443
Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 443-452
FUNDAMENTAL THEOREM OF WIENER CALCULUS
CHULL PARK
Dept. of Math. & Stat.
Miami University
Oxford, OH 45056
DAVID SKOUG
Dept. of Math. & Stat.
University of Nebraska
Lincoln, NE 68588-0323
LAWRENCE SMOLOWITZ
Dept. of Math. & Comp. Sci.
Hobert & William Smith College
Geneva, NY 14456
(Received April 12, 1989)
ABSTRACT. In this paper we define and develop a theory of differentiation in Wiener
C[0,T]. We then proceed to establish a fundamental theorem of the integral calculus
for C[0,T]. First of all, we show that the derivative of the indefinite Wiener integral exists
space
and equals the integrand functional.
C[0,T]
Secondly, we show that certain functionals defined on
are equal to the indefinite integral of their Wiener derivative.
KEY WORDS AND PHRASES. Wiener (measure, integral, derivative, absolute continuity),
Lebesgue absolute continuity, fundamental theorem.
1980 MATHEMATICS SUBJECT CLASSIFICATION.
1.
28C20, 46G05.
INTRODUCTION.
Consider the Wiener measure space
(C[0,T],
5
mw)
where
C[0,T]
is the space of
all continuous functions x on [0,T] vanishing at the origin. For each partition
rn
< n T, let Xr:C[0,T
{tl,...,tn} of [0,T] with 0 O < <
n
defined by Xr(x
be the a-algebra of Borel sets
x(r) (x(tl),...,X(tn)). Let
n be
r
in [R
n.
Then a set of the type
Co PARK, D. SKOUG AND L. SMOLOWITZ
444
{x
is called a Wiener interval
E
C[0,T]
Xr(X
B}
E
_=
(or a Borel cylinder). It
mw(I
n
XI(B), B
is well known that
(1.1)
K(
B
where
II
{ tj-tj-lj=l
2r(tj-tj-1)}-l/2exPI-1/2j=l (r/.i-r/j_l)2 }
n
K(r’lT)
(l,...,r/n),
with
The measure mw is a probability measure defined on
of all Wiener intervals and mw is extended to the Caratheodory extension
the algebra
of 3. Let
r/0
0.
r be the a-algebra generated by the set {XI(B)
n}
B
with r
Then, by the definition of conditional expectation, see Doob [1], Tucker [2] and
fixed.
Yeh
and
(1.2)
[3], for
each Wiener integrable function
ttr(B)
F(x),
F(x)mw(dx)=
E(Flr)mw (dx)
XI(B
x-lr (B)
Y)Px r(d)
E(F(x)]Xr(X)
B
B e 2
n,
PX r(B) mw(x-l(B)), and E(F(x)lXr(X ) is a Lebesgue measurable function
which is unique up to null sets in 1 n. Also, using (1.1) and (1.3) and choosing
where
of
F(x)
1, we see that
PX (d)= K(r,)d
(1.4)
T
or
dP X
r
Next, for each F E
e Rn
K(r,)
d/
LI(C[0,T],mw)
(1.5)
and each partition r of
Fr
C[0,T],
let
(1.6)
E(FIr)
and
F()
Then,
{Fr}
E(F(x)[Xr(X
for almost all x e
(1.7)
for almost all x e
lim
Ilrll 0
E(F(y) lXr(y
x(r))
lim
Ilrll 0
F(x(r))
(1.9)
C[0,T].
