 2:
(ð48Þ
Since H . 1/2, it is then sufficient to choose a ¼ b ¼ 1.
4. Discrete approximation for the stochastic integral
We can now prove a discrete approximation for the stochastic integral defined as
divergence introduced in Definition 2.7.
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By Proposition 2.6 it follows that the integral of F with respect to fBm coincides the
Skorohod integral of K*HF with respect to the Bm. Hence one can provide a discretization
of the integral d H(F) by using a discrete approximation of dðK*HFÞ: However, this is not
satisfactory because the discretization depends on the Hurst index of the Brownian motion
with respect to which the integral is defined. Therefore, we define first a discretization of
the process and then a procedure for obtaining the approximation of the integral.
4.1 Resolution of [0,T]
We introduce first the setting and some basic notions needed to prove the main result of
this paper. For further details we refer to Malliavin [21]. Consider the measure space
ð½0; T�;B; lÞ, where B denotes the Borelian s-algebra and l the Lebesgue measure.
Definition 4.1. A resolution of L 2([0,T]) is an increasing family of s-algebras generated
by a finite number of sets
B1 , B2 , · · · , Bn , · · · , B:
such that <n[N
L2ð½0; T�;Bn; lÞ is dense in L2([0,T]). We denote with Gn the finite set of
generators of Bn, that we assume to be a partition of the interval [0,T].
Given a resolution {Bn}, consider the set GN U <Nn¼0Gn and the set G ¼ <
n[NGn. We
obtain on RG a coherent system of marginal laws uN and by Kolmogorov theorem there
exists a probability measure u on RG with this family of marginal laws. Hence we can
associate a probability space ðRG; uÞ to a resolution {Bn}.
Proposition 4.2. The probability spaces associated to different resolutions are
canonically isomorphic.
Proof. For the proof see Ref. [21]. A
4.2 Fock space
We denote with R the set of all resolutions of L2([0,T ]).
Definition 4.3. The Fock space is the set of all pairs ðVðrÞ;Cr;r0 Þ where r; r0 [ R, V(r)
denotes the probability space associated to the resolution r and Cr;r0 is the canonical
isomorphism between V(r) and V(r 0).
All properties of a Fock space do not depend on the choice of the resolution. For
further details, see Ref. [21]. Consider a subset g [ B. The space L 2([0,T ]) can be
decomposed as follows
L2ð½0; T�Þ ¼ L2ðgÞ%L2ðg cÞ:
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This decomposition induces a similar one on the relative Fock spaces
FocðL2ð½0; T�ÞÞ ¼ FocðL2ðgÞÞ £ FocðL2ðg cÞÞ:
Definition 4.4. We denote by
E g c
: L2ðFocðL2ð½0; T�ÞÞÞ! L2ðFocðL2ðg cÞÞÞ ð49Þ
the natural projection from L 2(Foc(L 2([0,T]))) onto L 2(Foc(L 2(g c))).
We recall the construction of the Skorohod–Zakai–Nualart–Pardoux integral (in
short, SZNP-integral) given in Ref. [21] for F [ L 2(V,L 2([0,T ])). We define
InðFÞ UXg[Gn
E g c
ðEBnðFÞÞðgÞBðgÞ;
where {Bn} is a resolution of L 2([0,T ]) as introduced in Definition 4.1, E g c
denotes the
projection (49) and B(g) is the usual Wiener integral of the indicator function xg with
respect to Bm, i.e.
BðgÞ U
ðT0
xgðsÞdBs; g [ B:
The SZNP-integral is defined as the limit
I ðFÞ U limn!1
InðFÞ in L2ðVÞ:
By Malliavin [21] we have the following theorem
Theorem 4.5. Let F [ L1;2: Then the SZNP-integral exists and
limn!1
InðFÞ ¼ dðFÞ in L2ðVÞ: ð50Þ
Proof. For the proof we refer to Malliavin [21]. A
Let F [ L1;2. Since by Proposition 3.6, L1;2 # D1;2H ðHÞ and by Proposition 3.4,
D1;2H ðHÞ # domdH , the stochastic integral d H(F) is well defined.
Definition 4.6. Let F [ L1;2. We define
. IHn ðFÞ UP
g[GnE g c
ðEBn ðK*HFÞðgÞÞBðgÞ;
. JHn ðFÞ UP
g[GnEBnðE g c
ðK*HðE
BnðFÞÞÞðgÞBðgÞÞ;
where {Bn} is a resolution of L2([0,T]) as introduced in Definition 4.1 and E g c
denotes the
projection (49).
