Integration with respect to fractal functions and …...Integration with respect to fractal functions and stochastic calculus. I M. Za¨hle Mathematical Institute, University of Jena,

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Integration with respect to fractal functionsand stochastic calculus. I

M. ZaÈ hle

Mathematical Institute, University of Jena, Ernst-Abbe-Platz 1-4, D-07740 Jena,Germany. e-mail: [email protected]

Received: 14 January 1998 /Revised version: 9 April 1998

Abstract. The classical Lebesgue±Stieltjes integralR b

a f dg of real orcomplex-valued functions on a ®nite interval �a; b� is extended to alarge class of integrands f and integrators g of unbounded variation.The key is to use composition formulas and integration-by-part rulesfor fractional integrals and Weyl derivatives. In the special case ofHoÈ lder continuous functions f and g of summed order greater than 1convergence of the corresponding Riemann±Stieltjes sums is proved.

The results are applied to stochastic integrals where g is replaced bythe Wiener process and f by adapted as well as anticipating randomfunctions. In the anticipating case we work within Slobodeckij spacesand introduce a stochastic integral for which the classical ItoÃ formularemains valid. Moreover, this approach enables us to derive calcula-tion rules for pathwise de®ned stochastic integrals with respect tofractional Brownian motion.

Mathematical Subject Classi®cation (1991): Primary 60H05; Second-ary 26A33, 26A42

0. Introduction

In order to motivate our paper we recall some well-known facts fromStieltjes integration.

Throughout the paper we consider Borel measurable real (orcomplex-valued) functions on R, most often vanishing outside a given®nite interval �a; b�.

Probab. Theory Relat. Fields 111, 333±374 (1998)

If such a function g has ®nite variation on �a; b� then it may berepresented by g � g1 ÿ g2 where g1 and g2 are monotone. Denote the®nite Borel measures associated with g1 and g2 by l1 and l2, rep-ectively. The Lebesgue±Stieltjes integral of a function f with respect tog is de®ned by

(L-S)

Z b

af �x� dg�x� :�

Z b

af �x� dl1�x� ÿ

Z b

af �x� dl2�x� �1�

provided that f is Lebesgue integrable with respect to the variationmeasure l :� l1 � l2 on �a; b�.

In the special case f being continuous this integral agrees with theRiemann±Stieltjes integral given by

(R-S)

Z b

af �x� dg�x� :� lim

D!0

Xi

f �x�i ��g�xi�1� ÿ g�xi�� 2�

where convergence holds uniformly in all ®nite partitions PD :�fa �: x0 � x�0 � x1 � � � � � xn � x�n � xn�1 � bg with maxijxi�1xij < D.The assumption on g ensures the absolute convergence of the aboveRiemann±Stieltjes sums.

In general, the Riemann±Stieltjes integral of f with respect to g isdetermined if the uniform convergence in (2) holds (but not neces-sarily the absolute convergence). As a corollary of the Banach±Steinhaus theorem the following was shown: If for some g theconvergence (2) holds for all continuous f then g must be of ®nitevariation (see, e.g. [10]).

In stochastic calculus based on martingale theory the absoluteconvergence of the Riemann±Stieltjes sums is replaced by convergencein mean square or, more generally, in probability. In this approach g isa random process being a semimartingale. Again, one cannot choosearbitrary (random) continuous functions f as integrands unless g has®nite variation. However, the class of square integrable adaptedrandom functions provides an appropriate solution. In particular, ifthe Wiener process W plays the role of g one turns to classical ItoÃcalculus. The so-called Skorohod integrals extend this construction tocertain anticipating integrands f .

In the present paper we extend the Stieltjes integrals to functions ofunbounded variation via fractional calculus. Recall that if f or g aresmooth on �a; b� the Lebesgue±Stieltjes integral may be rewritten as

(L-S)

Z b

af dg �

Z b

af �x�g0�x� dx �3�

and

334 M. ZaÈ hle

(L-S)

Z b

af dg � ÿ

Z b

af 0�x�g�x� dx� f �bÿ�g�bÿ� ÿ f �a��g�a�� 4�

respectively.(Throughout the paper we denote f �a�� :� limd&0f �a� d�;

g�bÿ� :� limd&0g�bÿ d� supposing that the limits exist.) The mainidea of our approach consists in replacing the ordinary derivatives byappropriate fractional derivatives in the sense of Riemann andLiouville and using their Weyl representation. We put

fa��x� :� 1�a;b��x��f �x� ÿ f �a�� 5�

gbÿ�x� :� 1�a;b��x��g�x� ÿ g�bÿ�� 6�and de®ne the integral byZ b

af dg � �ÿ1�a

Z b

aDa

a�fa��x�D1ÿabÿ gbÿ�x� dx

� f �a��g�bÿ� ÿ g�a��for certain 0 � a � 1 provided that f and g satisfy some fractionaldi�erentiability conditions in Lp-spaces, where �ÿ1�a � eipa. (In thecase of real-valued g the function �ÿ1�aD1ÿa

bÿ gbÿ�x� is real-valued.)The paper is organized as follows:In section 1 some background from fractional calculus is summa-

rized.The integral mentioned above is introduced in Section 2. We show

that for HoÈ lder continuous g and step functions f the integral agreeswith the corresponding Riemann±Stieltjes sums. Theorem 2.4 pro-vides general conditions when our integral coincides with the Le-besgue±Stieltjes integral. The additivity of the integral as function ofthe boundary is proved in Theorem 2.6.

Because of the choice of left and right sided fractional derivativesthe above integral seems to be directed forward. Therefore we con-struct in Section 3 a backward integral in a similar way. It turns outthat both the integrals coincide. As a corollary we obtain an inte-gration-by-part formula for these integrals.

The special case of HoÈ lder continuous functions f and g of sum-med order greater than 1 is studied in Section 4. As a basic result weprove the convergence of the Riemann±Stieltjes sums (2) to our in-tegral. This implies that the classical chain rule for the change ofvariables remains valid. We further prove that the integral as functionof the boundary is HoÈ lder continuous of the same order as g. Thisleads to an analogue to Lebesgue integration with respect to a mea-

Integration with respect to fractal functions and stochastic calculus. I 335

sure which is absolutely continuous with respect to a reference mea-sure via densities.

The second part of the paper, i.e. Section 5, deals with applicationsto stochastic calculus.

In Section 5.1 we demonstrate on the example of fractionalBrownian motion BH that our integral may be used in order to con-struct (stochastic) integrals for almost all realizations of stochasticprocesses without semimartingale properties. As long as we assumeHoÈ lder continuity (or fractional di�erentiability) of the randomintegrands f of order greater than one minus that of the integrator wedo not need any adaptedness. (This makes it possible to investigatestochastic di�erential equations with respect to fractional Brownianmotion of order greater than 1/2.)

In Section 5.2 we replace g by the Wiener process W and show thatfor adapted random L2-functions f of ``fractional degree of di�eren-tiability'' greater than 1/2 our integral agrees with the classical ItoÃintegral. For the more general class of functions having fractionalderivatives in some L2-sense of all orders less than 1/2 we proveconvergence in probability of the integrals

I1ÿ�f � �ÿ1�1=2ÿ�=2Z b

aD1=2ÿ�=2

a� f �x�D1=2ÿ�=2bÿ Wbÿ�x� dx

to the ItoÃ integral If as �& 0. Su�cient conditions for mean squareconvergence are also provided.

Finally, in section 5.3 we extend these results to anticipating ran-dom functions f . We ®rst de®ne the anticipating integral

�A�Z 1

0

f dW �X1n�0

eIn�1f n � nZ 1

0

eInÿ1f n��; t; tÿ� dt� �

in terms of the ItoÃ -Wiener chaos expansion f �P10 eInf n. For adaptedf this integral coincides with the ItoÃ integral. In the anticipating casewe introduce the Slobodecki-type spaces Wa

2;��0; 1� of random func-tions f and show that for a > 1=2 (where the integral agrees with theextended Stratonovitch integral

R 10 f � dW ) it is equal to the fractional

integralR 10 f dW considered before. If f 2Wa

2;� for any a < 1=2 andthe above integrals I1ÿ�f converge in the mean square then the limitagrees with �A� R 10 f dW . A su�cient condition for this convergence isthat f lies additionally in the space L1;2

C which is introduced in thetheory of Skorohod integrals d�f �. For such f we obtain

�A�Z 1

0

f dW � d�f � �Z 1

0

Dtf �tÿ� dt

336 M. ZaÈ hle

with Malliavin derivative Dt. In general, this integral di�ers from theStratonovitch integral, but we also get

�A�Z 1

0

cf dW � c �A�Z 1

0

f dW

for random constants c.

1. Fractional integrals and derivatives

Let Ln be Lebesgue measure in Rn. Integration with respect toL�dx� will be denoted by dx. For p � 1 let Lp � Lp�a; b� be thespace of complex-valued functions on R such that jjf jjLp

�ÿ R ba jf �x�jp dx

�1=p<1. (Similarly for p � 1.) Sometimes the values

of f in a neighborhood of �a; b� are of interest. Functions which agreeat Lebesgue-almost all points are usually identi®ed.

An exhaustive survey on classical fractional calculus may be foundin Samko, Kilbas and Marichev [11]. We recall some important no-tions and results presented there.

