Integration with respect to fractal functions and stochastic …¤hle... · 2011. 11. 23. · for fractional integrals and Weyl derivatives. In the special case of Ho¨lder continuous

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 ba

f x dgx :Z b

af x dl1x ÿ

Z ba

f x dl2x 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 ba

f x dgx : limD!0

Xi

f x�i gxi1 ÿ gxi 2

where convergence holds uniformly in all ®nite partitions PD :fa : x0 � x�0 � x1 � � � � � xn � x�n � xn1 bg with maxijxi1xij < 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 ba

f dg Z b

af xg0x dx 3

and

334 M. ZaÈ hle

(L-S)

Z ba

f dg ÿZ b

af 0xgx dx f bÿgbÿ ÿ f aga 4

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

gbÿ : limd&0gbÿ 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

fax : 1a;bxf x ÿ f a 5

gbÿx : 1a;bxgx ÿ gbÿ 6and de®ne the integral byZ b

af dg ÿ1a

Z ba

DaafaxD1ÿabÿ gbÿx dx

f agbÿ ÿ gafor certain 0 � a � 1 provided that f and g satisfy some fractionaldierentiability conditions in Lp-spaces, where ÿ1a eipa. (In thecase of real-valued g the function ÿ1aD1ÿabÿ gbÿx 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 dierentiability) 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 dierential 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 dieren-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 ÿ11=2ÿ�=2Z b

aD1=2ÿ�=2a f xD1=2ÿ�=2bÿ Wbÿx dx

to the ItoÃ integral If as �& 0. Sucient 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

AZ 10

f dW X1n0

eIn1f n n Z 10

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 Wa2;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 2Wa2; for any a < 1=2 and

the above integrals I1ÿ�f converge in the mean square then the limitagrees with A R 10 f dW . A sucient condition for this convergence isthat f lies additionally in the space L1;2C which is introduced in thetheory of Skorohod integrals df . For such f we obtain

AZ 10

f dW df Z 10

Dtf tÿ dt

336 M. ZaÈ hle

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

AZ 10

cf dW c AZ 10

f dW

for random constants c.

1. Fractional integrals and derivatives

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

a jf xjp dx�1=p

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

Iaaf x :1

CaZ x

axÿ yaÿ1f y dy 7

Iabÿf x :ÿ1ÿaCa

Z bxy ÿ xaÿ1f 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 formulaIaabÿIbabÿ

f Iababÿ

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 xIaagx dx ÿ1a

Z ba

gxIabÿf x dx : 10

Fractional dierentiation may be introduced as an inverse operation.For our purposes it is sucient 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 IaabÿLp be the class of functions f which may be

represented as an Iaabÿ-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 yxÿ ya1 dy

Z bx�

f x ÿ f yy ÿ xa1 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 IaaLp IabÿLp � Lq with 1=q 1=p ÿ a. If ap > 1 any f 2 IaabÿLp is HoÈ ldercontinuous of order aÿ 1=p on the interval a; b.

It can be shown that the function u in the above representationf Iaa

bÿu 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

Daaf x : 1a;bx1

C1ÿ addx

Z xa

f yxÿ ya dy 11�

Dabÿf x : 1a;bxÿ11aC1ÿ a

ddx

Z bx

f yy ÿ xa dy

�: 12

The corresponding Weyl representation reads

Daaf x 1

C1ÿ af xxÿ aa a

Z xa

f x ÿ f yxÿ ya1 dy

!1a;bx 13

�Dabÿf x

ÿ1aC1ÿ a

f xbÿ xa a

Z bx

f x ÿ f yy ÿ xa1 dy

!1a;bx

�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

Daf x :a

C1ÿ aZ 10

f x ÿ f xÿ yya1

dy 130

Daÿf x :aÿ1a

C1ÿ aZ 10

f x ÿ f x yya1

dy : 140

Since f vanishes outside a; b we obtainDaabÿ

f x 1a;bxDaÿf x :Note that in the literature the factor ÿ1a 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 IaabÿLp,

IaabÿDaa

bÿf f : 15

We also haveDaabÿIaabÿ

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

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

lima!1

Daabÿ

f x f 0x: 17Here the relationship

lim�!0

I�abÿ

hx hx ±± 18is used which holds for arbitrary h 2 L1 at all points x 2 a; b wherethe left (right) limit, hxÿhx 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 dierentiable in theLp-sense.