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By Theorem 4.5 we have that
limn!1
IHn ðFÞ ¼ d K*HF
� �in L2ðVÞ; ð51Þ
and by Proposition 2.6 we get
d K*HF
� �¼ dHðFÞ:
The purpose is now to prove that JHn ðFÞ provides the desired discretization of d H(F).
Theorem 4.7. Let F [ L1;2. Then
limn!1
JHn ðFÞ ¼ dHðFÞ in L2ðVÞ:
Proof. Let F [ L1;2. We need to prove
limn!1
IHn ðFÞ2 JHn ðFÞ� �
¼ 0 in L2ðVÞ:
By (16) the chaos expansion of F is given by
Ft ¼X10
Iqð f q;tÞ;
where fq,t [ L 2([0,T]q þ 1) are symmetric functions for every q [ N:By Proposition 3.2 we get
K*HF
� �t¼X1q¼0
Iq K*H;· f q;·
� �ðtÞ:
We have that
EBn K*HF
� �t¼X1q¼0
Iq EBn K*H;· f q;·
� �ðtÞ
� �; ð52Þ
EBn K*HF
� �ðgÞ ¼
X1q¼0
Iq EBnð f q;·ÞðgÞ� �
; ð53Þ
E g c
EBn K*HF
� �ðgÞ ¼
X10
Iq EBn K*H;· f q;·
� �ðgÞx
^qg c
� �: ð54Þ
Here we writeK*H;· f q;· to indicate onwhich variable the operatorK
*H is operating. ByNualart
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[23] we obtain that the product of two Wiener integrals can be written as
Iq EBn K*H;·f q;·
� �� �ðgÞx
^qg c BðgÞ ¼ Iqþ1 EBn K*
H;·f q;·
� �ðgÞxgx
^qg c
� �:
Hence IHn ðFÞ can be rewritten as
IHn ðFÞ ¼X1q¼0
Iqþ1
Xg[Gn
EBn K*H;·f q;·
� �xgx
^qg c
!
and analogously
JHn ðFÞ ¼X1q¼0
Iqþ1
Xg[Gn
EBn K*H;· E
Bn ð f q;·Þ� �� �
xgx^qg c
!:
It follows that the difference between IHn ðFÞ and JHn ðFÞ is of the form
IHn ðFÞ2 JHn ðFÞ ¼X1q¼0
Iqþ1
Xg[Gn
EBn ðK*H;·f q;·Þ2 EBn K*
H;· EBn ð f q;·Þ
� �� �h ixgx
^qg c
!:
Taking the norms, we get
IHn ðFÞ2JHn ðFÞ 2
L 2ðVÞ¼X1q¼0
ðqþ1Þ!
£Xg[Gn
EBn K*H;· f q;·
� �2EBn K*
H;· EBnðf q;·Þ
� �� �h ixgx
^qg c
2
L 2ð½0;T�qþ1Þ
:
Let An : ½0; T�qþ1 ! R be the function
An U EBn K*H;· f q;·
� �2 EBn K*
H;· EBnð f q;·Þ
� �� �:
Since for every n [ N; the sets {g}g[Gnare a partition of the interval [0,T], we obtain
Xg[Gn
Anxgx^qg c
2
L 2ð½0;T�qþ1Þ
¼
ð½0;T�qþ1
Xg[Gn
Anxgx^qg c
!2
dt1 . . . dtqdt
¼
ð½0;T�qþ1
Xg[Gn
A2nxgx
^qg c
!dt1 . . . dtqdt
#
ð½0;T�qþ1
Xg[Gn
A2nxg
!dt1 . . . dtqdt
¼
ð½0;T�qþ1
Anðt1; . . . tq; tÞXg[Gn
xgðtÞ
!dt1 . . . dtqdt
¼ kAnk2L 2ð½0;T�qþ1Þ:
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Hence it follows
IHn ðFÞ2 JHn ðFÞ 2
L 2ðVÞ#X1q¼0
ðqþ 1Þ!kAnk2L 2ð½0;T�qþ1Þ:
Put AqðnÞ U ðqþ 1Þ!kAnk2L 2ð½0;T�qþ1Þ: To prove that
P1q¼0AqðnÞ goes to zero in L 2(V)
for n ! 1, it is sufficient to verify the following conditions:
AqðnÞ! 0 as n!1 ;q $ 0; ð55Þ
X1q¼0
AqðnÞ converges uniformly in n: ð56Þ
Condition (55) is equivalent to
limn!1
kAnk2L 2ð½0;T�qþ1Þ ¼ 0: ð57Þ
We compute the norm
kAnk2L 2ð½0;T�qþ1Þ ¼
ð½0;T�qþ1
A2nðt1; . . . ; tq; tÞdt1 . . . dtqdt
¼
ð½0;T�q
ðT0
A2nðt1; . . . ; tq; tÞdt
� �dt1 . . . dtq:
To prove (57), we define the sequence
Gnðt1; . . . ; tqÞ U
ðT0
A2nðt1; . . . ; tq; tÞdt ð58Þ
and show that it verifies the hypothesis of Lebesgue Theorem on bounded convergence.