For f 2 L1 and a > 0 the left- and right-sided fractional Riemann±Liouville integrals of f of order a on �a; b� is given at almost all x by

Iaa�f �x� :� 1

C�a�Z x

a�xÿ y�aÿ1f �y� dy �7�

Iabÿf �x� :� �ÿ1�

ÿa

C�a�Z b

x�y ÿ x�aÿ1f �y� dy ; �8�

respectively, where C denotes the Euler function.They extend the usual n-th order iterated integrals of f for

a � n 2 N. We have the ®rst composition formula

Iaa��bÿ��Ib

a��bÿ�

f � � Ia�ba��bÿ�

f : �9�

If f 2 Lp, g 2 Lq, p � 1, q � 1, 1=p � 1=q � 1� a, where p > 1 andq > 1 for 1=p � 1=q � 1� a, then the ®rst integration-by-parts ruleholds: Z b

af �x�Ia

a�g�x� dx � �ÿ1�aZ b

ag�x�Ia

bÿf �x� dx : �10�

Fractional di�erentiation may be introduced as an inverse operation.For our purposes it is su�cient to work with a class of functionswhere this inversion is well-determined and the Riemann±Liouvillederivatives agree with the (more general) version in the sense of Weyland Marchaud:


For p � 1 let Iaa��bÿ��Lp� be the class of functions f which may be

represented as an Iaa��bÿ�-integral of some Lp-function u. If p > 1 this

property is equivalent to f 2 Lp and Lp-convergence of the integralsZ xÿ�

a

f �x� ÿ f �y��xÿ y�a�1 dy

Z b

x��

f �x� ÿ f �y��y ÿ x�a�1 dy

!

as function in x 2 �a; b� as �& 0 putting f �y� � 0 if x j2 �a; b� (cf. [11],x13). Moreover aÿ 1=p, for ap < 1 one knows that Ia

a��Lp� �Iabÿ�Lp� � Lq with 1=q � 1=p ÿ a. If ap > 1 any f 2 Ia

a��bÿ��Lp� is HoÈ lder

continuous of order aÿ 1=p on the interval �a; b�.It can be shown that the function u in the above representation

f � Iaa��bÿ�

u is unique in Lp on �a; b� and for 0 < a < 1 it agrees a.e. with

the left-(right-)sided Riemann±Liouville derivative of f of order a

Daa�f �x� :� 1�a;b��x� 1

C�1ÿ a�ddx

Z x

a

f �y��xÿ y�a dy �11��

Dabÿf �x� :� 1�a;b��x� �ÿ1�

1�a

C�1ÿ a�ddx

Z b

x

f �y��y ÿ x�a dy

�: �12�

The corresponding Weyl representation reads

Daa�f �x� � 1

C�1ÿ a�f �x��xÿ a�a � a

Z x

a

f �x� ÿ f �y��xÿ y�a�1 dy

!1�a;b��x� �13�

�Da

bÿf �x� � �ÿ1�aC�1ÿ a�

f �x��bÿ x�a � a

Z b

x

f �x� ÿ f �y��y ÿ x�a�1 dy

!1�a;b��x�

��14�

where the convergence of the integrals at the singularity y � x holdspointwise for almost all x if p � 1 and in the Lp-sense if p > 1. (Themore familiar in®nite versions of the Weyl derivatives are given by

Da�f �x� :� a

C�1ÿ a�Z 10

f �x� ÿ f �xÿ y�ya�1 dy �130�

Daÿf �x� :� a�ÿ1�a

C�1ÿ a�Z 10

f �x� ÿ f �x� y�ya�1 dy : �140�

Since f vanishes outside �a; b� we obtainDa

a��bÿ�

f �x� � 1�a;b��x�Da��ÿ�

f �x� :Note that in the literature the factor �ÿ1�a is usually omitted, thoughit was originally used by Liouville. It appears appropriate for our

338 M. ZaÈ hle

integral construction and plays also a role in fractional Fouriertransformations (c.f. [14]).

Recall that by construction for f 2 Iaa��bÿ��Lp�,

Iaa��bÿ��Da

a��bÿ�

f � � f : �15�We also have

Daa��bÿ��Ia

a��bÿ�

f � � f �16�which is valid for general f 2 L1.

Straightforward calculation shows that for f continuously di�er-entiable in a neighborhood of x 2 �a; b�,

lima!1

Daa��bÿ�

f �x� � f 0�x�: �17�Here the relationship

lim�!0

I�a��bÿ�

h�x� � h�x ±±�� 18�

is used which holds for arbitrary h 2 L1 at all points x 2 �a; b� wherethe left (right) limit, h�xÿ��h�x�� exists, i.e., at Lebesgue-almost all x.If h 2 Lp, p � 1, we can take in (18) the Lp-limit, too. In particular, in(17) Lp-convergence holds for all f 2 Lp which are di�erentiable in theLp-sense.

Furthermore, (11) and (12) imply

lima!0

Daa��bÿ�

f �x� � f �x� �19�which is also true in the Lp-sense if p > 1. For completeness denote

D0a��bÿ�

f �x� � f �x�; I0a��bÿ��Lp� � Lp; and D1

a��bÿ�

f �x� � f 0�x�if the latter derivative exists.The following two formulas play an essential role for our integrationconcept:

Daa��bÿ��Db

a��bÿ�

f � � Da�ba��bÿ�

f �20�

if f 2 Ia�ba��bÿ��L1�; a � 0; b � 0; a� b � 1 (second composition formula),

�ÿ1�aZ b

aDa

a�f �x� g�x� dx �Z b

af �x�Da

bÿg�x� dx �21�

provided that 0 � a � 1, f 2 Iaa��Lp�, g 2 Ia

bÿ�Lq�, p � 1, q � 1,1=p � 1=q � 1� a (second integration-by-parts rule).

2. An extension of Stieltjes integrals

The calculation rules (3) and (4) for Lebesgue±Stieltjes integrals withrespect to smooth functions, the composition formula (20) and the


integration-by-part rule (21) suggest the following notion. (In order toavoid the restrictive condition f �a�� 0 or g�bÿ� � 0 at some placeswe introduce the auxiliary functions fa� and gbÿ as in (5) and (6)assuming that the right- and left-sided limits always exist when theyappear in the formulae.)

De®nition. The (fractional) integral of f with respect to g is de®ned byZ b

af �x� dg�x� � �ÿ1�a

Z b

aDa


� f �a��g�bÿ� ÿ g�a�� 22�provided that fa� 2 Ia

a��Lp�, gbÿ 2 I1ÿabÿ �Lq� for some 1=p � 1=q � 1;

0 � a � 1.

2.1. Proposition The de®nition is correct, i.e. independent of the choiceof a.

Proof. If the conditions are ful®lled for �a; p; q� and �a0; p0; q0� witha0 � a� b > a then we get

�ÿ1�a0Z b

aDa0

a�fa��x�D1ÿa0bÿ gbÿ�x� dx

�(20) �ÿ1�a�ÿ1�bZ b

aDb

a��Daa�fa��x�D1ÿ�a�b�

bÿ gbÿ�x� dx

�(21) �ÿ1�aZ b

aDa

a�fa��x�Dbbÿ�D1ÿ�a�b�

bÿ gbÿ��x� dx

�(20) �ÿ1�aZ b

aDa

a�fa��x�D1ÿabÿ gbÿ�x� dx :

In order to check the conditions of (21) use (16) and (9). (

Remark. For ap < 1 we have fa� 2 Iaa��Lp� i� f 2 Ia

a��Lp� and f �a��exists. In this case the derivatives satisfy the relation

Daa�fa��x� � Da

a��f ÿ f �a��1�a;b��x�

� Daa�f �x� ÿ 1

C�1ÿ a�f �a��xÿ a�a 1�a;b��x�

(cf. Proposition 2.2 below) and (22) may be rewritten asZ b

af �x� dg�x� � �ÿ1�a

Z b

aDa

a�f �x�D1ÿabÿ gbÿ�x� dx : �220�

340 M. ZaÈ hle

which is determined for general f 2 Iaa��Lp� bounded in a�. For a � 0

and a � 1 the integral (22) may be transformed intoZ b

af �x� dg�x� �

Z b

af �x�g0�x� dx �23�

andZ b

af �x� dg�x� � ÿ

Z b

af 0�x�g�x� dx� f �bÿ�g�bÿ� ÿ f �a��g�a��

�24�which agrees with the corresponding Lebesgue±Stieltjes integrals (3)and (4), respectively.

Our next aim is to show that for functions g as above with ®nitevariation the integral (22) agrees with the Lebesgue±Stieltjes integralof the functions f under consideration. First we let f be the indicatorfunction of a subinterval �c; d� � �a; b�.2.2. Proposition. If g is HoÈlder continuous on �a; b� of some order thenwe have

(i)R b

a 1�c;d��x� dg�x� � g�d� ÿ g�c�(ii)R b

a 1�c;b��x� dg�x� � g�bÿ� ÿ g�c� .Note that on the right hand side g�c� has to be replaced by g�a�� ifc � a.

Proof. The fractional derivatives of the function 1�c;d��x� may be cal-culated by means of (130):

Daa�1�c;d��x� � 1�a;b��x� a

C�1ÿ a�Z 10

1�c;d��x� ÿ 1�c;d��xÿ y�ya�1 dy

� aC�1ÿ a�

"1�c;d��x�

Z 10

1ÿ 1�c;d��xÿ y�ya�1 dy

ÿ 1�a;b�n�c;d��x�Z 10

1�c;d��xÿ y�ya�1 dy

�� 1

C�1ÿ a��1�c;d��x�a

Z 1xÿc

1

ya�1 dy

ÿ 1�d;b��x�Z xÿc

xÿd

1

ya�1 dy�:

Thus,


Daa� 1�c;d��x� �

1

C�1ÿ a� 1�c;b��x� 1

�xÿ c�a ÿ 1�d;b��x� 1

�xÿ d�a� �

:

�25�Similarly,

Daa� 1�c;b��x� �

1

C�1ÿ a� 1�c;b��x�1

�xÿ c�a : �26�

Taking the Iaa�-integral of the right-hand side we can see that

1�c;d� 2 Iaa��Lp� i� ap < 1. Further, if k is the HoÈ lder exponent of the

function g then gbÿ lies in I�bÿ�Lq� for any q and � < k. Hence, theconditions of (22) are ful®lled for arbitrary a � 1ÿ �, � < k, i.e.,Z b

a1�c;d��x� dg�x� � �ÿ1�1ÿ�

Z b

aD1ÿ�

a� 1�c;d��x� D�bÿgbÿ�x� dx

� �ÿ1�1ÿ� 1

C��Z bÿc

0

x�ÿ1D�bÿgbÿ�c� x� dx

ÿZ bÿd

0

x�ÿ1D�bÿgbÿ�d � x� dx

�� ÿ1�1ÿ��ÿ1��

�I�bÿ�D�

bÿgbÿ��c� ÿ I�bÿ�D�bÿgbÿ��d�

�� ÿ g�c� ÿ g�d��

according to (15).The arguments for (ii) are similar. (

By linearity this result extends to step functions: Let P �fa � x0 < x1 < � � � < xn < bg be any partition of �a; b� and fP :�Pnÿ1

i�0 fi1�xi;xi�1� � fn1�xn;b� for some complex values fi.

