Multiple fractional integral with Hurst parameter less than 12

Stochastic Processes and their Applications 116 (2006) 463–479

Multiple fractional integral with

Hurst parameter less than

1

₂

Xavier Bardina

, Maria Jolis

Departament de Matema`tiques, Universitat Auto`noma de Barcelona, 08193-Bellaterra, Barcelona, Spain

Received 6 April 2005; received in revised form 12 September 2005; accepted 29 September 2005 Available online 14 November 2005

Abstract

We construct a multiple Stratonovich-type integral with respect to the fractional Brownian motion with Hurst parameterH_o1

2. This integral is obtained by a limit of Riemann sums procedure in the

Sole´ and Utzet [Stratonovich integral and trace, Stochastics Stochastics Rep. 29 (2) (1990) 203–220] sense. We also define the suitable traces to obtain the Hu–Meyer formula that gives the Stratonovich integral as a sum of Itoˆ integrals of these traces. Our approach is intrinsic in the sense that we do not make use of the integral representation of the fractional Brownian motion in terms of the ordinary Brownian motion.

r2005 Elsevier B.V. All rights reserved.

Keywords:Fractional Brownian motion; Hu–Meyer formula; Itoˆ-type multiple integral; Stratonovich multiple integral

1. Introduction

The aim of this work is to construct a multiple integral of Stratonovich type with respect

to the fractional Brownian motion with Hurst parameterHo1₂.

The notion of multiple integral (with respect to the Brownian motion) was introduced

first by Wiener[20]who called it polynomial chaos. Later, Itoˆ[12]modified the definition

of Wiener and constructed a multiple integral that possesses the property of orthogonality of different order integrals and allows to obtain the chaos decomposition of any square integrable functional of the Wiener process.

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In an important paper, Hu and Meyer [9]introduced a different multiple integral, that they called ‘‘of Stratonovich type’’, in order to give a new approach to the Feynmann integrals. The authors also proposed a formula (called Hu–Meyer formula) that gives the relationship of the Stratonovich integral with the Itoˆ integrals of some functions called the

traces, that involve integrals of f on the diagonals. Later, several authors presented different definitions of the multiple Stratonovich integrals and the traces. See, for instance,

Johnson and Kallianpur[13]and Sole´ and Utzet [19]. We cite also the paper of Nualart

and Zakai [16] that considered a multiple Ogawa-type integral, that can be seen as a

multiple Stratonovich integral in the sense of Sole´–Utzet and, as the authors pointed out, is

the integral closest in spirit to that introduced by Wiener[20].

As Hu and Meyer[10]pointed out, all the Stratonovich integrals introduced by different

authors can be always interpreted as the limit of the multiple integrals of the integrand function with respect to some absolutely continuous approximations of the Brownian motion. In this sense, one expect that the multiple Stratonovich integral satisfies the rules of the ordinary calculus.

The first construction of multiple integrals with respect to general Gaussian processes

was raised by Huang and Cambanis [11]. The aim of that paper was to construct an

integral that preserves the properties of the Itoˆ integral. In particular, the integrals of different orders were mutually orthogonal.

For the fractional Brownian motion with parameterH41

2defined on the interval½0;T,

BH, Dasgupta and Kallianpur[4]constructed a multiple integral that satisfies that for any

functionfof the form

f ¼X m

am1Am

1Amn witham2Rfor allm,

without any restriction on the intervalsAm_i , its integral equals to

X m amBHðAm1Þ. . .B H_ðAm nÞ, whereBH_ðAm

i Þdenotes the increment of the fractional Brownian motion on the interval

Am_i . The corresponding Hu–Meyer formula is proved in Dasgupta and Kallianpur [5].

Bardina et al. [3] proved that this integral is the limit in L2_{ðOÞ of the Riemann sums}

considered by Sole´ and Utzet[19], that is

IS_nðfÞ ¼L2ðOÞ lim jpj!0 X i1;...;in 1 jDp i1j jD p inj Z Dp i₁Dpin fðx1;. . .;xnÞdx1. . .dxn ! BHðDp_i1Þ BHðDp_inÞ, wherefDp

ijg are the intervals determined by a partitionp of the interval½0;T. As usual,

given an intervalD, we denote byjDjits length.

We cite also the paper of Duncan et al.[7]where, following the ideas of Hu and Meyer

[10], a multiple Stratonovich integral with respect to the fractional Brownian motion of

parameterH41

2is defined.

The fractional Brownian motion with Hurst parameter H41

2 is a particular case of a

Gaussian process with covariance function of bounded variation on ½0;T2. For such

functions related with the measure induced by the covariance function (see[14]). Observe

that the fractional Brownian motion with Hurst parameterHo1

2cannot be included in this

setting.