Indeed
Gnðt1; . . . ; tqÞ ¼ kAnk2L 2ð½0;T�Þ ¼ EBn K*
H;· f q;·
� �2 EBn K*
H;· EBn ð f q;·Þ
� �� � 2L 2ð½0;T�Þ
¼ EBn K*H;· f q;·
� �2 K*
H;· EBnð f q;·Þ
� �� �h i 2L 2ð½0;T�Þ
# K*H;· f q;· 2 K*
H;· EBn ð f q;·Þ
� � 2L 2ð½0;T�Þ
¼ kK*H;· f q;· 2 EBnð f q;·Þ� �
k2L 2ð½0;T�Þ
¼ f q;· 2 EBn ð f q;·Þ 2
H # kH f q;· 2 EBn ð f q;·Þ 2
L 2ð½0;T�Þ:
where kH ¼ T 2H21/(H 2 (1/2)) as in (6). The last term tends to zero when n ! 1 and then
limn!1
Gnðt1; . . . ; tqÞ ¼ 0; for a:e:2 ðt1; . . . ; tqÞ:
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We need now to show that there exists a function G [ L 1([0,T]q) such that
jGnj # G; for a:e:2 ðt1; . . . ; tqÞ:
The following relation holds
jGnðt1; . . . ; tqÞj ¼ Gnðt1; . . . ; tqÞ # cH f q;· 2 EBn ð f q;·Þ 2
L 2ð½0;T�Þ
# cHk f q;·kL 2ð½0;T�Þ þ kEBnð f q;·ÞkL 2ð½0;T�Þ2 # 4cHk f q;·k2L 2ð½0;T�Þ:
Set Gðt1; . . . ; tqÞ U 4cHk f qk2L 2ð½0;T�Þ. We haveð
½0;T�qGðt1; . . . ; tqÞdt1 . . . dtq ¼ 4cH
ð½0;T�q
k f qk2L 2ð½0;T�Þdt1 . . . dtq
¼ 4cH
ð½0;T�q
ðT0
f 2qðt1; . . . ; tq; tÞdt
� �dt1 . . . dtq
¼ 4cHk f qk2L 2ð½0;T�qþ1Þ , 1;
ð59Þ
since F [ L1;2. Hence the norm kGkL 1ð½0;T�qÞ is finite and by the Lebesgue theorem applied
to Gn(t1, . . . ,tq) we can conclude that kAnk2L 2ð½0;T�qþ1Þ converges to zero when n ! 1. This
ends the proof of (57) and hence of (55).
To prove (56) we use the following criterion of uniform convergence if
X1q¼0
supnjAqðnÞj , 1;
thenP1
q¼0AqðnÞ converges uniformly in n. We have by (58) and (59) that
jAqðnÞj ¼ ðqþ 1Þ!kAnk2L 2ð½0;T�qþ1Þ ¼ ðqþ 1Þ!Gnðt1; . . . ; tqÞ # ðqþ 1Þ!4cHk f qk
2L 2ð½0;T�qþ1Þ:
Hence
supnjAqðnÞj # 4cHðqþ 1Þ!k f qk
2L 2ð½0;T�qþ1Þ:
Since F [ L1;2, the seriesP
q¼0k f qk2L 2ð½0;T�qþ1Þ is convergent and then alsoP1
q¼0 supnjAqðnÞj , 1. This ends the proof. A
Acknowledgements
We wish to express our gratitude to Paul Malliavin for interesting discussions and remarks.
Notes
1. Fax: þ39-0512094490. Email: [email protected]. Fax: þ39-0512094490. Email: [email protected]
References
[1] J.P. Aubin, Applied Functional Analysis, John Wiley and Sons, New York, NY, 1979.[2] E. Alos, A. Leon, and D. Nualart, Stratonovich stochastic calculus with respect to to fractional
Brownian motion with Hurst parameter less than 1/2, Taiwanese J. Math. 5 (2001),pp. 609–632.