2.3. Corollary. If g is HoÈlder continuous on �a; b� we haveZ b

afP�x� dg�x� �

Xnÿ1i�0

fi �g�xi�1� ÿ g�xi�� fn �g�bÿ� ÿ g�xn�� :

We now turn to comparison with the Lebesgue-Stieltjes integral underthe condition that g has bounded variation. In the complex case theLebesgue±Stieltjes integral considered in the introduction may be un-derstood in the real vector-valued sense via coordinate representation.

2.4. Theorem. Suppose that g has bounded variation with variationmeasure l and f and g satisfy the conditions of (22).

342 M. ZaÈ hle

(i) IfR b

a Iaa��jDa

a�fa�j��x�l�dx� <1 then we haveR ba f �x� dg�x� � (L-S)

R ba f �x� dg�x�.

(ii) The integrals also agree if f is bounded and right-( or left-)con-tinuous at l±a.a. points.

Remark. As a special case of (ii) we obtain for any continuous f asabove Z b

af �x� dg�x� � (R-S)

Z b

af �x� dg�x� :

Proof of the theorem. The condition of (i) and Fubini justify thechanges of the order of integration in the following Lebesgue±Stieltjesintegrals. By (15) we get

(L-S)

Z b

af dg � (L-S)

Z b

aIaa��Da

a�fa��x� dg�x�� f �a��g�bÿ� ÿ g�a�� :

The integral on the right-hand side equals

1

C�a�Z b

a

Z x

a�xÿ y�aÿ1Da

a�fa��y� dy dg�x�

� 1

C�a�Z b

aDa

a�fa��y�Z b

y�xÿ y�aÿ1 dg�x� dy

�Z b

aDa

a�fa��y� 1ÿ aC�a�

Z b

y

Z 1x�zÿ y�aÿ2 dz dg�x�

� �dy :

The expression in the brackets may be rewritten by

1ÿ aC�a�

Z 1y

Z min�z;b�

ydg�x� �zÿ y�aÿ1 dz

� ÿ 1ÿ aC�a�

Z 1y

g�y� ÿ g�min�z; b��zÿ y�1ÿa�1 dz

� ÿ 1ÿ aC�a�

Z b

y

g�y� ÿ g�z��zÿ y�1ÿa�1 dzÿ 1

C�a�g�y� ÿ g�bÿ��bÿ y�1ÿa

� �ÿ1�aD1ÿabÿ gbÿ�y�

according to (14). Substituting this under the above integral we inferthat the primary integral equals �ÿ1�a R b

a Daa�fa��y�D1ÿa

bÿ gbÿ�y�dy � f �a��g�bÿ� ÿ g�a��, i.e. (i) is proved.

In order to show (ii) note that both integrals agree withÿ R b

a f 0�x�g�x� dx� f �bÿ�g�bÿ� ÿ f �a��g�a�� if f is smooth. We will


approximate general f by smooth functions so that both types ofintegrals converge to those of f . Let �a; p� satisfy the conditions of(22). Then Da

a�fa� is an Lp-function.Let k (or kÿ) be a nonnegative smooth function vanishing outside

�0; 1� (or �ÿ1; 0�) such thatR 10 k�x� dx � 1 (or

R 0ÿ1 kÿ�x� dx � 1�. By

k�ÿ�N �x� :� N k�ÿ��Nx� �27�

we get a standard familiy of smoothing kernels converging to the d-function as N !1.

For the convolution fN :� fa� � kN we obtain

Daa�fN � 1�a;b��Da

a�fa�� kN : �28�The right-hand side converges in Lp to Da

a�fa� as N !1. Further,fN �a�� 0. Hence, by the HoÈ lder inequality we obtain

�ÿ1�aZ b

aDa


� limN!1�ÿ1�a

Z b

aDa

a�fN �x�D1ÿabÿ gbÿ�x� dx

� limN!1�ÿZ b

af 0N �x�gbÿ�x� dx�

� limN!1�L-S�

Z b

afN �x� dgbÿ�x�

� limN!1�L-S�

Z b

afN �x� dg�x� :

The right-sided continuity of f at x yields

limN!1

fa� � kN �x� � fa��x� :

Therefore Lebesgue's bounded convergence theorem implies

limN!1�L-S�

Z b

afa� � kN �x� dg�x�

� �L-S�Z b

afa��x� dg�x�

� �L-S�Z b

af �x� dg�x� ÿ f �a��g�bÿ� ÿ g�a��

which leads to the assertion.The case of left-sided continuity is similar. Here it is appropriate to

use the kernel kÿ. (

344 M. ZaÈ hle

Remark. It turns out that in (ii) the Lebesgue±Stieltjes integral doesnot depend on the choice of right- or left-sided limits of f . This comesfrom the conditions of (22). In case of discontinuous f they force acertain HoÈ lder continuity of g.

At the end of this section we will show that our integral (22) is anadditive function of the boundary. Let a � x < y < z � b.

2.5. Theorem.

(i)R y

x f dg � R ba 1�x;y�f dg

if for both the integrals the conditions of de®nition (22) are ful-®lled.

(ii)R y

x f dg� R zy fdg � R z

x fdgÿ f �y��g�y�� ÿ g�yÿ��if all summands are determined as in (22).

Proof. Let kÿN be the family of smoothing kernels introduced in (27).We ®rst will approximate the function gbÿ by the smooth functionsgN :� gbÿ � kÿN so that

D1ÿabÿ gN � 1�a;b��D1ÿa

bÿ gbÿ� � kÿN �29�and gN�bÿ� � 0. Then we obtain by Lq convergence for x > a (the casex � a is similar)Z b

a1�x;y�f dg � �ÿ1�a

Z b

aDa

a�1�x;y�f �u�D1ÿabÿ gbÿ�u� du

� limN!1�ÿ1�a

Z b

aDa

a�1�x;y�f �u�D1ÿabÿ gN�u� du

� limN!1

Z b

a1�x;y��u�f �u�g0N �u� du

� limN!1

Z y

xf �u� gbÿ � �kÿN �0�u� du

� limN!1

Z y

xf �u� gyÿ � �kÿN �0�u� du :

The last equality follows from the asymptotic equivalence of thefunctions gbÿ � �kÿN �0 and gyÿ � �kÿN �0 on the interval �x; y�. Further, theconditions of (22) are also ful®lled for the interval �x; y� for some�a0; p0; q0�. Therefore we may continue the above equations bylim

N!1

Z y

xDa0

x�fx��u� D1ÿa0yÿ gyÿ � kÿN �u� du� f �x��g�yÿ� ÿ g�x��

�Z y

xDa0

x�fx��u�D1ÿa0yÿ gyÿ�u� du� f �x��g�yÿ� ÿ g�x��

Z y

xf dg :


Thus (i) is proved.For (ii) we use similar arguments in order to getZ y

xf dg�

Z z

yf dgÿ f �x��g�yÿ� ÿ g�x�� ÿ f �y��g�zÿ� ÿ g�y��

� limN!1

Z y

xfx��u� g � �kÿN �0�u� du�

Z z

yfy��u� g � �kÿN �0�u� du

� �� lim

N!1

Z z

xfx��u� g � �kÿN �0�u� duÿ

�f �x�� ÿ f �y��

Z z

yg � kÿN �u� du

��Z z

xf dgÿf �x��g�zÿ�ÿg�x��ÿ�f �x��ÿf �y��g�zÿ�ÿg�yÿ�� :

(

Remark. For ap < 1, f 2 Iaa��Lp� being bounded in x� and y�,

gbÿ 2 I1ÿabÿ (where g is continuous), g�a�� existing, 1

p � 1q � 1 we get

similarly Z y

xf dg�

Z z

yf dg �

Z z

xf dg

by means of �220�.

3. Backward integrals and integration by parts

The construction (22), i.e.,Z b

af dg � �ÿ1�a

Z b

aDa

a�fa��x�D1ÿabÿ gbÿ�x� dx� f �a��g�bÿ� ÿ g�a��

is directed because of the choice of left-sided derivatives of f andright-sided derivatives of g. We will also call this expression the for-ward integral of f with respect to g. Similarly, we may introduce thebackward integralZ b

adg�x� f �x� :� �ÿ1�ÿa0

Z b

aDa0

bÿfbÿ�x�D1ÿa0a� ga��x� dx

� f �bÿ��g�bÿ� ÿ g�a�� 30�if fbÿ 2 Ia0

bÿ�Lp0 �, ga� 2 I1ÿa0a� �Lq0 � for some 1=p0 � 1=q0 � 1, 0 � a0 � 1.

Then the backward versions of �220�±(26) may be proved by com-pletely analogous arguments. In particular, for indicator functions for smooth functions f or g the forward and backward integrals agree.Generally, the following holds.

346 M. ZaÈ hle

3.1. Theorem. If f and g satisfy the conditions of (22) and (30) then wehave

(i)R b

a f dg � R ba dg f .

(ii)R b

a f dg � f �bÿ�g�bÿ� ÿ f �a��g�a�� ÿ R ba g df

(integration-by-part formula).

Proof. Using the approximations (28) and (29) for the left- and right-sided derivatives in the forward, as well as the backwardintegrals we infer from convergences in Lp; Lq and Lp0 ; Lq0 , respec-tively,

R ba f dg � limN!1

R ba f � kN �x� g � �kÿN �0�x� dx � R b

a dg f , i.e.,(i). (ii) is a consequence, since by de®nition,Z b

af dg � ÿ

Z b

adf g� f �a��g�bÿ� ÿ g�a��

� g�bÿ��f �bÿ� ÿ f �a��

� ÿZ b

ag df � f �bÿ�g�bÿ� � f �bÿ�g�bÿ� ÿ f �a��g�a�� :

Remark. Let Hk � Hk�a; b� be the space of functions being HoÈ ldercontinuous of order k on the interval �a; b�. Then the conditions ofTheorem 3.1 are ful®lled if f 2 Hk, g 2 Hl, k� l > 1. (In this case wemay choose p � q � 1 for a < k, 1ÿ a < l.) In the next section wewill study this situation in more detail.