The case of the fractional Brownian motion with arbitrary Hurst parameter was

considered by Perez-Abreu and Tudor[17]. They use the fact that the fractional Brownian

motion of Hurst index Hcan be written asBH_t ¼R₀tKHðt;sÞdWs whereW is a standard

Brownian motion andKH is a deterministic Volterra-type kernel. Then, they define both

Itoˆ and Stratonovich (in the sense of Johnson and Kallianpur[13]) multiple integrals with

respect to BH _{by tensorizing this identity. These integrals are constructed as the}

corresponding multiple integrals with respect to the ordinary Brownian motion of some operator acting on the integrand function. In this way, they obtain easily some properties of the multiple integrals, in particular the Hu–Meyer formula, by transferring the same properties known when the integrator is the Wiener process. Nevertheless, the hypotheses of some of their results involve the transferring operator and are difficult to verify.

Our aim here is to present a more explicit construction of the Stratonovich multiple integral, without using the integral representation of the fractional Brownian motion in terms of the Wiener process. We define the integral of Stratonovich type as an integral in the Sole´–Utzet sense. Here, we point out that we have imposed some uniformity on the length of the intervals of the partitions, stated as condition (P) (see Section 2, before Lemma 2.3). This condition is used in our treatment of the hyper-singular integrals that appear along the paper.

In the paper of Hu[8]a Stratonovich integral with respect to the fractional Brownian

motion with arbitrary Hurst parameter is also defined without use of the integral representation of this process in terms of the Wiener process. Unlike our construction, in that paper, the Stratonovich integral is defined as the sum of Itoˆ integrals of some traces

inspired by those of Johnson and Kallianpur[13].

The paper is organized as follows. In Section 2, we give the notations and known results on the first order integral and prove that it can be given as a limit of certain Riemann sums, with sequences of partitions satisfying condition (P). The third section is

devoted to adopt the results of Huang and Cambanis[11]on the multiple Itoˆ integral to

our situation. Section 4 deals with the construction of the multiple Stratonovich integral, the definition of the traces and the proof of Hu–Meyer formula. Finally, as an example, we

adopt these results for a function of class Cb, withb412H, in the case of the double

integral.

2. The integral of order 1

LetBH _{¼ fB}H

t ; t2 ½0;Tgbe a fractional Brownian motion with Hurst parameterHo12.

That is,BH is a centered Gaussian process with covariance function given by

RHðs;tÞ ¼EðBH_s BH_t Þ ¼1₂ðt2Hþs2H jtsj2HÞ.

We will introduce the integral of first order of a deterministic function with respect to

BH. Consider the setSof all step functions on½0;Tof the form

f ¼X N

j¼1

where ½aj;bjÞ ½0;T and f_j2R. Clearly, S is a vectorial space. For a function of the

above type, define its integral with respect toBH in the natural way as follows:

I1ðfÞ ¼ XN

j¼1

f_jðBH_bj BH_ajÞ.

It is easily verified that this is a good definition, in the sense that it does not depend on

the particular representation of the simple functionf, and thatI1 is linear. Moreover, for

allf 2Swe have thatEðI1ðfÞÞ ¼0.

If we introduce in Sthe scalar product

Cðf;gÞ ¼E½I1ðfÞI1ðgÞ,

(it is positive defined because the covariance RH _{is positive defined), we can extend the}

integral in a standard way to the space, that we denote byLH_T, given by the completion of

S with respect to the scalar product C. The space LH_T is completely known and is

characterized by means of fractional derivatives (see, for instance,[6,2,18]). Nevertheless,

we will use another simpler characterization ofLHT more suitable in order to raise multiple

integration.

This characterization is based on the following expression of the covariance of two integrals of simple functions.

Lemma 2.1. For f;g2S Cðf;gÞ ¼ 1 2Hð12HÞ Z T 0 Z T 0 ðfðxÞ fðyÞÞðgðxÞ gðyÞÞ jxyj22H dxdy þH Z T 0 fðxÞgðxÞ 1 x12Hþ 1 ðTxÞ12H dx. ð1Þ

Proof. By bilinearity, it suffices to check (1) for f ¼1½0;sÞ and g¼1½0;tÞ. We must see

that in this case the right-hand side of (1) equals to RHðs;tÞ. Suppose, for instance,

that 0osptpT. Then 1 2 Hð12HÞ Z T 0 Z T 0 ðfðxÞ fðyÞÞðgðxÞ gðyÞÞ jxyj22H dxdy ¼Hð12HÞ Z T t Z s 0 ðyxÞ2H2dxdy ¼1 2½ðTsÞ 2H _ðtsÞ2H_þt2H_T2H_.