Stochastics: An International Journal of Probability and Stochastics Processes 425
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by [
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rn Q
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slan
d] a
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57 0
7 O
ctob
er 2
014
[3] E. Alos, O. Mazet, and D. Nualart, Stochastic calculus with respect to Gaussian processes,Ann. Probab 29 (2000), pp. 766–801.
[4] E. Alos and D. Nualart, Stochastic integration with respect to the fractional Brownian motion,Stochast. Stochast. Rep. 75 (2003), pp. 129–152.
[5] T.O. Androshchuk and Y.S. Mishura, The sequence of semimartingales that converges to theprocess of capital in a mixed Brownian-fractional Brownian model, Preprint University ofKiev, 2005.
[6] F. Biagini, B. Øksendal, A. Sulem, and N. Wallner, An introduction to white noise theory andMalliavin calculus for fractional Brownian motion, Proc. Roy. Soc. 460 (2004), pp. 347–372.
[7] F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional BrownianMotion and Applications, Springer, New York, NY, 2006, forthcoming book.
[8] F. Biagini and B. Øksendal, Forward integrals and an Ito formula for fractional Brownianmotion, preprint University of Oslo, June, 2004.
[9] P. Carmona, L. Coutin, and G. Montseny, Stochastic integration with respect to fractionalBrownian, Annales de I’Institut Henri Poincare-PR 39(1) (2003), pp. 27–68.
[10] P. Cheridito and D. Nualart, Stochastic integral of divergence type with respect to the fractionalBrownian motion with Hurst parameter H , 1/2, Annales de I’Institut Henri Poincare 41(2005), pp. 1049–1081.
[11] L. Decreusefond, A Skorohod–Stratonovich Integral for the Fractional Brownian Motion,Proceedings of the 7-th Workshop on stochastic analysis and related fields Birkhauser, Basel,2000.
[12] L. Decreusefond, Stochastic integration with respect to fractional brownian motion, Theoryand Applications of Long-range Dependence, Birkhauser, Basel, 2003.
[13] L. Decreusefond and A.S. Ustunel, Stochastic analysis of the fractional Brownian motion,Potential Anal. 10 (1999), pp. 177–214.
[14] T.E. Duncan, Y. Hu, and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion,I. Theory, SIAM Journal on Control and Optimization 38 (2000), pp. 582–612.
[15] T.E. Duncan, J. Jakubowski, and B. Pasik-Duncan, An approach to stochastic integration forfractional Brownian motion in a Hilbert Space, Fifteenth International Symposium on Math.Theory Networks and Systems, South Bend, 2002.
[16] R.J. Elliott and J. Van der Hoek, A general fractional white noise theory and applications tofinance, Math. Finance 13 (2003), pp. 301–330.
[17] Y. Hu and B. Øksendal, Fractional white noise calculus and applications to finance, Infin.Dimens. Anal. Quantum Prob. Relat. Top. 6 (2003), pp. 1–32.
[18] S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, Cambridge, 1997.[19] C. Kluppelberg and C. Kuhn, Fractional Brownian motion as a weak limit of Poisson shot noise
processes with applications to finance, Stochastic Process. Appl. 113 (2004), pp. 333–351.[20] S.J. Lin, Stochastic analysis of fractional Brownian motion, Stochast. Stochast. Rep. 55 55
(1995), pp. 121–140.[21] P. Malliavin, Stochastic Analysis, Springer, New York, NY, 1997.[22] Y. Mishura and E. Valkeila, An isometric approach to generalized stochastic integrals, J.
Theoret. Probab. 13 (2000), pp. 673–693.[23] D. Nualart, The Malliavin Calculus and Related Topics, Springer, New York, NY, 1995.[24] D. Nualart, Stochastic integration with respect to fractional Brownian motion and applications,
Contemp. Math. 336 (2003), pp. 3–39.[25] P.E. Protter, Stochastic Integration and Differential Equations: A New Approach, Springer,
New York, NY, 1990.[26] M. Zahle, On the link between fractional and stochastic calculus, in Stochastic Dynamics,
Bremen 1997, H. Crauel and M. Gundlach, eds., Springer, New York, NY, 1999, pp. 305–325.[27] M. Zahle, Integration with respect to fractal functions and stochastic calculus I, Probab.
Theory Related Fields 111 (1998), pp. 333–374.[28] M. Zahle, Integration with respect to fractal functions and stochastic calculus II,
Mathematische Nachrichten 225 (2001), pp. 145–183.
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