For a given partition r
on
(1.8)
F(x)
Fr(X
C[0,T]. Furthermore,
F(x)
Ix(r)]
E(FIXr)(
is a martingale, and by the martingale convergence theorem,
lim
Ix]
)
[0,T] by
rn of
[0,T] and
x e
C[0,T],
define the polygonal function
445
FUNDAMENTAL THEOREM OF WIENER CALCULUS
[x](t)
x(tj_l)
+
Similarly, for each
t-t j_
< < tj,
1,...,n.
tj-tj_ (x(tj)- x(tj_l)), tj_
(r/1,...,rn) E [n, define the polygonal function [i}] of
on
[0,T] by
t-t
j_
r/j_l + tj-tj_ (j- /j_l) tj_ < < t.j,
Then both functions Ix] and ] are continuous on [0,T],
each subinterval [tj_1,tj], and [x](tj) = x(tj) and [](tj)
For x,y e C[0,T], we use the convention:
[](t)
x _< y if and only if
1...,n with
z]0
0.
their graphs are line segments on
r/j
at each
E r.
tj
x(t) _< y(t) for every
[0,T]
< y if and only if x(t) < y(t) for every
e (0,T]
and
x
The main purpose of this paper is to define and develop a theory of differentiation in
Wiener space
on
C[0,T], and then
C[0,T]; namely, that the
to establish a fundamental theorem of the integral calculus
Wiener derivative of the indefinite integral
[ F(y)mw(dY
is
y_<x
F(x), and that
a Wiener absolutely continuous function can be expressed as the indefinite
integral of its Wiener derivative. This study was initiated by Smolowitz [4].
we incorporate some recent results of Park and Skoug
simplify the concepts and results of Smolowitz
2.
[5]
to improve and substantially
[4].
THE WIENER DERIVATIVE.
Our first objective is to define the Wiener derivative
"x y_<xF(y)mw(dY)
for F
[5]
In this paper
LI(C[0,T],mw).
.x(.)
so that
F(x)
We stm’t by quoting the following theorem from Park and Skoug
which plays an important role in this paper.
LI(C[0,T],mw). Then for any Borel set
F(x)mw(dX [ Ex[F(x -[x] + ]]Px (d)
THEOREM A. Let F E
[
#r(B)
B
n,
(2.1)
where
Ex[F(x- Ix] + ])1
[
F(x- Ix] +
CI0,T]
[])mw(dx).
In view of (1.3) and (2.1), we may conclude that
)
E(F(x)IXr(X)
for almost all
terms of an ordinary Wiener integral.
F()
Ex[F(x- [x] + [])]
(2.2)
in n; i.e., we may express the conditional expectation E(FIXr)(
E(FIXr)(
is in
Note that for F e
LI(Nn,PX (d).
T
in
LI(C[0,T],mw)
Also note that for each x e
C[0,T]
and each
446
C. PARK, D. SKOUG AND L. SMOLOWITZ
{tl,...,tn}
x(tl)," ",X(tn).
of
partition r
[0,T]
[0,T], F(x(r))
E(F(y)lXr(y
DEFINITION 1. Let F E
LI(C[0,T],mw).
define the operator
by
x(r)
x(r))
is a function of
For each partition r
{tl,...,tn}
On
if it exists.
derivative of
F(x(r))
.x(r)F(x) 0X(n)..
0X(tl) / K(r,x(r))
Furthermore, if .x(r)F(x) exists for each partition r, then the Wiener
F(x)
of
(2.3)
is defined by
lim
x(r)F(x)
if the limit exists.
Our first theorem is the first half of the fundamental theorem of Wiener calculus.
THEOREM 1. Let F E
LI(C[0,T],mw).
Then
"x y_<xJ F(y)mw(dY)= F(x)
for almost all x e
C[0,T].
PROOF. For x e C[0,T] let G(x) denote the indefinite Wiener integral
[ F(y)mw(dY
G(x)
y_<x
Ix(Y
where
is the indicator function
1
0
Ix(Y)
Then using (1.7), (2.4), (2.2), (1.3),
G()
Ey[Ix(Y)F(Y)]
y(t) _< x(t) for all
e [0,W]
otherwise.