Furthermore, (11) and (12) imply

lima!0

Daabÿ

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

D0abÿ

f x f x; I0abÿLp Lp; and D1abÿf x f

0xif the latter derivative exists.The following two formulas play an essential role for our integrationconcept:

DaabÿDba

bÿf Daba

bÿf 20

if f 2 IababÿL1; a � 0; b � 0; a b � 1 (second composition formula),

ÿ1aZ b

aDaaf x gx dx

Z ba

f xDabÿgx dx 21

provided that 0 � a � 1, f 2 IaaLp, g 2 IabÿLq, 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 gbÿ 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 ba

f x dgx ÿ1aZ b

aDaafaxD1ÿabÿ gbÿx dx

f agbÿ ÿ ga 22provided that fa 2 IaaLp, 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

ÿ1a0Z b

aDa

0afaxD1ÿa

0bÿ gbÿx dx

(20) ÿ1aÿ1bZ b

aDbaDaafaxD1ÿabbÿ gbÿx dx

(21) ÿ1aZ b

aDaafaxDbbÿD1ÿabbÿ gbÿx dx

(20) ÿ1aZ b

aDaafaxD1ÿabÿ gbÿx dx :

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

Remark. For ap < 1 we have fa 2 IaaLp i f 2 IaaLp and f aexists. In this case the derivatives satisfy the relation

Daafax Daaf ÿ f a1a;bx

Daaf x ÿ1

C1ÿ af axÿ aa 1a;bx

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

f x dgx ÿ1aZ b

aDaaf xD1ÿabÿ gbÿx dx : 220

340 M. ZaÈ hle

which is determined for general f 2 IaaLp bounded in a. For a 0and a 1 the integral (22) may be transformed intoZ b

af x dgx

Z ba

f xg0x dx 23

andZ ba

f x dgx ÿZ b

af 0xgx dx f bÿgbÿ ÿ f aga

24which 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 1c;dx dgx gd ÿ gc(ii)R b

a 1c;bx dgx gbÿ ÿ gc .Note that on the right hand side gc has to be replaced by ga ifc a.Proof. The fractional derivatives of the function 1c;dx may be cal-culated by means of (130):

Daa1c;dx 1a;bxa

C1ÿ aZ 10

1c;dx ÿ 1c;dxÿ yya1

dy

aC1ÿ a

"1c;dx

Z 10

1ÿ 1c;dxÿ yya1

dy

ÿ 1a;bnc;dxZ 10

1c;dxÿ yya1

dy�

1C1ÿ a

�1c;dxa

Z 1xÿc

1

ya1dy

ÿ 1d;bxZ xÿc

xÿd

1

ya1dy�:

Thus,


Daa 1c;dx 1

C1ÿ a 1c;bx1

xÿ ca ÿ 1d;bx1

xÿ da� �

:

25Similarly,

Daa 1c;bx 1

C1ÿ a 1c;bx1

xÿ ca : 26

Taking the Iaa-integral of the right-hand side we can see that1c;d 2 IaaLp i ap < 1. Further, if k is the HoÈ lder exponent of thefunction g then gbÿ lies in I�bÿLq for any q and � < k. Hence, theconditions of (22) are ful®lled for arbitrary a 1ÿ �, � < k, i.e.,Z b

a1c;dx dgx ÿ11ÿ�

Z ba

D1ÿ�a 1c;dx D�bÿgbÿx dx

ÿ11ÿ� 1C�

�Z bÿc0

x�ÿ1D�bÿgbÿc x dx

ÿZ bÿd0

x�ÿ1D�bÿgbÿd x dx�

ÿ11ÿ�ÿ1��

I�bÿD�bÿgbÿc ÿ I�bÿD�bÿgbÿd�

ÿ gc ÿ gd 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 :Pnÿ1i0 fi1xi;xi1 fn1xn;b for some complex values fi.2.3. Corollary. If g is HoÈlder continuous on a; b we haveZ b

afPx dgx

Xnÿ1i0

fi gxi1 ÿ gxi fn gbÿ ÿ gxn :

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 IaajDaafajxldx

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 Daafa is an Lp-function.

