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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 Itoà 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
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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 Itoà 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 Itoà -Wiener chaos
expansion f P10 eInf n. For adaptedf this integral coincides with
the Itoà 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
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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:
Integration with respect to fractal functions and stochastic
calculus. I 337
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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
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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 with respect to fractal functions and stochastic
calculus. I 339
-
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,
Integration with respect to fractal functions and stochastic
calculus. I 341
-
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 :
Integration with respect to fractal functions and stochastic
calculus. I 345
-
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.
Integration with respect to fractal functions and stochastic
calculus. I 347
-
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
efPDx ÿ f x ÿ efPDy ÿ f yxÿ ya1 dy
aZ x
xi
efPDx ÿ f x ÿ efPDy ÿ f yxÿ ya1 dy
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
Integration with respect to fractal functions and stochastic
calculus. I 349
-
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
Integration with respect to fractal functions and stochastic
calculus. I 351
-
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 theItoà sense. A link between both these approaches
will be established inthe next section.
5.2 A new representation of the Itoà 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 Itoà 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 ;
Integration with respect to fractal functions and stochastic
calculus. I 355
-
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 Itoà -Wiener chaos
expansion of random L2-func-tions (see Itoà [3]).
We ®rst will establish a link between our integrals in the sense
of(22) and the symmetric multiple Itoà -integrals of deterministic
func-tions f 2 L20; 1n arising in the Itoà -Wiener chaos expansion.
Theiterated Itoà 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 Itoà 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
Integration with respect to fractal functions and stochastic
calculus. I 359
-
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 Itoà 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 ItoñWiener
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
Integration with respect to fractal functions and stochastic
calculus. I 365
-
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 Itoà ±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
Itoà ±Wiener chaos expansion f s P1n0eInf n�; s for ®xed s the
Mall-iavin derivative of this random variable is given by
Integration with respect to fractal functions and stochastic
calculus. I 369
-
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
Integration with respect to fractal functions and stochastic
calculus. I 371
-
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 Itoà 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 theItoà 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
associe s a 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] Itoà , 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.
Integration with respect to fractal functions and stochastic
calculus. I 373
-
[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