(2.2) and the Fubini theorem, we obtain
E(G(u)[Xr(u)= )
Eu(Ey[Iu(Y)F(y)]lX(u )
Eu[Ey[Iu_[ul + 1 (y) F(y)I]
Eu[ nEy(Iu-[u]+](Y)F(y)’Xr(y) )Pxr(d)
nEu[Ey[Iu_[u] +[]
But
Iu_[u]+](y-[y]+ff]
(y-[y] + ])F(y-[y] + ])]]Px
is zero unless
y(t)- [y](t) +
for all
[](t) _< u(t)- [u](t) + [](t)
e [0,T]. But (2.6) implies that
j
for
r(d)
y(tj)- [y](tj) + [](tj) _< u(tj)- [u](tj)
1,...,n. Hence we can write
+
](tj)
(2.)
FUNDAMENTAL THEOREM OF WIENER CALCULUS
n
()
447
1
Eu[Ey[Iu-[u]+[](Y-[Y]+[])
F(y-[y]+ ])]]K(r,)dl
and so for each x E
C[0,T],
x(t n)
dn
x(t 1)
I
Eu[Ey[Iu-[u]+[x(r)] (y-[y]+ if])
F(y-[y]+ )]]U(r,)dl dn.
Hence
on
c(tn)
G!.x)/ll
Eu[Ey[Iu_[u]+[x(r)](y-[y]+[x(t) ])
F(y-[y]+[x(r)])]]U(r,x(r))
x(r),
Eu,y(iu(Y)F(y)[Xv(y
Applying (1.9) to (2.7) yields
(G(x))
aim
I111o
for almost all x in
C[0,T]
LI(C[0,T],mw)
x(r)
(2.8)
x(v))
which ncludes the proof of Threm 1.
COROLLARY 1. If
is in
_
x(r),Xr(U
Eu,y(Iu(Y)F(y)lXr(Y
F(x)
x(r))S(r,x(r)).
Xv(u
{t;,...,t}
{tl,...,tn}
r
and if F(y):
f(Y(tl),...,Y(tm)
then
and
x y_xF(y)mw(dy
C[0,T].
PROOF. Using (2.7)
Xr(U)
F(X)Su,y(Iu(Y)lXr(Y)
F(y)mw(dY)
y_<x
F(x)=
x(r))
f(x(tl) ,. ..,x(t))
for almost all x in
x(r)
y_x
and
F(Y)mw(dY)
Under the conditioning
Xr(Y
(2.4)
we see that
Eu,y(Iu(Y)F(Y)IXr(Y)
x(r), F(y) becomes
x(r),
Xr(u
x(r)).
f(x(t[),...,x(t)) which equals F(x).
Therefore,
As
x(r) F(y)mw(dY)= F(X)Eu,y(Iu(Y)lXr(Y) Xr(U) x(r)).
y_<x
by (1.9) for almost all x in
Ilrll 0, Eu,y(Iu(Y)lXr(y Xr(u x(r)) Ix(X
C[0,T].
Thus Corollary 1 is established.
COROLLARY 2. Let
F(x)
r"
{t,’",t}
f(x(tl),...,x(t)) be in Ll(C[0,Wl,mw).
be any partition of
Then
.xF(X)
[0,W],
0.
and let
C. PARK, D. SKOUG AND L. SMOLOWITZ
448
PROOF. Let r be a partition of [0,T] properly containing
.x(r) F(x)
Thus
(x(r))
E(Fly)lXr(Y
0, and so
xF(X)
x(r))=
Then
r’.
f(x(t),...,x(t)).
0.
LEBESGUE AND WIENER ABSOLUTE CONTINUITY.
3.
In this section we show that certain functions defined on C[0,T] are equal to the
indefinite integral of their Wiener derivative.
For
V(,I,k)
(al,...,an)
and
I
(bl,...,bn)
IR n with a <
be the collection of all points of the form
a or b and exactly k of the v are a ’s.
k
in
1,...,n, let
bi,
(Vl,...,Vn)
where each v is either
For any function f defined on
V(,t,k)
for
0,1,...,n, let
n
A
,I
f
A function of n variables
f(l) + E (-1) k E
f()
k=l
eV(,,k)
(3.1)
f(ul,...,Un)
the sense of Vitali (see
is said to be Lebesgue absolutely continuous in
Clarkson and Adams [6,7] and Hobson [8]) on the region f [Rn if,
given e > 0, there is a
a
(a
> 0 such that if k
i=l
N
y"
k=l
[Ag(k),(k)f[
(k)
<
’ where mL(.