Let k (or kÿ) be a nonnegative smooth function vanishing outside0; 1 (or ÿ1; 0) such that R 10 kx dx 1 (or R 0ÿ1 kÿx dx 1. By

kÿN x : N kÿNx 27

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

For the convolution fN : fa � kN we obtainDaafN 1a;bDaafa � kN : 28

The right-hand side converges in Lp to Daafa as N !1. Further,fN a 0. Hence, by the HoÈ lder inequality we obtain

ÿ1aZ b

aDaafaxD1ÿabÿ gbÿx dx

limN!1ÿ1a

Z ba

DaafN xD1ÿabÿ gbÿx dx

limN!1ÿZ b

af 0N xgbÿx dx

limN!1L-S

Z ba

fN x dgbÿx

limN!1L-S

Z ba

fN x dgx :

The right-sided continuity of f at x yields

limN!1

fa � kN x fax :

Therefore Lebesgue's bounded convergence theorem implies

limN!1L-S

Z ba

fa � kN x dgx

L-SZ b

afax dgx

L-SZ b

af x dgx ÿ f agbÿ ÿ ga

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 b

a 1x;yf dgif for both the integrals the conditions of de®nition (22) are ful-®lled.

(ii)R y

x f dgR z

y fdg R z

x fdgÿ f ygy ÿ gyÿif 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 1a;bD1ÿabÿ gbÿ � kÿN 29and gNbÿ 0. Then we obtain by Lq convergence for x > a (the casex a is similar)Z b

a1x;yf dg ÿ1a

Z ba

Daa1x;yf uD1ÿabÿ gbÿu du

limN!1ÿ1a

Z ba

Daa1x;yf uD1ÿabÿ gNu du

limN!1

Z ba1x;yuf ug0N u du

limN!1

Z yx

f u gbÿ � kÿN 0u du

limN!1

Z yx

f u gyÿ � kÿN 0u 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 somea0; p0; q0. Therefore we may continue the above equations bylim

N!1

Z yx

Da0

xfxu D1ÿa0

yÿ gyÿ � kÿN u du f xgyÿ ÿ gx

Z y

xDa

0xfxuD1ÿa

0yÿ gyÿu du f xgyÿ ÿ gx

Z yx

f dg :


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

xf dg

Z zy

f dgÿ f xgyÿ ÿ gx ÿ f ygzÿ ÿ gy

limN!1

Z yx

fxu g � kÿN 0u duZ z

yfyu g � kÿN 0u du

� � lim

N!1

Z zx

fxu g � kÿN 0u duÿ�

f x ÿ f yZ z

yg � kÿN u du

�Z z

xf dgÿf xgzÿÿgxÿf xÿf ygzÿÿgyÿ :

(

Remark. For ap < 1, f 2 IaaLp being bounded in x and y,gbÿ 2 I1ÿabÿ (where g is continuous), ga existing, 1p 1q � 1 we getsimilarly Z y

xf dg

Z zy

f dg Z z

xf dg

by means of 220.

3. Backward integrals and integration by parts

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

f dg ÿ1aZ b

aDaafaxD1ÿabÿ gbÿx dx f agbÿ ÿ ga

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

adgx f x : ÿ1ÿa0

Z ba

Da0

bÿfbÿxD1ÿa0

a gax dx f bÿgbÿ ÿ ga 30

if fbÿ 2 Ia0bÿLp0 , ga 2 I1ÿa0

a 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 b

a dg f .

(ii)R b

a f dg f bÿgbÿ ÿ f aga ÿR b

a 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 0x dx

R ba dg f , i.e.,

(i). (ii) is a consequence, since by de®nition,Z ba

f dg ÿZ b

adf g f agbÿ ÿ ga

gbÿf bÿ ÿ f a

ÿZ b

ag df f bÿgbÿ f bÿgbÿ ÿ f aga :

Remark. Let Hk Hka; 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 :Xni0

f xi1xi;xi1

in the following sense.