On the other hand,

H Z T 0 1½0;sÞðxÞ1½0;tÞðxÞ 1 x12Hþ 1 ðTxÞ12H dx ¼1 2½s 2H _ðTsÞ2H_þT2H_{: &}

Remark 2.2. It can be seen in Jolis [15] that formula (1) can be derived from the cross second order derivative in the distributional sense ofRHðs;tÞ.

We can also obtain easily the following expression forCðf;gÞ, withf;g2S: Cðf;gÞ ¼1 2Hð12HÞ Z Z R2 ðfðxÞ fðyÞÞðgðxÞ gðyÞÞ jxyj22H dxdy, (2) with fðxÞ ¼ fðxÞ if x2 ½0;T; 0 otherwise:

From (1) and (2) one can guess thatLH_T will be equal to the space

f 2L2ð½0;TÞ: Z Z R2 ðfðxÞ fðyÞÞ2 jxyj22H dxdyo1 . To prove this claim we need some notations and results.

Along the paper we will make use of sequences of partitions, denoted by P¼ ðpm_Þ

m.

These sequences will be (finite) partitions of the interval½0;Tor (infinite) partitions ofR. We will assume that all these sequences satisfy the following condition:

CP :¼ sup m2N sup i;j jDm_i j jDm_j j ! o1. (P)

Here, and from now on, given a partitionp, we denote byDia generic intervalðti;tiþ1of

the partition and by jDij its length. Condition (P) is related with our treatment of the

hyper-singularity of the integrals that will appear. Concretely, it is used to prove the following technical lemma.

Lemma 2.3. Let P¼ ðpm_Þ

m be a sequence of partitions ofRsatisfying condition (P).Then, there exists a constant C (depending only on CP and on H) such that for any i;j2N

with iaj and anyðu;vÞ 2Dm_i Dm_j we have that

1 jDm_i j jDm_j j Z Z Dm iDmj 1 jxyj22H dxdypCjuvj 2H2_.

Proof. By commodity of notation, along the proof, we will suppress the superscriptsm.

Suppose, for instance, thatioj. We consider two possible situations. First, assume that

iþ1oj, in this case

Z tjþ1

tj

Z tiþ1

ti

ðyxÞ2H2dxdypðtjtiþ1Þ2H2jDij jDjj.

Since tjþ1ti¼ jDjj þtjtiþ1þ jDijpð2CPþ1Þðtjtiþ1Þ, we can bound the last

expression by 1 2CPþ1 2H2 ðtjþ1tiÞ2H2jDij jDjjp 1 2CPþ1 2H2 ðvuÞ2H2jDij jDjj, for anyðu;vÞ 2DiDj.

Suppose now thatj¼iþ1. Therefore, we can proceed as follows: Z Z DiDj 1 jxyj22H dxdy ¼ Z tjþ1 tj Z tj ti ðyxÞ2H2dxdy p 1 12H Z tjþ1 tj ðytjÞ2H1dy¼ 1 2H 1 12H ðtjþ1tjÞ2H ¼ 1 2H 1 12H ðtjþ1tjÞ2H2ðtjþ1tjÞ2 p 1 2H 1 12H CPjDjj2H2jDij jDjj, ð3Þ

where condition (P) on the partitions is used in the last step. Using again condition (P), we have that tjþ1ti¼ jDjj þ jDijpðCPþ1ÞjDjj, or equivalently that jDjjX 1 CPþ1 ðtjþ1tiÞ.

By substituting this inequality in (3) we obtain

Z Z DiDj 1 jxyj22H dxdy p 1 2H 1 12H 1 CPþ1 2H2 CPðtjþ1tiÞ2H2jDij jDjj p 1 2H 1 12H 1 CPþ1 2H2 CPðvuÞ2H2jDij jDjj,

for anyðu;vÞ 2DiDj. This inequality finishes the proof. &

Remark 2.4. Actually, it is necessary to impose some condition on the partitions in order to obtain the conclusion of the above lemma. We can see this fact by considering, for a

fixedDi, the functions

FðtÞ ¼ Z t tiþ1 Z tiþ1 ti ðyxÞ2H2dxdy and

GðtÞ ¼ ðttiÞ2H2jDijðttiþ1Þ,

defined fortXt_iþ1.

Any constantCcannot exist such that

because, as it is easily checked, lim

t#tiþ1

FðtÞ GðtÞ¼ 1.