(a{k),...,a(nk)).
A function
1,2,...
are disjoint
< a for any N, then
k=l
denotes n-dimensional Lebesgue measure, and
n-dimensional rectangles contained in 2 with
N
.
k
g
mL(Ik)
f(ul,...,Un)is
said to be Lebesgue absolutely
(in the sense of Hardy-Krause; see Berkson and Gillespie [9], and Clarkson and
Adams [6,7]) on a region f c IRn if for each k
1,...,n-l, whenever n-k variables are
continuous
fixed then f, as a function of its remaining k variables, is Lebesgue absolutely continuous in
the sense of Vitali on f
R k. When we merely state "Lebesgue absolutely continuous", it
is always meant in the sense of Hardy-Krause.
It is well known that if
n
Vitali on R
on R.
[ai,bi],
i=l
Furthermore
I
then
f(ul,...,Un)
is Lebesgue absolutely continuous in the sense of
0nf(ul,...,Un)/0Ul.--0Un exists a.e.
on R and is integrable
[0nf(ul,...,Un)/0Ul...0Un]dUl...dun
A
10nf(ul,...,Un)/0Ul...0UnldUl...dUn
Var(f,R)
R
,f
f,
(3.2)
and
R
where Var(f,R) denotes the total variation of f over R.
(3.3)
FUNDAMENTAL THEOREM OF WIENER CALCULUS
449
Let G(x) be any Wiener integrable function on C[0,T]. Then, by definition,
G()
E(GIXr)()
is a function of
which is integrable with respect to
PX (d)= K(r,)d.
T
DEFINITION 2. A Wiener integrable function G(x) defined on C[0,T] is said to be
Wiener absolutely continuous provided that for each s > 0 there exists a 5 > 0 such that
_<
a! k) < x(s!k)) _< !k), N
{x e C[0,T]
if the sequence k
a!k) < k) _<
(R)
1,...,m
mw(Ik)
are disjoint Wiener intervals with
k=l
(k)}
with
< 5 for any N, then
N
k=l
The following propositions can be easily established.
PROPOSITION A. If G(x) is Wiener absolutely continuous on C[0,T], then for
partition r of
(0,T], G()
is Lebesgue absolutely continuous on
1 rl, where
rl
ever)’
denotes the
number of points in r.
LI(C[0,T],mw).
PROPOSITION B. Let F
I F(y)mw(dY
y_<x
G(x)
Then the indefinite Wiener integral
C[0,T].
is Wiener absolutely continuous on
Our next theorem is the second half of the fundamental theorem of Wiener Calculus.
THEOREM 2. Let G e
(i)
xG(x)
(ii)
G(x)
(iii) If
satisfy the conditions:
C[0,T] and belongs
absolutely continuous on C[0,T],
sequence in C[0,T] such that Xk(S0)
exists for almost all x e
is Wiener
{Xk}.kl
point s o e
LI(C[0,T],mw)
(0,T],
is a
then
G(Xk)
to
LI(C[0,T],mw),
for some fixed
as k
.
0 as k
Then
J
G(u)
(a.4)
xG(x)mw(dX)
x_<u
for almost all u in
C[0,T].
5(/3) >
PROOF. For given s > 0 let 5
continuity of
G(x), and also
0 be the value for the Wiener absolute
assume that
mw(S)
<
, [ I.xG(x)lmw(dX
< /3.
S
Let {r
(k)} be a sequence of partitions of [0,T] such that IIr(k)ll
im
r(k))G(x)
k x(
By Egoroff’s threm, there exists a set
sitive integer k 0 such that if k
k0,
G(x)
for Mmost M1 x e
C c C[0,T]
then
with
mw(C)
0
k
.