4. The case of HoÈ lder continuous functions

4.1 Approximation by step functions

For arbitrary partitions PD as before any HoÈ lder continuous functionf on �a; b� may be approximated by the special step functions

efPD :�Xn

i�0f �xi�1�xi;xi�1�

in the following sense.

4.1.1. Theorem. If f 2 Hk for some 0 < k � 1 then we have

(i) limD!0 supPDkefPD ÿ f kL1�a;b� � 0

(ii) limD!0 supPDkDa

a��bÿ��fPD� a�

�bÿ�ÿ Da

a��bÿ�

f a��bÿ�kL1�a;b� � 0

for any a < k.


Proof. (i) is obvious.

For (ii) we will prove only the left-sided version. (The right-sided caseis analogous.) Let H�k� be the HoÈ lder constant of f . By de®nition,

C�1ÿ a�jDaa��efPD�a��x� ÿ Da

a�fa��x�j

�efPD�x� ÿ f �x��xÿ a�a � a

Z x

a

efPD�x� ÿ f �x� ÿ �efPD�y� ÿ f �y��xÿ y�a�1 dy

�� :

The L1-norm of the ®rst summand of the last sum may be estimated byH�k��bÿ a�1ÿaDk.

For x 2 �xi; xi�1� the second summand, say SPD�x�, may be splittedinto

aXiÿ1k�0

Z xk�1

xk


� aZ x

xi


� aXiÿ1k�0

Z xk�1

xk

f �xi� ÿ f �x� ÿ �f �xk� ÿ f �y��xÿ y�a�1 dy

� aZ x

xi

f �xi� ÿ f �x� ÿ �f �xi� ÿ f �y��xÿ y�a�1 dy :

Therefore the HoÈ lder continuity of f leads here to the estimation

jSPD�x�j � H�k�Xn

i�01�xi;xi�1��x�

��xÿ xi�ka

Z xi

a

1

�xÿ y�a�1 dy

� aXiÿ1k�0�xk�1 ÿ xk�k

Z xk�1

xk

1

�xÿ y�a�1 dy

� aZ x

xi

�xÿ y�k�xÿ y�a�1 dy

�

� H�k�Xn

i�01�xi;xi�1��x�

��xÿ xi�kÿa � a

Xiÿ1k�0�xk�1 ÿ xk�k

�Z xk�1

xk

1

�xÿ y�a�1 dy � akÿ a

�xÿ xi�kÿa�:

Hence,

348 M. ZaÈ hle

kSPDkL1 � H�k��

kkÿ a

Xn

i�0

Z xi�1

xi

�xÿ xi�kÿa dx

� aXn

i�0

Xiÿ1k�0�xk�1 ÿ xk�k

Z xi�1

xi

Z xk�1

xk

1

�xÿ y�a�1 dy dx�

� H�k��

kkÿ a

1

kÿ a� 1

Xn

i�0�xi�1 ÿ xi�kÿa�1

�Xnÿ1k�0�xk�1 ÿ xk�k

Z xk�1

xk

Z b

xk�1

1

�xÿ y�a�1 dx dy�

� H�k� kkÿ a

1

kÿ a� 1� 1

1ÿ a

� �Xn

i�0�xi�1 ÿ xi�kÿa�1

� H�k� kkÿ a

1

kÿ a� 1� 1

1ÿ a

� ��bÿ a�Dkÿa

which completes the proof of (ii). (

4.2 Interpretation as Riemann±Stieltjes integral

4.2.1. Theorem. If f 2 Hk, g 2 Hl for some k� l > 1 the Riemann±Stieltjes integral (R-S)

R ba f dg in the sense of (2) exists and agrees with

the forward and backward integralsR b

a f dg andR b

a dg f in the sense of(22) and (30).

Proof. Let fPD andefPD be the step functions used in (2) and Theorem

4.1.1, respectively. We estimate the di�erence of their Riemann±Stieltjes sums by

supPD

Xn

i�0f �x�i ��g�xi�1� ÿ g�xi�� ÿ

Xn

i�0f �xi��g�xi�1� ÿ g�xi��

��

� supPD

Xn

i�0jf �x�i � ÿ f �xi�jjg�xi�1� ÿ g�xi�j

� H�k�H�l� supPD

Xn

i�0�xi�1 ÿ xi�k�l

� H�k�H�l��bÿ a�Dk�lÿ1

where H�k� and H�l� are the HoÈ lder constants of f and g, respec-tively. Therefore it is enough to prove the convergence of the Riem-ann±Stieltjes sums

Pni�0 f �xi��g�xi�1� ÿ g�xi�� to

R ba f dg which agrees


withR b

a dg f by Theorem 3.1 (i). According to corollary 2.3 these sumsmay be interpreted as the forward integralsZ b

a

efPD dg � �ÿ1�aZ b

aDa

a��efPD�a��x�D1ÿabÿ gbÿ�x� dx

� f �a��g�bÿ� ÿ g�a��for any 1ÿ l < a < k. By Theorem 4.1.1 (i) the right-hand side tendsto

�ÿ1�aZ b

aDa

a�fa��x�D1ÿabÿ gbÿ�x� dx� f �a��g�bÿ� ÿ g�a��

Z b

af dg

as D! 0 uniformly in the partitions PD since D1ÿabÿ gbÿ is bound-

ed. (

4.3 A change-of-variable formula

It is well-known that the chain rule

dF �f �x�� F 0�f �x�� df �x�of classical real di�erentiation theory does not hold for functions f ofHoÈ lder exponent 1/2 arising as sample paths of stochastic processeswhich are semimartingales (cf. section 5). However, it follows fromTheorem 4.2.1 that for functions of HoÈ lder exponent greater than 1/2the classical formula remains valid in the sense of Riemann±Stieltjesintegration:

4.3.1. Theorem. Let f 2 Hk�a; b� and F 2 C1�R� be real-valued func-tions such that F 0 � f 2 Hl�a; b� for some k� l > 1. Then we have forany y 2 �a; b�

F �f �y�� ÿ F �f �a�� Z y

aF 0�f �x�� df �x� :

Proof. For arbitrary partitions PD as above the mean value theoremfor F and the continuity of f imply

F �f �y�� ÿ F �f �a�� Xn

i�0F �f �xi�1�� ÿ F �f �xi��

�Xn

i�0F 0�f �exi��f �xi�1� ÿ f �xi��

350 M. ZaÈ hle

for some exi 2 �xi; xi�1�. The last expression tends toR y

a F 0�f �x� df �x� asD! 0 by Theorem 4.2.1. (

Remark. The conditions of this theorem are satis®ed if f 2 Hk�a; b�for some k > 1=2 and F is a C1-function with Lipschitz derivative.

A more general variant of Theorem 4.3.1 for F 2 C1�R� �a; b��and F 01�f ��; �� 2 Hl�a; b�; k� l > 1, reads

F �f �y�; y� ÿ F �f �a�; a� �Z y

aF 01�f �x�; x� df �x� �

Z y

aF 02�f �x�; x� dx

�31�where F 01 and F 02 are the partial derivatives of F with respect to the ®rstand second variable, respectively. The proof is left to the reader.

Example. If f 2 Hk�a; b� for some k > 1=2 we may choose in 4.3.1F �u� � u2 and obtainZ y

af �x� df �x� � 1

2�f �y�2 ÿ f �a��2� : �32�

4.4 The integral as function of the boundary

An immediate consequence of the interpretation as Riemann±Stieltjesintegral for f 2 H k, g 2 Hl, k� l > 1, is the additive dependence onthe boundary which has already been proved in Theorem 2.5 bymeans of smoothing.

SinceR y

x f dg � �ÿ1�a R yx Da

x�fx�u�D1ÿayÿ gy�u� du� f �x��g�y� ÿ g�x��

if 1ÿ l < a < k, a < x < y < b, and the derivatives in the last integralare bounded we may estimate j R y

x f dgj � const�y ÿ x� � const�y ÿ x�land obtain that the integral as function of the upper or lower boun-dary is HoÈ lder continuous of order l:

4.4.1. Proposition. Under the above conditions we have

1�a;b�

Z ��a

f dg 2 Hl�a; b� and 1�a;b�

Z b

��f dg 2 Hl�a; b� :

In particular, for h 2 Hk; g 2 Hl; k� l > 1, we may consider theintegrals

u�x� :�Z x

ah�y� dg�y� 1�a;b��x�


and

w�x� :� ÿZ b

xh�y� dg�y� 1�a;b��x�

as functions from Hl�a; b�.4.4.2. Theorem. Under the above conditions we haveZ b

af �x�h�x� dg�x� �

Z b

af �x� du�x� �

Z b

af �x� dw�x� :

Proof. For the step functions efPD�x� �Pn

i�0 f �xi�1�xi;xi�1��x� we getfrom Corollary 2.3 and additivity of the integralZ b

a

efPD�x� du�x� �Xn

i�0f �xi��u�xi�1� ÿ u�xi��

�Xn

i�0f �xi�

Z xi�1

xi

h�y� dg�y�

�Xn

i�0

Z xi�1

xi

efPD�y�h�y� dg�y�

�Z b

a

efPD�y�h�y� dg�y� :

Theorem 4.2.1 implies

limD!0

Z b

a

efPD�x� du�x� �Z b

af �x� du�x� :

In order to show

limD!0

Z b

a

efPD�y�h�y� dg�y� �Z b

af �y�h�y� dg�y�

recall that according to the proof of Theorem 4.2.1Z b

a

efPD�y�h�y� dg�y� ÿZ b

a

efPD�y�ehPD�y� dg�y�� const Dkÿa

and

limD!0

Z b

a

efPD�y�ehPD�y� dg�y� �Z b

af �y�h�y� dg�y� :

Hence,

352 M. ZaÈ hle

Z b

af �x� u�dx� �

Z b

af �y�h�y� dg�y� :

The arguments for w instead of u are similar. (

5. Applications to stochastic calculus

5.1 Integration with respect to fractional Brownian motion

A modern presentation of the theory of stochstic integration withrespect to semimartingales may be found in Protter [10] and in Win-kler and v. WeizsaÈ cker [13]. These books also contain many referencesto related literature. Semimartingales provide the most general class ofstochastic processes for which a stochastic calculus has been devel-oped. In particular, stochastic di�erential equations are treated.