Given a locally integrable functionf :I !R(whereIis a finite interval orI ¼R), and a

partitionpof Iwe define fp¼X i 1 jDij Z Di fðxÞdx1Di.

We can now prove a result of approximation of the Wiener integral by Riemann sums

and the characterization ofLH_T that we will use in the sequel.

We introduce forf :½0;T!Rmeasurable, the following (possibly infinite) quantity:

kfk_H ¼ 1 2 Hð12HÞ Z Z R2 ðfðxÞ fðyÞÞ2 jxyj22H dxdy 1=2 .

Theorem 2.5. The domain of the simple integral I1 is given by

LH_T ¼ ff 2L2ð½0;TÞ:kfkHo1g. If we endow this space with the scalar product

Cðf;gÞ ¼1 2Hð12HÞ Z Z R2 ð¯fðxÞ ¯fðyÞÞð¯gðxÞ ¯gðyÞÞ jxyj22H dxdy then integral I1 is an isometry betweenLHT and a subspace of L2ðOÞ.

Moreover,for any f 2LH_T and any sequenceP¼ ðpm_Þ

m of partitions of ½0;T verifying condition(P)and with normjpm_{jtending to zero,we have that}

kfpmfk_H!0 as m! 1. (4) As a consequence, I1ðfÞ ¼L2ðOÞ lim m!1 X i 1 jDm_ij Z Dm i fðxÞdx ! BHðDm_i Þ. (5)

Proof. We must see that V:¼ff 2L2ð½0;TÞ:kfk_Ho1g with the scalar product C is a

Hilbert space and thatSis dense therein. The proof of completeness is standard and then

omitted. Thus, in order to finish the proof we will only check (4).

Recall that the fractional Sobolev space W1=2H;2ðRÞ (see, for instance, [1]) can be

defined as W1=2H;2ðRÞ ¼ f 2L2ðRÞ: Z Z R2 ðfðxÞ fðyÞÞ2 jxyj22H dxdyo1 . Then V¼ ff 2L2ð½0;TÞ:f 2W1=2H;2ðRÞg.

We introduce inW1=2H;2ðRÞthe following seminorm: kfk¼ Z Z R2 ðfðxÞ fðyÞÞ2 jxyj22H dxdy 1=2 .

Since for any f 2V we have ¯f2W1=2H;2ðRÞ, there exists ffngn2N, sequence of

C1-functions with support contained in½a;b ¼ ½1;Tþ1, such thatkfnfkconverges

to 0 whenn tends to1.

It can be seen that, for anyf 2C1with compact support, the sequencefpm converges in

the seminormk k tof.

Next, we will see that forf 2W1=2H;2_{ðRÞand a sequencePof partitions ofRsatisfying}

condition (P) there exists a constantC, only depending onCPand on H, such that

kfpmkpCkfk. (6)

We can extend the sequence of partitionsðpm_Þ

mto another sequence of partitions ofR

containing the points a and b (that we continue denoting by ðpm_Þ

m), satisfying also

condition (P) with the same constantCPand withjpmjtending to zero whenmtends to1.

We have that Z Z R2 ðfpmðxÞ fpmðyÞÞ2 jxyj22H dxdy ¼ Z Z R2 1 jxyj22H X iaj 1 jDij Z Di fðuÞdu 1 jDjj Z Dj fðuÞdu !2 1DiDjðx;yÞ 0 @ 1 Adxdy ¼ Z Z R2 1 jxyj22H X iaj 1 jDij jDjj Z Z DiDj ðfðuÞ fðvÞÞdudv !2 1DiDjðx;yÞ 0 @ 1 Adxdy pX iaj 1 jDij jDjj Z Z DiDj ðfðuÞ fðvÞÞ2dudv Z Z DiDj 1 jxyj22H dxdy ¼X iaj Z Z DiDj Ci;jðfðuÞ fðvÞÞ2dudv, ð7Þ with Ci;j¼ 1 jDij jDjj Z Z DiDj 1 jxyj22H dxdy.

By Lemma 2.3, there exists a constantC(depending onCPand onH) such that for any

ðu;vÞ 2DiDj we have that

Ci;j¼ 1 jDij jDjj Z Z DiDj 1 jxyj22H dxdypC 2_juvj2H2_,

and then, we can majorize (7) by C2 X iaj Z Z DiDj ðfðuÞ fðvÞÞ2 juvj22H dudvpC 2_kfk2 ,

and inequality (6) is proved.