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

(i) limD!0 supPD kefPD ÿ f kL1a;b 0(ii) limD!0 supPD kDaabÿfPD abÿ ÿ D

aabÿ

f abÿkL1a;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 Hk be the HoÈ lder constant of f . By de®nition,

C1ÿ ajDaaefPDax ÿ DaafaxjefPDx ÿ f xxÿ aa a

Z xa

efPDx ÿ f x ÿ efPDy ÿ f yxÿ ya1 dy

�� :

The L1-norm of the ®rst summand of the last sum may be estimated byHkbÿ a1ÿaDk.

For x 2 xi; xi1 the second summand, say SPDx, may be splittedinto

aXiÿ1k0

Z xk1xk


aZ x

xi


aXiÿ1k0

Z xk1xk

f xi ÿ f x ÿ f xk ÿ f yxÿ ya1 dy

aZ x

xi

f xi ÿ f x ÿ f xi ÿ f yxÿ ya1 dy :

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

jSPDxj � HkXni0

1xi;xi1x�xÿ xika

Z xia

1

xÿ ya1 dy

aXiÿ1k0xk1 ÿ xkk

Z xk1xk

1

xÿ ya1 dy

aZ x

xi

xÿ ykxÿ ya1 dy

�

� HkXni0

1xi;xi1x�xÿ xikÿa a

Xiÿ1k0xk1 ÿ xkk

�Z xk1

xk

1

xÿ ya1 dy a

kÿ a xÿ xikÿa�:

Hence,

348 M. ZaÈ hle

kSPDkL1 � Hk�

kkÿ a

Xni0

Z xi1xixÿ xikÿa dx

aXni0

Xiÿ1k0xk1 ÿ xkk

Z xi1xi

Z xk1xk

1

xÿ ya1 dy dx�

Hk�

kkÿ a

1

kÿ a 1Xni0xi1 ÿ xikÿa1

Xnÿ1k0xk1 ÿ xkk

Z xk1xk

Z bxk1

1

xÿ ya1 dx dy�

� Hk kkÿ a

1

kÿ a 11

1ÿ a� �Xn

i0xi1 ÿ xikÿa1

� Hk kkÿ a

1

kÿ a 11

1ÿ a� �

bÿ aDkÿa

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 and efPD be the step functions used in (2) and Theorem4.1.1, respectively. We estimate the dierence of their Riemann±Stieltjes sums by

supPD

Xni0

f x�i gxi1 ÿ gxi ÿXni0

f xigxi1 ÿ gxi��

�� sup

PD

Xni0jf x�i ÿ f xijjgxi1 ÿ gxij

� HkHl supPD

Xni0xi1 ÿ xikl

� HkHlbÿ aDklÿ1

where Hk and Hl 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

Pni0 f xigxi1 ÿ gxi 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 ÿ1a Z ba

DaaefPDaxD1ÿabÿ gbÿx dx f agbÿ ÿ ga

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

ÿ1aZ b

aDaafaxD1ÿabÿ gbÿx dx f agbÿ ÿ ga

Z ba

f 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 0f x df xof classical real dierentiation 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 Hka; b and F 2 C1R be real-valued func-tions such that F 0 � f 2 Hla; b for some k l > 1. Then we have forany y 2 a; b

F f y ÿ F f a Z y

aF 0f 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 Xni0

F f xi1 ÿ F f xi

Xni0

F 0f exif xi1 ÿ f xi

350 M. ZaÈ hle

for some exi 2 xi; xi1. The last expression tends to R ya F 0f x df x asD! 0 by Theorem 4.2.1. (Remark. The conditions of this theorem are satis®ed if f 2 Hka; bfor 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 C1R� a; band F 01f �; � 2 Hla; b; k l > 1, reads

F f y; y ÿ F f a; a Z y

aF 01f x; x df x

Z ya

F 02f x; x dx31

where F 01 and F02 are the partial derivatives of F with respect to the ®rst

and second variable, respectively. The proof is left to the reader.