An important problem, e.g., in ®nance mathematics is to determinesimilar di�erential equations for fractional Brownian motion as anappropriate noise model for real stock-market processes with depen-dent increments. The study of fractional Brownian motion BH as a real-valued Gaussian process on �0;�1� with stationary increments ofmean zero and variance

E�BH �t � s� ÿ BH �t��2 � s2H ;

(where 0 < H < 1) goes back to Kolmogorov and Jaglom (cf. thereferences in [5]). A representation in terms of a Fourier transform ofordinary Brownian motion B � B1=2 was given in Hunt [2]. The nameof the process was created in Mandelbrot and van Ness [5] who calledthe parameter H indicating a certain scaling self-similarity the Hurstcoe�cient of the motion. For more details see Kahane [4].

One can show that BH has a version with sample paths of HoÈ lderexponent H , i.e. of HoÈ lder continuity of all orders k < H on any ®niteinterval �0; T � with probability 1. The quadratic variation on �a; b�equals

limD!0

Xi

�BH �ti�1� ÿ BH �ti��2 �1 if H < 1=2bÿ a if H � 1=20 if H > 1=2

8<:where convergence holds uniformly in PD with probability 1 ifH 6� 1=2 and in the mean squared if H � 1=2. Therefore except ofthe case H � 1=2 (ordinary Brownian motion or Wiener process)fractional Brownian motion cannot form a semimartingale so thatclassical stochastic integration does not work. However, the


HoÈ lder continuity of BH ensures the pathwise existence of ourintegrals (22) Z t

0

f �s� dBH �s�; 0 < t � T ; �33�

with probability 1 for any measurable random function f on �0; T �such that f0� 2 Ia

0��L1�0; T �� with probability 1 for some a > 1ÿ H .Note, that we do not need here any assumption of adaptedness. In thespecial case f 2 Hk�0; T � with probability 1 for some k > 1ÿ H wemay use the interpretation as Riemann±Stieltjes integral and exploitthe change-of-variable formula (31), the HoÈ lder continuity of the in-tegral as function of the boundary 4.4.1 and the integration rule 4.4.2.In particular, we may choose f �t� � r�X �t�; t� for some real-valuedLipschitz function r and any random function X whose sample pathslie in Hk�0; T � with probability 1. For H > 1=2 this makes it possibleto investigate (stochastic) di�erential equations.

Example. Consider the linear equation

dX �t� � a X �t� dBH�t� � b X �t� dt �34�which means

X �t� � X �0� � aZ t

0

X �s� dBH �s� � bZ t

0

X �s� ds

for some random constants a and b, where H > 1=2.Its unique solution reads

X �t� � X �0� expfaBH �t� � btg �35�

Proof. The change-of-variable formula (31) implies that (35) is a so-lution of (34). Let Y �t� be any other solution as above with the sameinitial condition Y �0� � X �0�. For simplicity we assume here thatX �0� 6� 0 and show that Y agrees with X . (For the case X �0� � 0 thisfollows from a more general uniqueness result contained in a relatedPh. D. Thesis which is in preparation.) We consider only (®xed)sample paths denoted by the same symbol Y which are HoÈ lder con-tinuous of order greater than 1=2. In this case there are some numbersC > c > 0 such that c < j�t�j < C and sgnY �t� � sgnY �0� for 0 < t � �with su�ciently small � > 0. For these t we may apply Theorem 4.3.1to a smooth function F with F �x� � ln x if x 2 �c;C� and tof �t� � jY �t�j if t 2 �0; �� and obtain ln jY �t�j ÿ ln jY �0�j � aB�t� � btaccording to Theorem 4.4.2. This yields Y �t� � X �t� for 0 � t � �. Inthe same way one can show that for any t > 0 with Y �t� � X �t� thereexists a right-sided neighborhood where the functions coincide.

354 M. ZaÈ hle

We next consider the following example for the application of thechange-of-variable formula 4.3.1:Z y

xBH �t� dBH �t� � 1

2�BH �y�2 ÿ BH �x�2�; 0 � x < y <1 ; �36�

with probability 1 provided that H > 1=2. This re¯ects the fact thatthe quadratic veriation of BH vanishes. Note that for H � 1=2 in theexponent of (35) as well as on the right-hand side of (36) an additionallinear term arises. Here the stochastic integrals are determined in theItoÃ sense. A link between both these approaches will be established inthe next section.

5.2 A new representation of the ItoÃ integral for random functionsfrom Ia

0��L2�

In this section the integrator g is replaced by the Wiener processW � B1=2 and the random integrand f is assumed to be adapted withrespect to the ®ltration given by W . If f 2 L2�0; T � with probability 1the classical ItoÃ integral

Itf � (Ito )

Z t

0

f dW

is determined. We write If � IT f .

5.2.1. Theorem. If f is adapted and f 2 Ia0��L2� with probability 1 for

some a > 1=2 then the integralsR t0 fdW , 0 < t < T , in the sense of (22)

are determined and agree with the continuous version of the ItoÃ integralswith probability 1.

Proof. For the special case of smooth f both the integrals agree with

ÿZ t

0

f 0�s�W �s� ds� f �t�W �t� :

Arbitrary realizations f 2 Ia0��L2� will again be approximated by the

smooth functions fN � f � kN with the smoothing kernels kN given by(27). Using that W 2 I1ÿa

tÿ �L2� with probability 1 we obtain from theproof of Theorem 2.4

limN!1

Z t

0

fN dW �Z t

0

f dW ; 0 < t < T ;


with probability 1. On the other hand the almost sure L2�0; T �-con-vergence of fN to f implies the convergence of ItfN to Itf in proba-bility for any ®xed t. This yields the assertion. (

For applications to stochastic di�erential equation in the ItoÃ sensethe choice a > 1=2 in Theorem 5.2.1 is too restrictive. Since the samplepaths of W do not belong to I1=20� �L2� the approach (22) does not workfor a � 1=2. However, we may approximate the ItoÃ integrals of afunction f with ``fractional degree of di�erentiability'' 1=2 by ourintegrals (22) for some regularization of f :

5.2.2. Corollary. Let f be an adapted random function such thatf 2 Ia

0��L2� for any a < 1=2 with probability 1. Then we have the fol-lowing convergence in probability:

lim�&0

Z t

0

I�0�f dW � lim�&0�ÿ1�1=2

Z t

0

D1=2ÿ�=20� f �s�D1=2ÿ�=2

tÿ Wtÿ�s� ds

� (Ito )

Z t

0

f dW :

Proof. First note that �I�0�f �0� � I �0�f 2 Ia0��L2� for any 1=2 < a <

1=2� �. Therefore R t0 I�0�f dW is determined by

�ÿ1�1=2ÿ�=2Z t

0

D1=2��=20� I �0�f �s�D1=2ÿ�=2

tÿ Wtÿ�s� ds

� �ÿ1�1=2ÿ�=2Z t

0

D1=2ÿ�=20� f �s�D1=2ÿ�=2

tÿ Wtÿ�s� ds :

According to Theorem 5.2.1 we haveZ t

0

I�0�f dW � It�I�0�f �with probability 1 for any � > 0. The almost sure L2�0; t�-convergenceof I�0�f to f as �& 0 implies the convergence in probability of the ItoÃintegrals It�I�0�f � to Itf . (

Next we will state a su�cient condition for the above convergencein terms of square means. For, we introduce the classes Ia0��L2� ofmeasurable random functions f such that

Ef �0��2 <1 �37�

E

Z T

0

�f �t� ÿ f �0��2t2a

dt <1 �38�

and the random functions

356 M. ZaÈ hle

h��t;x� :�Z tÿ�

0

f0��t;x� ÿ f0��s;x��t ÿ s�a�1 ds �39�

converge in L2 :� L2��0; T � � X, L� P� as �& 0.Note that (37) and (38) imply

E

Z T

0

f �t�2 dt <1: �40�

We put

Ibÿ0��L2� :�\a<b

Ia0��L2�: �41�

5.2.3. Corollary. For any adapted f 2 I1=2ÿ0� �L2� we have

E

Z t

0

I �0�f dW ÿ Itf� �2

� E

Z t

0

I�0�f �s� ÿ f �s�ÿ �2 ds

which converges to zero as �& 0.

Proof. Completely analogous arguments as in the deterministic case (s.[11], Theorem 13.2) show that f 2 Ia0��L2� implies f0� 2 Ia

0��L2� (andhence f 2 Ia

0��L2� if a < 1=2) with probability 1. Since the spacesIa0��L2� are decreasing in a it follows that for any f 2 I1=2ÿ0� �L2� wehave f 2 Ta<1=2 Ia

0��L2� with probability 1. Consequently, the condi-tions of Corollary 5.2.2 are ful®lled andZ t

0

I�0�f dW � It�I�0�f �

with probability 1. The L2-property (40) of f and the isometric be-haviour of the ItoÃ integral yield the asserted equation. The conver-gence

lim�&0

E

Z t

0

�I �0�f �s� ÿ f �s��2 ds � 0

may be shown similarly as in the deterministic case. (

Note that the Wiener process itself is an element of I1=2ÿ0� �L2�.

Moreover, we have the following.

5.2.4. Theorem. If f is adapted and f 2 I1=2ÿ0� �L2� then any measurableversion X �t� of the ItoÃ-integral (ItoÃ )

R t0 f dW , t 2 �0; T �, is again an

element of I1=2ÿ0� �L2�.