Finally, givenf 2Vandd40, we takegof classC1_{with compact support contained in}

½1;Tþ1 such that kf gkpd. Given this g, we can take m02N such that for any

mXm₀ it holds thatkggpmkpd. Then, by using (6), we obtain formXm₀,

kf fpmk_H¼ kf fp m kpkf gkþ kggp m kþ kðgfÞp m k pð2þCÞd: &

3. The multiple Itoˆ-type integral

From Huang and Cambanis[11], the multiple Itoˆ-type integral of ordernwith respect to

BH, denoted byIn, is defined on the tensor product spaceðLHTÞn, that is the completion of

the set of functions that are linear combinations of functions of the form f₁ f_n

(withf_i2LH_T; i¼1. . .;n) with respect to the inner product given by

Cnðf;gÞ ¼Cðf₁;g₁Þ. . .Cðf_n;g_nÞ,

forf ¼f₁ f_n andg¼g₁ g_n.

Next, we state in the following theorem the main properties of the Itoˆ-type multiple integral (see [11]).

Theorem 3.1. The multiple Itoˆ-type integrals have the following properties: For f,

g2 ðLHTÞn,and a;b2R,

(a) Inðaf þbgÞ ¼aInðfÞ þbInðgÞ.

(b) InðfÞ ¼Inðf~Þ,where we denote byf the symmetrized of f~ . (c) EðInðfÞInðgÞÞ ¼n!Cnðf~;gÞ~.

(d) For f 2 ðLH_TÞn and g2 ðL_THÞm,EðInðfÞImðgÞÞ ¼0,if nam. (e) If h2LH_T with khkH¼1 then

InðhnÞ ¼n!HnðI1ðhÞÞ. (8)

In this last assertion we are denoting by Hnthe nth normalized Hermite polynomial. Recall that this polynomial is defined as

HnðxÞ ¼ð1Þ n n! e x2₌₂ dn dxn ðe x2₌₂ Þ if nX1, and H0ðxÞ ¼1.

As in the one-dimensional case define

fðy₁;. . .;y_nÞ ¼ fðy1;. . .;ynÞ if ðy1;. . .;ynÞ 2 ½0;T n_;

0 otherwise:

Observe that using the compact form ofCgiven by (2), forfandg tensor products of

elements ofLH_T, we have that

Cnðf;gÞ ¼ Z Z R2n Yn i¼1

ðf_iðxiÞ f_iðy_iÞÞðg_iðxiÞ g_iðy_iÞÞ jxiyij22H dx1 dxndy1 dyn ¼ Z Z R2n Dn_ðx1;...;xnÞfðy1;. . .;ynÞDnðx1;...;xnÞgðy1;. . .;ynÞ Qn i¼1jxiyij22H dx1 dxndy1 dyn.

Here,Dn_{ðx1;...;xnÞ}hðy₁;. . .;y_nÞdenotes then-dimensional increment of the functionhbetween the pointsðx1;. . .;xnÞandðy1;. . .;ynÞ, that is

Dn_ðx1...;xnÞhðy₁;. . .;y_nÞ ¼ X If1;...;ng ð1ÞjIjhðzI₁;. . .;zI_nÞ, where fori¼1;. . .;n, zI_i ¼ xi if i 2I; y_i if i2Ic; (

and we denote byjIjthe cardinal of the setI.

By linearity the expression found for Cnðf;gÞ is also valid with f and g being linear

combinations of tensor products. We introduce also the notation

kfk_H;n¼ Z Z R2n ðDn_ðx1;...;xnÞfðy1;. . .;ynÞÞ2 Qn i¼1jxiyij22H dx1 dxndy1 dyn !1=2 . Given a partitionpof½0;Tand a function f 2L1ð½0;TnÞ, define

fp¼ X i1;...;in 1 jDi1j jDinj Z Di₁Din fðx1;. . .;xnÞdx1 dxn1Di₁Din.

With these notations we can establish the following proposition.

Proposition 3.2. Given f :½0;Tn!R such that kfkH;no1 and a sequence ðpmÞm of partitions of½0;Tverifying condition(P)and with normjpm_{jtending to zero,we have that}

kfpmfkH;n!0 as m! 1. Moreover,ðLH_TÞn coincides with the space

ff :½0;Tn!R:kfk_H;no1g,

endowed with the inner product Cnðf;gÞ ¼ Z Z R2n Dn_{ðx1;...;xnÞ}fðy₁;. . .;y_nÞDn_{ðx1;...;xnÞ}gðy₁;. . .;y_nÞ Qn i¼1jxiyij22H dx1 dxndy1 dyn. Proof. To prove thatðLH_TÞn ¼ ff :½0;Tn!R:kfkH;no1g, we can see the completeness

completeness follows in an easy way, taking into account thatkfk2_H;n contains the summand Z Z ½0;Tn f2ðx1;. . .;xnÞ Yn i¼1 1 ðxiÞ12H þ 1 ðTxiÞ12H ! dx1 dxn.