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

af x df x 1

2f y2 ÿ f a2 : 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 ÿ1aR y

x DaxfxuD1ÿayÿ gyu du f xgy ÿ gx

if 1ÿ l < a < k, a < x < y < b, and the derivatives in the last integralare bounded we may estimate j R yx f dgj � consty ÿ x consty ÿ xland 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

1a;b

Z �a

f dg 2 Hla; b and 1a;bZ b�

f dg 2 Hla; b :

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

ux :Z x

ahy dgy 1a;bx


and

wx : ÿZ b

xhy dgy 1a;bx

as functions from Hla; b.4.4.2. Theorem. Under the above conditions we haveZ b

af xhx dgx

Z ba

f x dux Z b

af x dwx :

Proof. For the step functions efPDx Pni0 f xi1xi;xi1x we getfrom Corollary 2.3 and additivity of the integralZ b

a

efPDx dux Xni0

f xiuxi1 ÿ uxi

Xni0

f xiZ xi1

xihy dgy

Xni0

Z xi1xi

efPDyhy dgyZ b

a

efPDyhy dgy :Theorem 4.2.1 implies

limD!0

Z ba

efPDx dux Z ba

f x dux :

In order to show

limD!0

Z ba

efPDyhy dgy Z ba

f yhy dgy

recall that according to the proof of Theorem 4.2.1Z ba

efPDyhy dgy ÿ Z ba

efPDyehPDy dgy�� const Dkÿaand

limD!0

Z ba

efPDyehPDy dgy Z ba

f yhy dgy :

Hence,

352 M. ZaÈ hle

Z ba

f x udx Z b

af yhy dgy :

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 dierential equations are treated.

An important problem, e.g., in ®nance mathematics is to determinesimilar dierential 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

EBH t s ÿ BH t2 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 Hurstcoecient 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; bequals

limD!0

Xi

BH ti1 ÿ BH ti2 1 if H < 1=2bÿ a if H 1=20 if H > 1=2

8

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 Ia0L10; 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 Hk0; 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 rX t; t for some real-valuedLipschitz function r and any random function X whose sample pathslie in Hk0; T with probability 1. For H > 1=2 this makes it possibleto investigate (stochastic) dierential equations.

Example. Consider the linear equation

dX t a X t dBHt b X t dt 34which means

X t X 0 aZ t0

X s dBH s bZ t0

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 < jtj < C and sgnY t sgnY 0 for 0 < t � �with suciently 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 tj if t 2 0; � and obtain ln jY tj ÿ ln jY 0j aBt 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

2BH y2 ÿ BH x2; 0 � x < y 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 Ia0L2

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 L20; T with probability 1the classical ItoÃ integral

Itf (Itô )Z t0

f dW

is determined. We write If IT f .

5.2.1. Theorem. If f is adapted and f 2 Ia0L2 with probability 1 forsome a > 1=2 then the integrals

R 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 t0

f 0sW s ds f tW t :

Arbitrary realizations f 2 Ia0L2 will again be approximated by thesmooth functions fN f � kN with the smoothing kernels kN given by(27). Using that W 2 I1ÿatÿ L2 with probability 1 we obtain from theproof of Theorem 2.4

limN!1

Z t0

fN dW Z t0

f dW ; 0 < t < T ;


with probability 1. On the other hand the almost sure L20; 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 dierential 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 dierentiability'' 1=2 by ourintegrals (22) for some regularization of f :

5.2.2. Corollary. Let f be an adapted random function such thatf 2 Ia0L2 for any a < 1=2 with probability 1. Then we have the fol-lowing convergence in probability:

lim�&0

Z t0

I�0f dW lim�&0ÿ11=2

Z t0

D1=2ÿ�=20 f sD1=2ÿ�=2tÿ Wtÿs ds

(Itô )Z t0

f dW :

Proof. First note that I�0f 0 I �0f 2 Ia0L2 for any 1=2 < a <1=2 �. Therefore R t0 I�0f dW is determined by