Proof. Let 0 < a < 1=2. The isometry property of the ItoÃ integral leadsto

E

Z T

0

X �t�tÿ2a dt �Z T

0

tÿ2aEZ t

0

f �s�2 ds dt

� E

Z T

0

f �s�2Z T

stÿ2a dt ds

� const E

Z T

0

f �t�2 dt <1 :

It remains to prove that

E

Z T

0

Z tÿ�0

tÿ�

X �t� ÿ X �s��t ÿ s�a�1 ds

!2

dt

tends to zero as �& 0 uniformly in �0 < �. Here we put X �s� � 0 ifs < 0. This expression equals

Z T

0

E

Z �

�0

X �t� ÿ X �t ÿ s�sa�1 ds

� �2

dt

� 2

Z T

0

Z �

�0

Z u

�0sÿ�a�1�uÿ�a�1�E�X �t�

ÿ X �t ÿ s��X �t� ÿ X �t ÿ u�� ds du dt

� 2

Z T

0

Z �

�0

Z u

�0sÿ�a�1�uÿ�a�1�E

Z t

tÿsf �v�2 dv ds du dt

in view of the almost sure equality

X �t� ÿ X �t ÿ s� � (Ito )

Z T

0

1�tÿs;t��x�f �x� dW �x�

and the isometry property of the ItoÃ integral. The right-hand side isequal to

2

Z �

�0

Z u

�0sÿ�a�1�uÿ�a�1�s ds du E

Z T

0

f �v�2 dv

� 2

1ÿ a

Z �

�0uÿ�a�1�u1ÿa du E

Z T

0

f �v�2 dv � const �1ÿ2a

which yields the assertion. (

358 M. ZaÈ hle

5.3. Anticipating integrals

Our aim is now to extend the results of the preceding section to an-ticipating (i.e. non-adapted) functions f , where the ItoÃ integral isextended to a new version of stochastic integral. This concept isclosely related to Skorohod and extended Stratonovitch integrals.(For introduction, survey and further literature to related stochasticcalculus cf. Nualart [6], Nualart and Pardoux [8], Pardoux [9].) Ourmain tool will be the classical approach to Skorohod integration (seeSkorohod [12]) via ItoÃ -Wiener chaos expansion of random L2-func-tions (see ItoÃ [3]).

We ®rst will establish a link between our integrals in the sense of(22) and the symmetric multiple ItoÃ -integrals of deterministic func-tions f 2 L2��0; 1�n� arising in the ItoÃ -Wiener chaos expansion. Theiterated ItoÃ integral of such an f is given by

Inf :� (Ito )

Z 1

0

Z tn

0

� � �Z t2

0

f �t1; . . . ; tn� dW �t1� . . . dW �tn� :

(It is well-determined for tensor products f � f1 � � � fn and maybe extended to general f by linearity and the corresponding isometry.)By means of symmetrization

ef �t1; . . . ; tn� :� 1

n!

Xp2Sn

f �tp�1�; . . . ; tp�n��

(for the permutation group Sn) one turns to the concept of symmetricn-th order ItoÃ integral eInf :� n! Inef : �42�Its isometry property reads

EeImfeIng � n!R�0;1�n ef �t�eg�t�Ln�dt� if m � n

0; else :

��43�

For 0 < a < 1 let eIa0��n;L2� be the class of those functions f from

L2��0; 1�n�1� being symmetric in the ®rst n arguments for which thefunctions

h��t1; . . . ; tn; t� :�Z tÿ�

0

f �t1; . . . ; tn; t� ÿ f �t1; . . . ; tn; s��t ÿ s�a�1 ds �44�

converge in L2��0; 1�n�1� as �& 0, where h��t1; . . . ; tn; t� � 0 if t < 0.(Completely analogous arguments as in the proof of Theorem 13.2

in [11] show that eIa0��n; L2� is exactly the class of those functions on


�0; 1�n�1 with the required symmetry which are representable as Ia0�-

integral with respect to the last variable of some L2��0; 1�n�1�-func-tion. Moreover, if a > 1=2 these functions (up to equivalence) areHoÈ lder continuous in the last argument, cf. [11], Theorem 3.6. Put

f0��t1; . . . ; tn; t� :� 1�0;1��t��f �t1; . . . ; tn; t� ÿ �f �t1; . . . ; tn; 0��assuming everywhere that the right-sided limit exists.

For f0� 2 eIa0��n;L2� the symmetric multiple ItoÃ integralseInÿ1f ��; s; t�, eInf ��; t�, and eIn�1f make sense because of the L2-prop-

erties. In view of the isometry (43) the random functionseInf ��; t�0� � eInf0��; t� are elements of the class Ia0��L2� introduced inSection 5.2. Therefore the integrals

R 10eInf ��; t� dW �t� in the sence of

(22) are determined with probability 1 for any a > 1=2.

5.3.1. Theorem.Z 1

0

eInf ��; t� dW �t� � eIn�1f � nZ 1

0

eInÿ1f ��; t; t� dt

with probability 1 if f0� 2 eIa0��n; L2� for some a > 1=2.

Proof. By de®nition we have with probability 1Z 1

0

eInf ��; t� dW �t� � �ÿ1�aZ 1

0

Da0�eInf0��; ��t�D1ÿa

1ÿ W1ÿ�t� dt

�eInf ��; 0��W �1� :As before, we approximate f by functions being smooth in the lastargument

fN �t1; . . . ; tn; t� :� f �t1; . . . ; tn; �� kN �t�and obtain by (43) eInfN��; t� � eInf ��; �� kN �t� ;eInÿ1fN ��; s; t� � eInÿ1f ��; s; �� kN �t� :Then we have

l.i.m.N!1

Z 1

0

eInfN��; t� dW �t� �Z 1

0

eInf ��; t� dW �t�

(cf. Section 2) and by (43)

l.i.m.N!1

eIn�1fN � eIn�1f

and similar estimates as in the proof of Theorem 3.6 in [11] yield

360 M. ZaÈ hle

l.i.m.N!1

Z 1

0

eInÿ1fN ��; t; t� dt �Z 1

0

eInÿ1f ��; t; t� dt

(where l.i.m. means convergence in the mean square) since thesmoothing kernels kN are concentrated on 0; 1N

� �and f is continuous in

the last argument. Therefore is enough to consider fN instead of f . Wesplit the multiple integral on the left-hand side into a part not con-taining ``diagonal'' arguments and the remainder and approximateboth the summands by piecewise integration:Z 1

0

eInfN ��; t� dW �t�

� n! l.i.m.k!1

Xn

j�0

X0�i1<��<ij<i<ij�1<��<in�kZ i�1

k

ik

(Ito)

Z in�1k

ink

� � �Z i1�1

k

i1k

fN �t1; . . . ; tn; t� dW �t1� . . . dW �tn� dW �t�

� n! l.i.m.k!1

Xn

j�1

X0�i1<��<in�kZ ij�1

k

ijk

(Ito)

Z in�1k

ink

� � �Z i1�1

k

i1k

fN �t1; . . . ; tn; t� dW �t1� . . . dW �tn� dW �t�

Because of the choice of disjoint intervals of integration and the in-dependent increments of the Wiener process we may change the orderof integration in the ®rst summand and exploit thatZ i�1

k

ik

fN �t1; . . . ; tn; t� dW �t�

agreeswith the ItoÃ integral in the sense of the correspondingL2-equality.Therefore the ®rst limit yields eIn�1fN because of the symmetry of fN inthe ®rst n arguments. In the second summand we also may change theorder of integration except of the ``diagonal'' integrals arising fromZ ij�1

k

ijk

(Ito )

Z ij�1k

ijk

fN �t1; . . . ; tn; t� dW �tj� dW �t� :

By the smoothness property of fN we may use formula (22) with a � 1for the outer integral to obtain the expression

ÿZ ij�1

k

ijk

(Ito )

Z ij�1k

ijk

f 0N �t1; . . . ; tn; t� dW �tj�Wij�1k ÿ�t� dt

� (Ito )

Z ij�1k

ijk

fN �t1; . . . ; tn;ijk� dW �tj� W

ij � 1

k

� �ÿ W

ijk

� �� :


The ®rst summand may be neglected asymptotically as k !1 afterintegrating in tl; l 6� j, and summing up in view of the usual L2-esti-mations. Similar L2-arguments show that the second summand maybe replaced asymptotically by

fN t1; . . . ; tjÿ1;ijk; tj�1; . . . ; tn;

ijk

� �W

ij � 1

k

� �ÿ W

ijk

� �� 2

:

Using the quadratic variation of the Wiener process and the symmetryproperty of fN we obtain after integration and summation for thecorresponding limit the value

nZ 1

0

eInÿ1fN ��; t; t� dt :

(We now turn to anticipating integrals using the ItoÃ±Wiener chaosexpansion of random functions f 2 L2:

f �t� �X1n�0eInf n��; t� �45�

(where eI0f 0��; t� � Ef �t�) for unique f n 2 L2��0; 1�n�1� being sym-metric in the ®rst n-arguments. Recall that the Skorohod integral of fexists and is given by

d�f � � �S�Z 1

0

f dW :�X1n�0eIn�1f n �46�

if this series converges in the mean square.We introduce the extended Stratonovitch integral of f byZ 1

0

f � dW :�X1n�0

eIn�1f n � n2

Z 1

0

�eInÿ1f n��; t; tÿ� �eInÿ1f n��; t; t�� dt� �

�47�if this series converges in the mean square. One can show that undersome additional condition this integral agrees with the notion used inthe literature (cf. [6]).

Theorem 5.3.1 suggests the following new concept of anticipatingintegral:

�A�Z 1

0

f dW :�X1n�0

eIn�1f n � nZ 1

0


�48�

provided that the series converges in the mean square, where

362 M. ZaÈ hle

Z 1

0

eInÿ1f n��; t;ÿt� dt � l.i.m.�&0

Z 1

0

eInÿ1I�0�f n��; t; t� dt :

(The corresponding expression for t� in (47) is de®ned similarly.)If the f n (up to L2��0; 1�nÿ1�-equivalence) have no jumps at Le-

besgue-a.a. points on the diagonal given by the last two arguments weget

�A�Z 1

0

f dW �Z 1

0

f � dW :

For adapted f we have f n��; t; tÿ� � 0 at a.a. t and therefore

�A�Z 1

0

f dW � d�f � � (Ito )

Z 1

0

f dW

with probability 1. (The last equation is the well-known extensionproperty of the Skorohod integral.)

In general, the three integrals are di�erent and the existence of theStratonovitch integral or the anticipating integral (48) does not implythat of the Skorohod integral. Our de®nition will be justi®ed belowwhere we will study some relationships between these integrals.

It appears appropriate to work with the following Slobodeckij-typespacesWa

2;� �Wa2;��0; 1� of measurable random functions f such that

E f �0��2 <1 �49�

E

Z 1

0

�f �t� ÿ f �0��2t2a

dt <1 �50�

E

Z 1

0

Z 1

0

�f �t� ÿ f �s��2jt ÿ sj2a�1 ds dt <1 ; �51�

where 0 < a < 1. (For a survey on classical function spaces see, e.g.,[7].)