To conclude, it suffices to show that for a functionfsuch thatkfk_H;no1and a sequenceðpm_Þ m

of partitions of½0;Tverifying condition (P), we have thatkfpmfk_H;n!0, asjpmj !0. But

this fact follows by similar arguments to those used in Theorem 2.5. &

As a consequence of the above result we obtain the following corollary.

Corollary 3.3. Given f 2 ðLH_TÞn, and a sequence ðpm_Þ

m of partitions of ½0;T verifying condition(P)and with normjpm_{jtending to0,we have that}

InðfÞ ¼L2ðOÞ lim

m!1Inðf

pm

Þ.

4. The multiple Stratonovich integral

We will introduce now the multiple Stratonovich-type integral in the Sole´–Utzet sense.

Definition 4.1. Given an integrable functionf :½0;Tn!Rwe will say that there exists the fractional Stratonovich integral offif there exists the limit in L2ðOÞof

X i1;...;in 1 jDm i1j jD m inj Z Dm i₁Dmin fðt1;. . .;tnÞdt1 dtn ! BHðDm_i1Þ BHðDm_inÞ,

where the Dm_ij are the intervals defined by a sequence of partitions ðpm_Þ

m of ½0;T that

satisfies condition (P) and thatjpm_{jtends to zero.}

When this limit exists we will denote it byIS_nðfÞ.

From (5), it is easily seen that for f₁;. . .;f_n2L_TH, there exists IS_nðf₁ f_nÞ and moreover

IS_nðf₁ f_nÞ ¼Y n

i¼1

I1ðfiÞ.

Observe also that, if it exists, IS_nðfÞ ¼IS_nðf~Þ, where f~ is the symmetrized of f. By this

reason, we assume from now on thatfis a symmetric function.

We introduce fork¼1;2;. . .;½n₂andf :Rn!Rthe following notation:

Dk_{ðt1;t3;...;t2k1;½t1;t3;...;t2k1;Þ} fðt1;t3;. . .;t2k1;½t2;t4;. . .;t2k;Þ ¼ X If1;2;:::;kg ð1ÞjIjfðt1;t3;. . .;t2k1;zI1;. . .;zIk;Þ, where fori¼1;. . .;k zI_i ¼ t2i if i2I; t2i1 if i2Ic: (

That is, we fix thekcoordinatest1;t3;. . .;t2k1and then2klast coordinates of a point

ðt1;. . .;tnÞin Rn and with the free coordinates we construct thek-dimensional increment

Theorem 4.2 (Hu–Meyer’s formula for simple functions). Consider f ¼hnwith h2LH_T.Then, IS_nðfÞ ¼ X ½n₂ k¼0 n! k!ðn2kÞ!2kIn2k ðHð12HÞÞ k Z R2k Dk_{ðt1;t3;...;t2k} 1;½t1;t3;...;t2k1;Þ ¯fðt1;t3;. . .;t2k1;½t2;t4;. . .;t2k;Þ Qk i¼1jt2it2i1j22H dt1dt2 dt2k ! ,

with the convention that if k¼0 the corresponding term equals to InðfÞ.

Proof. Assumeha0. Using identity (8) and that

xn¼X ½n₂

k¼0

n!

k!2kHn2kðxÞ,

whereHm are the Hermite polynomials, it holds that

IS_nðfÞ ¼ khkn_HIS_n f khkn_H ¼ khkn_H I1 h khk_H n ¼ khkn_HX ½n 2 k¼0 n! k!2kHn2k I1 h khk_H ¼ khkn_HX ½n₂ k¼0 n! k!ðn2kÞ!2k In2k h khkH ðn2kÞ! ¼ X ½n 2 k¼0 n! k!ðn2kÞ!2k In2kðkhk 2k Hhðn2kÞÞ. But, forka0, khk2_Hkhðn2kÞ¼ 1 2Hð12HÞ Z R2 ð¯hðt1Þ ¯hðt2ÞÞ2 jt1t2j22H dt1dt2 k hðn2kÞ ¼ 1 2k ðHð12HÞÞ k Z R2k Yk i¼1 ð¯hðt2i1Þ ¯hðt2iÞÞ2 jt2i1t2ij22H dt1 dt2k ! hðn2kÞ. On the other hand, we have that

Z R2k Yk i¼1 ð¯hðt2i1Þ ¯hðt2iÞÞ2 jt2i1t2ij22H dt1 dt2k ¼ Z R2k Yk i¼1 ¯hðt2i1Þ¯hðt2i1Þ þ¯hðt2iÞ¯hðt2iÞ 2¯hðt2i1Þ¯hðt2iÞ jt2i1t2ij22H dt1 dt2k ð9Þ ¼2k Z R2k Yk i¼1 ð¯hðt2i1Þ¯hðt2i1Þ ¯hðt2i1Þ¯hðt2iÞÞ jt2i1t2ij22H dt1 dt2k, ð10Þ

by using that the symmetrization of each factor of the numerator of expression (10) gives the different factors of the numerator of expression (9).