ÿ11=2ÿ�=2Z t0

D1=2�=20 I�0f sD1=2ÿ�=2tÿ Wtÿs ds

ÿ11=2ÿ�=2Z t0

D1=2ÿ�=20 f sD1=2ÿ�=2tÿ Wtÿs ds :According to Theorem 5.2.1 we haveZ t

0

I�0f dW ItI�0f with probability 1 for any � > 0. The almost sure L20; t-convergenceof I�0f to f as �& 0 implies the convergence in probability of the ItoÃintegrals ItI�0f to Itf . (

Next we will state a sucient condition for the above convergencein terms of square means. For, we introduce the classes Ia0L2 ofmeasurable random functions f such that

Ef 02

h�t;x :Z tÿ�0

f0t;x ÿ f0s;xt ÿ sa1 ds 39

converge in L2 : L20; T � X, L� P as �& 0.Note that (37) and (38) imply

E

Z T0

f t2 dt

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

E

Z T0

X ttÿ2a dt Z T0

tÿ2aEZ t0

f s2 ds dt

EZ T0

f s2Z T

stÿ2a dt ds

� const EZ T0

f t2 dt

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 L20; 1n arising in the ItoÃ -Wiener chaos expansion. Theiterated ItoÃ integral of such an f is given by

Inf : (Itô )Z 10

Z tn0

� � �Z t20

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 : 1n!Xp2Sn

f tp1; . . . ; tpn

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

EeImfeIng n! R0;1n ef tegtLndt if m n0; else :

�43

For 0 < a < 1 let eIa0n;L2 be the class of those functions f fromL20; 1n1 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; st ÿ sa1 ds 44

converge in L20; 1n1 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 eIa0n; L2 is exactly the class of those functions on


0; 1n1 with the required symmetry which are representable as Ia0-integral with respect to the last variable of some L20; 1n1-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

f0t1; . . . ; tn; t : 10;1tf t1; . . . ; tn; t ÿ f t1; . . . ; tn; 0assuming everywhere that the right-sided limit exists.

For f0 2 eIa0n;L2 the symmetric multiple ItoÃ integralseInÿ1f �; s; t, eInf �; t, and eIn1f make sense because of the L2-prop-erties. In view of the isometry (43) the random functionseInf �; t0 eInf0�; t are elements of the class Ia0L2 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 10

eInf �; t dW t eIn1f n Z 10

eInÿ1f �; t; t dtwith probability 1 if f0 2 eIa0n; L2 for some a > 1=2.Proof. By de®nition we have with probability 1Z 1

0

eInf �; t dW t ÿ1a Z 10

Da0eInf0�; �tD1ÿa1ÿ W1ÿt dteInf �; 0W 1 :

As before, we approximate f by functions being smooth in the lastargument

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

l.i.m.N!1

Z 10

eInfN�; t dW t Z 10

eInf �; t dW t(cf. Section 2) and by (43)

l.i.m.N!1

eIn1fN eIn1fand similar estimates as in the proof of Theorem 3.6 in [11] yield

360 M. ZaÈ hle

l.i.m.N!1

Z 10

eInÿ1fN �; t; t dt Z 10

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

Xnj0

X0�i1

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; tj1; . . . ; tn;

ijk

� �W

ij 1k

� �ÿ W ij

k

� �� 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 10

eInÿ1fN �; t; t dt :(

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

f t X1n0eInf n�; t 45

(where eI0f 0�; t Ef t) for unique f n 2 L20; 1n1 being sym-metric in the ®rst n-arguments. Recall that the Skorohod integral of fexists and is given by

df SZ 10

f dW :X1n0eIn1f n 46

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

0

f � dW :X1n0

eIn1f n n2

Z 10

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:

AZ 10

f dW :X1n0

eIn1f n nZ 10

eInÿ1f n�; t; tÿ dt� � 48provided that the series converges in the mean square, where

362 M. ZaÈ hle

Z 10

eInÿ1f n�; t;ÿt dt l.i.m.�&0

Z 10

eInÿ1I�0f n�; t; t dt :(The corresponding expression for t in (47) is de®ned similarly.)