Remark: (49) and (50) imply

E

Z 1

0

f �t�2 dt <1 : �52�

In the theory of function spaces the deterministic versions of (51) and(52) de®ne the space W a

2 �0; 1�.Recall that the space Iaÿ0��L2� was de®ned in (41).

5.3.2. Proposition.

Wa2;� � Iaÿ0��L2� :


Proof. According to (39) ist is enough to show that for any b < a wehave

lim�&0

sup�0<�

E

Z 1

0

Z tÿ�0

tÿ�

f0��t� ÿ f0��s��t ÿ s�b�1

!2

dt � 0 :

Choose 0 < d < 2�aÿ b�. By the Chauchy±Schwarz inequality weobtain

E

Z 1

0

Z tÿ�0

tÿ�

f0��t� ÿ f0��s��t ÿ s�b�1

!2

dt

� E

Z 1

0

Z tÿ�0

tÿ�

f0��t� ÿ f0��s��t ÿ s�b�d

1

�t ÿ s�1ÿd

!2

dt

� const E

Z 1

0

Z tÿ�0

tÿ�

�f0��t� ÿ f0��s��2�t ÿ s�2b�2d

1

�t ÿ s�1ÿd ds dt

� const E

Z 1

0

Z tÿ�0

tÿ�

�f0��t� ÿ f0��s��2�t ÿ s�2a�1 �t ÿ s�2�aÿb�ÿd ds dt

� const ��ÿ �0�2�aÿb�ÿd�E

Z 1

0

Z 1

0

�f �t� ÿ f �s��2jt ÿ sj2a�1 ds dt

� E

Z �

0

f0��t�2Z 0

tÿ�

1

�t ÿ s�2a�1 ds dt�

� const�2�aÿb�ÿd const � E

Z �

0

�f �t� ÿ f �0��2t2a

dt

!� const �2�aÿb�ÿd ;

where the constants depend on d. (

Let now f �t� �P1n�0eInf n��; t� be the ItoÃ ±Wiener chaos expansionas above.

5.3.3. Proposition f 2Wa2;� implies f n

0� 2 eIb0��n; L2� for any b < a and

n 2 N.

Proof. By the de®nition of eIb0��n;L2� (cf. (44)) it su�ces to show that

lim�&0

sup�0<�

Z 1

0

� � �Z 1

0

Z tÿ�0

tÿ�

f n0��t1; . . . ; tn; t� ÿ f n

0��t1; . . . ; tn; s��t ÿ s�b�1 ds

!2

dt dt1 . . . dtn � 0 :

Similar estimations like in the proof of the preceding proposition leadto the following upper bound of these integrals for ®xed 0 < d <2�aÿ b�:

364 M. ZaÈ hle

const �2�aÿb�ÿd

�Z 1

0

. . .

Z 1

0

�f n�t1; . . . ; tn; t� ÿ fn�t1; . . . ; tn; 0��2t2a

dt1 . . . dtn dt

�Z 1

0

� � �Z 1

0

�f n�t1; . . . ; tn; t� ÿ fn�t1; . . . ; tn; s��2jt ÿ sj2a�1 dt1 . . . dtn ds dt

�:

It remains to show that the last two integrals are ®nite. Using theisometry (43) we infer

n!

Z 1

0

� � �Z 1

0

�f n�t1; . . . ; tn; t� ÿ f n�t1; . . . ; tn; 0��2t2a

dt1 . . . dtn dt

�Z 1

0

tÿ2aE�eIn�f n��; t� ÿ f n��; 0��2 dt

�Z 1

0

E�f �t� ÿ f �0��2t2a

dt ;

since f �t� ÿ f �0�� P1n�0eIn�f n��; t� ÿ f n��; 0�� and hence,E�f �t� ÿ f �0��2 �P1n�0 E�In�f n��; t� ÿ f n��; 0��2 by orthogonali-ty. In view of (50) the last integral is ®nite. Similarly one shows that

n!

Z 1

0

� � �Z 1

0

�f n�t1; . . . ; tn; t� ÿ f n�t1; . . . ; tn; s��2jt ÿ sj2a�1 dt1 . . . dtn ds dt

�Z 1

0

Z 1

0

E�f �t� ÿ f �s��2jt ÿ sj2a�1 ds dt

which is ®nite according to (51). (

We now are able to prove an extension of Theorem 5.2.1 to an-ticipating functions which justi®es the de®nition (48).

5.3.4. Theorem. If f 2Wa2;� for some a > 1=2 then the anticipating

integral �A� R 10 f dW in the sense of (48) exists and agrees with theintegral

R 10 f dW in the sense of (22) as well as with the extended

Stratonovitch integralR 10 f � dW .

Proof. Choose an arbitrary b 2 1=2; a� �. By Proposition 5.3.2,Z 1

0

f dW � �ÿ1�bZ 1

0

Db0�f0��t�D1ÿb

1ÿ W1�t� dt � f �0��W �1� :

Recall that


f0��t� �X1n�0eInf n

0��; t�; f �0�� X1n�0eInf n��; 0��

and therefore,

�A�Z 1

0

f dW � �A�Z 1

0

f0� dW � f �0��W �1� :

If we can show that

Db0�f0��t� �

X1n�0

Db0�eInf n

0��; ��t�

in the sense of L2-convergence of the series then the Cauchy±Schwarzinequality leads toZ 1

0

f0� dW �X1n�0

Z 1

0

eInf n0��; t� dW �t�

so that Proposition 5.3.3 and Theorem 5.3.1 yield the assertion.By construction, for any h 2 Ib0��L2� the derivative Db

0�h0� is theL2-limit of the random functions

Db0�;�h0��t� :� 1

C�1ÿ b�h�t� ÿ h�0��

tb� b

Z tÿ�

0

h0��t� ÿ h0��s��t ÿ s�b�1 ds

!as �& 0. For the special functions

hN �t� :�XN

n�0eInf n

0��; t�

we obtain by Fubini and the orthogonality EeInueImw � 0; n 6� m, theestimationZ 1

0

E�Db0�;�hN �t� ÿ Db

0�;�0hN �t��2 dt

�Z 1

0

XN

n�0E�Db

0�;�eInf n0��; ��t� ÿ Db

0�;�0eInf n0��; ��t��2 dt

�Z 1

0

X1n�0

E�Db0�;�eInf n

0��; ��t� ÿ Db0�;�0eInf n

0��; ��t��2 dt :

Moreover, for any � > 0,

Db0�;�f0� �

X1n�0

Db0�;�eInf n

0��; ��

366 M. ZaÈ hle

because of the corresponding boundedness property. Therefore theexpression on the right-hand side of the above estimation is equal toZ 1

0

E�Db0�;�f0��t� ÿ Db

0�;�0f0��t��2 dt :

In view of Proposition 5.3.2 the function f0� is an element of Ib0��L2�.

Hence, the last integral tends to zero as �& 0 uniformly in �0 < � andconsequently, the L2-convergence of Db

0�;�hN �t� as �& 0 is uniform inN . Thus we may change the order of the L2-limits and obtain

Db0�f0� � lim

�&0lim

N!1Db0�;�hN � lim

N!1lim�&0

Db0�;�hN

� limN!1

XN

n�0Db0�eInf n

0��; �� X1n�0

Db0�eInf n

0��; �� :

Finally, the equality

�A�Z 1

0

f dW �Z 1

0

f � dW

follows from the de®nition of the extended Stratonovitch integral andcontinuity of the functions f n in the last argument because of Prop-osition 5.3.3. (

Recall that the condition a > 1=2 is too restrictive concerning theapplication to stochastic di�erential equations. In order to extendCorollary 5.2.3 to anticipating f we introduce the class

W1=2ÿ2;� :�

\0<a<1=2

Wa2;� : �53�

By Proposition 5.3.2 , W1=2ÿ2;� � I

1=2ÿ2;� �L2�.

Similarly as in the deterministic case one proves that f 2W1=2ÿ2;�

implies I�0�f 2W�1=2��ÿ2;� . Therefore we may consider again the ``ap-proximating'' integralsZ 1

0

I�0�f dW � �ÿ1�1=2ÿ�=2Z 1

0

D1=2ÿ�=20� f �t�D1=2ÿ�=2

1ÿ W1ÿ�t� dt

as �& 0. Theorem 5.3.4 impliesZ 1

0

I�0�f dW � �A�Z 1

0

I�0�f dW :

Since f is anticipating convergence in L2 of these integrals as �& 0does not hold in general. However, we get the following main result ofthis section regarding that the mean square of �A� R 10 f dW in the senseof (48) may be computed by


Z 1

0

f 1�t; tÿ� dt� �2

�X1n�0�n� 1�!

ef n

� �n� 2�Z 1

0

f n�2��; t; tÿ� dt

2L2��0;1�n�1�

�54�

in view of the isometry property (43).

5.3.5. Theorem. Suppose that f 2W1=2ÿ2;� and

R 10 I �0�f dW converges in

the mean square as �& 0. Then the anticipating integral �A� R 10 f dW inthe sense of (48) exists and we have

E

Z 1

0

I �0�f dW ÿ �A�Z 1

0

f dW� �2

�Z 1

0

�I �0�f 1�t; tÿ� ÿ f 1�t; tÿ�� dt� �2

�X1n�0�n� 1�! I�0�ef n ÿ ef n � �n� 2�

Z 1

0

�I �0�f n�2��; t; tÿ� ÿ f n�2��; t; tÿ�� 2

L2��0;1�n�1

(where f k��; t; tÿ� is the L2-limit of I �0�f k��; t; t� as function in t as �& 0)and

l.i.m.�&0

Z 1

0

I�0�f dW � �A�Z 1

0

f dW :

Proof. Let 0 < � < 1=2. It follows from the Cauchy±Schwarz in-equality that

I�0�f �X1n�0

I�0�eInf n��; �� :

Then the isometry (43) yields the ItoÃ ±Wiener chaos expansion

I�0�f �X1n�0eInI�0�f n��; �� :

SinceR 10 I�0�f dW � �A� R 10 I�0�f dW it is enough to prove that

l.i.m.�&0

X1n�0

eIn�1I �0�f n � nZ 1

0

eInÿ1I �0�f n��; t; t� dt� �

�X1n�0

eIn�1f n � nZ 1

0


:

368 M. ZaÈ hle

(The asserted equation for the mean square distance follows from(54).) The series on the left-hand side is equivalent to the seriesZ 1

0

I�0�f 1�t; t� dt �X1n�0

eIn�1I �0�f n � �n� 2�Z 1

0

eIn�1I �0�f n�2��; t; t� dt� �

whose summands are pairwise orthogonal according to (43). By as-sumption, the limit in the mean square as �& 0 exists. The Hilbertspace arguments which we have used repeatedly show that this limitagrees with

lim�&0

Z 1

0

I�0�f 1�t; t� dt �X1n�0

l.i.m.�&0

eIn�1I�0�f n � �n� 2�Z 1

0

eIn�1I �0�f n�2��; t; t� dt

!