Finally, developing the product of the last expression and using the symmetry between each pair of variablesðt2i1;t2iÞwe obtain that

khk2_Hkhðn2kÞ ¼ ðHð12HÞÞk Z R2k Dk_{ðt1;t3;...;t2k1;½t1;t3;...;t2k1;Þ}¯fðt1;t3;. . .;t2k1;½t2;t4;. . .;t2k;Þ Qk i¼1jt2it2i1j22H dt1dt2 dt2k: &

Definition 4.3. Given a symmetric functionf 2 ðLH_TÞn we will say that it possesses trace of orderk,k2 f1; :::;½n₂gif there exists the limit, asm! 1, in ðLH_TÞðn2kÞ of

Tk_mfðÞ:¼ðHð12HÞÞk Z R2k Dk_{ðt1;t3;...;t2k} 1;½t1;t3;...;t2k1;Þf pm ðt1;t3;. . .;t2k1;½t2;t4;. . .;t2k;Þ Qk i¼1jt2it2i1j22H dt1 dt2k, whereðpm_Þ

m is a sequence of partitions of½0;Tthat satisfies condition (P) and the norm

jpm_{j tends to zero. As usual, we identifyðL}H

TÞ

0 _withR.

When this limit exists, we will denote it byTkfð:Þ.

Theorem 4.4 (General Hu–Meyer’s formula). Let f be a symmetric function inðLH_TÞn.If there exits its trace of order k for all k2 f1;. . .;½n₂gthen there exists the multiple fractional Stratonovich integral of f and

IS_nðfÞ ¼X ½n 2 k¼0 n! ðn2kÞ!k!2kIn2kðT k_fÞ,

with the convention that T0f ¼f.

Proof. The proof is a consequence of the fact that

X i1;...;in 1 jDm_i1j jDm_inj Z Dm i1D m in fðt1;. . .;tnÞdt1 dtn ! BHðDm_i1Þ BHðDm_inÞ ¼IS_nðfpmÞ ¼InðfpmÞ þX ½n 2 k¼1 n! ðn2kÞ!k!2k In2kðT k mfÞ,

using Theorem 4.2. Then, the result follows from Corollary 3.3, the definition of the traces

Tk and the continuity of the multiple Itoˆ-type integrals on theðLH_TÞðn2kÞ spaces. &

Finally, we can state the following result for the casen¼2.

Theorem 4.5. Let f 2 ðLH_TÞ2and assume that there exists40such that the restriction off~ to the setfðx;yÞ 2 ½0;T2 :jxyjogis of classCb,withb412H.Then,f is Stratonovich

integrable and we have IS₂ðfÞ ¼I2ðf~Þ þHð2H1Þ Z T 0 Z T 0 ~ fðx;yÞ f~ðx;xÞ jxyj22H dxdy þH Z T 0 fðx;xÞ 1 ðTxÞ12Hþ 1 x12H dx.

Proof. Along this proof we will denote byCall the multiplicative constants that appear in all the majorizations, although their actual value can vary from an expression to another one.

We can assume, without loss of generality, thatfis a symmetric function.

By Theorem 4.4 we know that if there exists the trace of order 1 of the functionf, then

there exists the multiple fractional Stratonovich integral offand

IS₂ðfÞ ¼I2ðfÞ þT1f.

So, we have to prove the existence of the limit, whenmtends to infinity, of T1_mf and

compute this limit. That is, we have to study the limit of

T1_mf ¼Hð12HÞ

Z Z R2

D1_ðt1;½t1Þfpmðt1;½t2Þdt1dt2,

whereðpm_Þ

m is a sequence of partitions of½0;Tthat satisfies condition (P) and the norm

jpm_{jtends to zero.}

The last expression is equal to

Hð12HÞ Z Z R2 fpmðt1;t1Þ fp m ðt1;t2Þ jt2t1j22H dt1dt2 ¼Hð2H1Þ Z Z ½0;T2 fpmðt1;t2Þ fp m ðt1;t1Þ jt2t1j22H dt1dt2 þH Z T 0 fpmðt1;t1Þ 1 ðTt1Þ12H þ 1 t12H 1 dt1.