If the f n (up to L20; 1nÿ1-equivalence) have no jumps at Le-besgue-a.a. points on the diagonal given by the last two arguments weget

AZ 10

f dW Z 10

f � dW :

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

AZ 10

f dW df (Itô )Z 10

f dW

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

In general, the three integrals are dierent 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-typespacesWa2; Wa2;0; 1 of measurable random functions f such that

E f 02

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

lim�&0

sup�0

const �2aÿbÿd�Z 1

0

. . .

Z 10

f nt1; . . . ; tn; t ÿ fnt1; . . . ; tn; 02t2a

dt1 . . . dtn dt

Z 10

� � �Z 10

f nt1; . . . ; tn; t ÿ fnt1; . . . ; tn; s2jt ÿ sj2a1 dt1 . . . dtn ds dt

�:

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

n!Z 10

� � �Z 10

f nt1; . . . ; tn; t ÿ f nt1; . . . ; tn; 02t2a

dt1 . . . dtn dt

Z 10

tÿ2aEeInf n�; t ÿ f n�; 02 dt�Z 10

Ef t ÿ f 02t2a

dt ;

since f t ÿ f 0 P1n0eInf n�; t ÿ f n�; 0 and hence,Ef t ÿ f 02 P1n0 EInf n�; t ÿ f n�; 02 by orthogonali-ty. In view of (50) the last integral is ®nite. Similarly one shows that

n!Z 10

� � �Z 10

f nt1; . . . ; tn; t ÿ f nt1; . . . ; tn; s2jt ÿ sj2a1 dt1 . . . dtn ds dt

�Z 10

Z 10

Ef t ÿ f s2jt ÿ sj2a1 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 anticipatingintegral 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 10

f dW ÿ1bZ 10

Db0f0tD1ÿb1ÿ W1t dt f 0W 1 :

Recall that


f0t X1n0eInf n0�; t; f 0 X1

n0eInf n�; 0

and therefore,

AZ 10

f dW AZ 10

f0 dW f 0W 1 :

If we can show that

Db0f0t X1n0

Db0eInf n0�; �tin the sense of L2-convergence of the series then the Cauchy±Schwarzinequality leads toZ 1

0

f0 dW X1n0

Z 10

eInf n0�; t dW tso that Proposition 5.3.3 and Theorem 5.3.1 yield the assertion.

By construction, for any h 2 Ib0L2 the derivative Db0h0 is theL2-limit of the random functions

Db0;�h0t :1

C1ÿ bht ÿ h0

tb b

Z tÿ�0

h0t ÿ h0st ÿ sb1 ds

!as �& 0. For the special functions

hN t :XNn0eInf n0�; t

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

0

EDb0;�hN t ÿ Db0;�0hN t2 dt

Z 10

XNn0

EDb0;�eInf n0�; �t ÿ Db0;�0eInf n0�; �t2 dt�Z 10

X1n0

EDb0;�eInf n0�; �t ÿ Db0;�0eInf n0�; �t2 dt :Moreover, for any � > 0,

Db0;�f0 X1n0

Db0;�eInf n0�; �

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

EDb0;�f0t ÿ Db0;�0f0t2 dt :

In view of Proposition 5.3.2 the function f0 is an element of Ib0L2.

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

b0;�hN t as �& 0 is uniform in

N . Thus we may change the order of the L2-limits and obtain

Db0f0 lim�&0

limN!1

Db0;�hN limN!1 lim�&0 Db0;�hN

limN!1

XNn0

Db0eInf n0�; � X1n0

Db0eInf n0�; � :Finally, the equality

AZ 10

f dW Z 10

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 dierential equations. In order to extendCorollary 5.2.3 to anticipating f we introduce the class

W1=2ÿ2; :

\0

Z 10

f 1t; tÿ dt� �2

X1n0n 1!

ef n n 2

Z 10

f n2�; t; tÿ dt

2

L20;1n154

in view of the isometry property (43).

5.3.5. Theorem. Suppose that f 2W1=2ÿ2; andR 10 I

�0f 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 10

I �0f dW ÿ AZ 10

f dW� �2

Z 10

I �0f 1t; tÿ ÿ f 1t; tÿ dt� �2

X1n0n 1!