�Z 1

0

f 1�t; tÿ� dt �X1n�0

eIn�1f n � �n� 2�Z 1

0

eIn�1f n�2��; t; tÿ� dt

!in view of the isometry property (43) and the corresponding L2-ver-sions of (18). Finally, the right-hand side is equivalent to

X1n�0

eIn�1f n � nZ 1

0


: (

5.3.6. Corollary. Under the conditions of Theorem 5.3.5 we have

�A�Z 1

0

cf dW � c �A�Z 1

0

f dW

for any bounded random variable c.

Proof. The de®nition (22) impliesZ 1

0

I�0��cf � dW � cZ 1

0

I �0�f dW :

Therefore Theorem 5.3.6 yields the assertion. (

In order to formulate a certain counterpart to Theorem 5.3.6 weneed some notions from the literature. Recall that in terms of ItoÃ ±Wiener chaos expansion f �s� �P1n�0eInf n��; s� for ®xed s the Mall-iavin derivative of this random variable is given by


Dtf �s� �X1n�1

n eInÿ1f n��; t; s�

provided that this series of random functions in t converges in L2 (cf.[6], [8], [9]). The space L1;2 of random functions is commonly used inthe literature in order to characterize the Skorohod integral as dualoperation to Malliavin derivation. For its de®nition we refer to [9]. Itis a subspace of the domain of de®nition of the Skorohod integral.L1;2

C denotes the space of those f 2 L1;2 for which the set of functions

fs! Dtf �s�; s 2 �0; 1�nftggt2�0;1�

is equicontinuous with values in L2�X;P� andess sup�s;t�2�0;1�2

E�Dtf �s��2 <1 :

For f 2 L1;2C denote

Dtf �tÿ� :� l.i.m.�&0

Dtf �t ÿ ��Dtf �t�� :� l.i.m.

�&0Dtf �t � ��

(cf. [9]).

5.3.7. Theorem.

(i) For f 2 L1;2C the anticipating integral (48) exists and equals

�A�Z 1

0

f dW � d�f � �Z 1

0

Dtf �tÿ� dt :

(ii) If f 2 L1;2C \W1=2ÿ

2;� then the convergence

l.i.m.�&0

Z 1

0

I �0�f dW � �A�Z 1

0

f dW

holds.

Remark. Similarly as in the proof below it can be shown that f 2 L1;2C

implies the existence of the extended Stratonovitch integral andZ 1

0

f � dW � d�f � � 1

2

Z 1

0

�Dtf �tÿ� � Dtf �t�� dt :

(This corresponds to Proposition 5.2 in [9].)

Proof of Theorem 5.3.7. (i) In order to show the mean square con-vergence of the series

370 M. ZaÈ hle

X1n�0

eIn�1f n � nZ 1

0


and the asserted equation we use the convergenceX1n�0eIn�1f n � d�f �

and prove thatX1n�0

nZ 1

0

eInÿ1f n��; t; tÿ� dt �Z 1

0

Dtf �tÿ� dt :

Regarding

Dtf �tÿ� � l.i.m.�&0

X1n�1

n eInÿ1f n��; t; t ÿ ��

�X1n�1

n eInÿ1f n��; t; tÿ�

at almost all t by (43) we still have to justify the change of the order ofsummation in n and integration in t. But this also follows from (43).(ii) It is not di�cult to check that f 2 L1;2

C implies I�0�f 2 L1;2C . Hence,

�A�Z 1

0

I �0�f dW � d�I�0�f � �Z 1

0

DtI �0�f �tÿ� dt :

Further, the above representation of Dtf �tÿ� in terms of the ItoÃ ±Wiener chaos expansion and (43) yield

DtI�0�f �tÿ� � I�0�Dtf ��tÿ� :From the corresponding L2-version of (18) we infer

l.i.m.�&0

Z 1

0

I �0�Dtf ��tÿ� dt �Z 1

0

Dtf �tÿ� dt :

Below we will show that

l.i.m.�&0

d�I �0�f � � d�f � :Consequently,

l.i.m.�&0

�A�Z 1

0

I �0�f dW � �A�Z 1

0

f dW :

If we additionally assume that f 2W1=2ÿ2;� then we may useZ 1

0

I�0�f dW � �A�Z 1

0

I �0�f dW


in view of Theorem 5.3.4. This leads to (ii).By de®nition of the Skorohod integral,

d�I�0�f � �X1n�0eIn�1�I �0�f n�

�XN

n�0eIn�1�I �0�f n� �

X1n�N�1

eIn�1�I�0�f n� :

For ®xed N the ®rst summand tends toPN

n�0eIn�1�f � as �& 0 by (43)and the corresponding L2-version of (18). The mean square of thesecond summand does not exceedX1

n�N�1�n� 1�!kI �0�ef nk2 � const

X1n�N�1

�n� 1�!kef nk2

for a certain constant independent of � because

kI�0�ef nk2 �Z 1

0

� � �Z 1

0

Z t

0

ef n�t1; . . . ; tn; s� 1

C�� t ÿ s��ÿ1 ds� �2

dt dt1 . . . dtn

� const

Z 1

0

� � �Z 1

0

Z t

0

ef n�t1; . . . ; tn; s�2 1

C�� t ÿ s��ÿ1

ds dt dt1 . . . dtn

� const kef nk2

according to the Cauchy±Schwarz inequality.Since kd�f �k2 �P1n�0�n� 1�! kef nk2 we obtain that the second

summand of the above sum tends to zero as N !1 uniformly in �.Thus,

l.i.m.�&0

d�I �0�f � � d�f � : (

Remark. 1. Recall that under various conditions an extended ItoÃformula for the change of variables in Skorohod integrals was proved.In distinction to the adapted case it contains an additional termconcerning Malliavin derivatives. (For Stratonovitch integrals theclassical chain rule from calculus remains valid.) We will show in partII of this paper that under appropriate conditions for the anticipatingintegral (48) the classical ItoÃ formula remains valid. This simpli®es thestudy of corresponding anticipating stochastic di�erential equations.

2. After ®nishing the manuscript we were referred to the paper ofCiesielski, Kerkyacharian and Roynette [1] which contains an exten-

372 M. ZaÈ hle

sion of the Riemann±Stieltjes integral to continuous functions fromcertain Besov spaces by means of their Schauder expansions and acorresponding limit procedure. The application to stochastic integralswith respect to the Wiener process leads to the (extended) Strato-novitch integral. For the case of fractional Brownian motion BH withH > 1=2 and the special integrands f of HoÈ lder exponent greater then1ÿ H it can be shown that our integral (22) agrees with that of theabove authors. This provides the convergence of the Riemann±Stieltjes sums and the corresponding calculation rules which have notbeen derived in [1]. (Concerning stochastic di�erential equations withrespect to BH the restriction to such f is natural, since the integral asfunction of the boundary has this property again.)

6. Postscript

In Part II of the paper our (stochastic) integral is studied in moredetail and further extended. In particular, we establish relationships toforward integrals existing in the literature. A pathwise approach tocertain (anticipative) SDE with random coe�cients by means of theItoÃ formula is presented. In the special case of adapted processes itagrees with known results.

References

[1] Ciesielski, Z., Kerkyacharian, G., Roynette, R.: Quelques espaces fonctionnels

associeÂ s aÂ des processus gaussiens, Studia Math. 107, 171±204 (1993)[2] Hunt, G.A.: Random Fourier Transforms, Trans. Amer. Math. Soc. 71, 38±69

(1951)[3] ItoÃ , K.: Multiple Wiener integral, J. Math. Soc. Japan 3, 157±169 (1951)

[4] Kahane, J.-P.: Some Random Series of Functions, 2nd edition, CambridgeUniversity Press 1985

[5] Mandelbrot, B.B., van Ness, J.W.: Fractional Brownian Motions, Fractional

Noises and Applications, SIAM Review 10, 422±437 (1968)[6] Nualart, D.: Non causal stochastic integrals and calculus, in: Stochastic Analysis

and Related Topics, H. Korezlioglu and A. S. Ustunel Eds., Lecture Notes in

Math. 1316, 80±129 (1986)[7] Nikol'ski, S.M. (Ed.): Analysis III ± Spaces of di�erentiable functions, Encyclop.

Math. Sciences 26 Springer 1991

[8] Nualart, D., and Pardoux, E.: Stochastic calculus with anticipating integrands,Probab. Th. Rel. Fields 73, 535±581 (1988)

[9] Pardoux, E.: Application of anticipating stochastic calculus to stochasticdi�erential equations, in: Stochastic Analysis and Related Topics II, H.

Korezlioglu and A. S. Ustunel Eds., Lecture Notes in Math. 1444, 63±105 (1988)[10] Protter, P.: Stochastic Integration and Di�erential Equations, Springer 1992.


[11] Samko, S.G., Kilbas, A.A., and Marichev, O.I.: Fractional Integrals and

Derivatives. Theory and Applications, Gordon and Breach 1993.[12] Skorohod, A.V.: On a generalization of a stochastic integral, Theory Probab.

Appl. 20, 219±233 (1975)

[13] Winkler, G., WeizsaÈ cker, H.v.: Stochastic Integrals, Vieweg, Advanced Lecturesin Mathematics, 1990

[14] ZaÈ hle, M.: Fractional di�erentiation in the self-a�ne case V ± The fractional

degree of di�erentiability, Math. Nachr. 185, 279±306 (1997)

374 M. ZaÈ hle

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