By the continuity offon a neighborhood of the diagonal it is clear that

lim m!þ1 Z T 0 fpmðx;xÞ 1 ðTxÞ12Hþ 1 x12H dx ¼ Z T 0 fðx;xÞ 1 ðTxÞ12Hþ 1 x12H dx. On the other hand, one can also see that

lim m!þ1 Z Z ½0;T2 fpmðx;yÞ fpmðx;xÞ jxyj22H dxdy ¼ Z Z ½0;T2 fðx;yÞ fðx;xÞ jxyj22H dxdy.

In fact, we will show that

Z Z

½0;T2

jfpmðx;yÞ fpmðx;xÞ fðx;yÞ þfðx;xÞj

jxyj22H dxdy (11)

converges to zero whenmtends to1.

Let W :¼fðx;yÞ 2 ½0;T2:jxyjog be the set on which f is Ho¨lder continuous and consider also the setV:¼fðx;yÞ 2 ½0;T2:jxyjo

2g.

We will study first the integral appearing in (11) outside the setV:

Z Z VC jfpmðx;yÞ fpmðx;xÞ fðx;yÞ þfðx;xÞj jxyj22H dxdy p 2 22H kfpmfk_L1_ð½0;T2_Þþ Z T 0 Z T 0 jfpmðx;xÞ fðx;xÞjdxdy .

The last expression converges to zero because fpm converges to f in L1_ð½0;T2_{Þ and}

fpmðx;xÞconverges uniformly to fðx;xÞ.

On the other hand, if we denote byfm_ij the integral mean

1 jDm_i j jDm_j j Z Dm iDmj fðu;vÞdudv,

it remains only to prove the convergence to zero of

X i;j Z Z V jfm_ij fm_ii fðx;yÞ þfðx;xÞj jxyj22H 1DmiðxÞ1D m jðyÞdxdy. (12)

Ifjpm_{jis small enough, we have that for anyD}m

i Dmj such thatðDimDmj Þ \V¼ ;then Dm_i Dm_j W.

We will divide the summation (12) in three parts: wheni¼j, whenjijj ¼1 and when

jijjX2.

First of all, wheni¼j, we have that expression (12) equals to

X i Z Z V jfðx;xÞ fðx;yÞj jxyj22H 1DimðxÞ1DmiðyÞdxdy pC Z Z S iðD m iDmiÞ jxyjb2þ2Hdxdy,

using the Ho¨lder-continuity of the functionfon the setW.

The last expression converges to zero becauseb2þ2H41 and the measure of the

integration set goes to zero.

Let us now consider the summation given by (12) when jijj ¼1. In fact, we can

on the setW we have that ifDm_i Dm_iþ1W jfm_ij fm_ii fðx;yÞ þfðx;xÞj1DmiðxÞ1D m iþ1ðyÞ pCðjDm_i jbþ jDm_iþ1jbÞ1Dm iðxÞ1D m iþ1ðyÞ pCjDm_i jb1Dm i ðxÞ1D m iþ1ðyÞ,

by using condition (P) in the last step. On the other hand,

Z Z Dm iDmiþ1 1 jxyj22H dxdypCjD m i j2H. Then, X jijj¼1 Z Z V jfm_ij fm_ii fðx;yÞ þfðx;xÞj jxyj22H 1Dmi ðxÞ1Dmj ðyÞdxdypC X i jDm_i jbþ2H

that converges to zero becausebþ2H140.

Finally, we study the summation given by expression (12) when jijjX2.

Consider gpmðx;yÞ ¼1Vðx;yÞ X jijjX2 jfm_ij fm_ii fðx;yÞ þfðx;xÞj jxyj22H 1Dmi ðxÞ1D m j ðyÞ.

Fixedðx;yÞ 2Dm_i Dm_j withjijjX_{2 it is clear, by continuity, thatg}pm_{ðx;yÞconverges}

to zero whenm! þ 1. Then, using the dominated convergence theorem, it is enough to

bound gpm

by an integrable function. But, using the Ho¨lder continuity of f on W and

condition (P) it is readily shown that

jgpmðx;yÞjpCjxyjb2þ2H

and the last function belongs toL1ð½0;T2Þ. This finishes the proof. &

Remark 4.6. We point out that iff 2Cbð½0;T2Þ, withb412H, then all the hypotheses of Theorem 4.5 hold.

Acknowledgements

This research was partially supported by DGES Grants BFM2003-01345 and BFM2003-00261.

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