I�0ef n ÿ ef n n 2

Z 10

I �0f n2�; t; tÿ ÿ f n2�; t; tÿ

2

L20;1n1

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

l.i.m.�&0

Z 10

I�0f dW AZ 10

f dW :

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

I�0f X1n0

I�0eInf n�; � :Then the isometry (43) yields the ItoÃ ±Wiener chaos expansion

I�0f X1n0eInI�0f n�; � :

SinceR 10 I

�0f dW A

R 10 I

�0f dW it is enough to prove that

l.i.m.�&0

X1n0

eIn1I �0f n nZ 10

eInÿ1I �0f n�; t; t dt� �X1n0

eIn1f n nZ 10

eInÿ1f n�; t; tÿ dt� � :

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�0f1t; t dt

X1n0

eIn1I �0f n n 2Z 10

eIn1I �0f n2�; 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 10

I�0f1t; t dt

X1n0

l.i.m.�&0

eIn1I�0f n n 2Z 10

eIn1I �0f n2�; t; t dt!

Z 10

f 1t; tÿ dt X1n0

eIn1f n n 2Z 10

eIn1f n2�; 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

X1n0

eIn1f n n Z 10

eInÿ1f n�; t; tÿ dt� � : (5.3.6. Corollary. Under the conditions of Theorem 5.3.5 we have

AZ 10

cf dW c AZ 10

f dW

for any bounded random variable c.

Proof. The de®nition (22) impliesZ 10

I�0cf dW cZ 10

I �0f 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 P1n0eInf n�; s for ®xed s the Mall-iavin derivative of this random variable is given by


Dtf s X1n1

n eInÿ1f n�; t; sprovided 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;2C denotes the space of those f 2 L1;2 for which the set of functions

fs! Dtf s; s 2 0; 1nftggt20;1is equicontinuous with values in L2X;P and

ess sups;t20;12

EDtf s2

X1n0

eIn1f n n Z 10

eInÿ1f n�; t; tÿ dt� �and the asserted equation we use the convergenceX1

n0eIn1f n df

and prove thatX1n0

nZ 10

eInÿ1f n�; t; tÿ dt Z 10

Dtf tÿ dt :

Regarding

Dtf tÿ l.i.m.�&0

X1n1

n eInÿ1f n�; t; t ÿ �X1n1

n eInÿ1f n�; t; tÿat 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 dicult to check that f 2 L1;2C implies I�0f 2 L1;2C . Hence,

AZ 10

I �0f dW dI�0f Z 10

DtI �0f tÿ dt :

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

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

l.i.m.�&0

Z 10

I �0Dtf �tÿ dt Z 10

Dtf tÿ dt :

Below we will show that

l.i.m.�&0

dI �0f df :Consequently,

l.i.m.�&0

AZ 10

I �0f dW AZ 10

f dW :

If we additionally assume that f 2W1=2ÿ2; then we may useZ 10

I�0f dW AZ 10

I �0f dW


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

dI�0f X1n0eIn1I �0f n

XNn0eIn1I �0f n X1

nN1eIn1I�0f n :

For ®xed N the ®rst summand tends toPN

n0eIn1f as �& 0 by (43)and the corresponding L2-version of (18). The mean square of thesecond summand does not exceedX1

nN1n 1!kI �0ef nk2 � const X1

nN1n 1!kef nk2

for a certain constant independent of � because

kI�0ef nk2 Z 10

� � �Z 10

Z t0

ef nt1; . . . ; tn; s 1C� t ÿ s�ÿ1 ds� �2

dt dt1 . . . dtn

� constZ 10

� � �Z 10

Z t0

ef nt1; . . . ; tn; s2 1C� t ÿ s�ÿ1ds dt dt1 . . . dtn

� const kef nk2according to the Cauchy±Schwarz inequality.

Since kdf k2 P1n0n 1! kef nk2 we obtain that the secondsummand of the above sum tends to zero as N !1 uniformly in �.Thus,

l.i.m.�&0

dI �0f df : (

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 dierential 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 dierential 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 coecients 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 dierentiable 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 stochasticdierential 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 Dierential 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 dierentiation in the self-ane case V ± The fractional

degree of dierentiability, Math. Nachr. 185, 279±306 (1997)

374 M. ZaÈ hle

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