Then
C[0,T].
>
5/2
and a
C. PARK, D. SKOUG AND L. SMOLOWITZ
450
x(r(k)) G(x)
xG(x)
< e/3 for every
Let
{x
Ck
C
Then
E
C[0,T]"
C k, and hence
[x(r(k))G(x
xG(x)l
< /3}, k _> k 0
f/2, and
mw(Ck) >_ mw(C) >
xG(X)lmw(dx)< /3 for
Jcklx(r(k))G(x)
mw(C
_<
k _> k 0
mw(C)
< g/2 for k _> k 0. Next consider fixed k,
k_)k 0 and let q denote the number of points in the partition r(k). Let
for some x C}
x(r
Ek { (}l"’"}q) e q"
The complements satisfy
(k))
Then,
K(r(k),)d
mw(C)
Ek
K(r(k),) is bounded in
Since
K(r(k),)d mw(C
on [R q and
Ek
a countable sequence of disjoint q-dimensional rectangles *
t
,
o0
such that Ek c
< 6/2, we can find
q (a! ) ig)], t= 1,2,...
i=1
ll It’ and
Ek
K(r(k),)d _< t=Z
. K(r(k),)d
Z
t=l
mw(It)
< 6,
t
where
t
{xe C[0,T]"
a!/) < x(s!k))5 ig) for each s!k)e r(k)}
Hence
C
[x(r(k))G(x)[mw(dx)=
/q) / &l" "Or/q d r/1" "dr/q
Ek
,r/q)/O/1 Ot/q dr/l" dr/q
r,
,
Var(G,t)
where the last equality follows from
(3.3). Now,
Var(G,If)
sup
ai,Pi
where the supremum is taken over all possible nets of
If,
and each net has total Lebesgue
FUNDAMENTAL THEOREM OF WIENER CALCULUS
451
measure equal to that of g, and so the corresponding Wiener intervals have total measure
equal to
mw(I).
E
Since
t=1
mw(It)
< (/3), by the Wiener absolute continuity of G, we
have
N
A_ G()I
=1
< s/3 for every N and every net.
ai’Pi
Thus, by taking the supremum over all nets, we get
N
E Var(G,If) <_ s/3 for every N,
=1
.
and hence
_< /3.
x(r(k))G(x)lmw(dx
Thus, for every k >_ k0,
i[0,W]l x(r(k))G(x)
ik
x(r(k)) G(x)
+
<
xG(x)Imw(dX)
"xG(x) Imw(dX)
I ’%(r(k))G(x)lmw(dx) +
[’xG(x)lmw(dx)
E
In particular
]-..(r(k))G(x)- xG(x)lmw(dX
[-]
where
<
for k _>
k0,
corresponds to r(k). Hence
k(R)aim[![x]_[u] %(r(k))G(x)mw(dx)[x]![u] .@xG(x)mw(dx)
Since
{x e C[0,T]
Ix] _< [u]}
{x e C[0,T]
x
_< u} as k
(R),
0.
an application of the
dominated convergence theorem yields
]![_
aim
xG(x)mw(dx
"@xG(x)mw(dx)
x_<u
Thus,
m
!
%( r(k)) G (X)mw(dx)
"xG(x)mw(dX)
x_<u
On the other hand, using (2.3), (3.2) and (3.1), we see that for any a e C[0,T] with
a<u,
452
C. PARK, D. SKOUG AND L. SMOLOWITZ
"x(r(k))G(x)mw(dx)
a(r(k))
<_x(Jk)
<_u( r (k)
x(r(kllG(x)mw dx)
u(sk)) u(sl k))
I I
a(sk)) a(s k))
Aa(r(k)),u(r(k)
a(s k))
s k
for every v E
(R)
-oo
as k
(3.6)
G
(u(Jk))) + t?=1 (<)
If we let
q) / Or/1. 0yq]d z/1.., d r/q
for
z
EV(a(r(k)),u(r(k)),t)
1, ..,q in
Var(k),ur k)),), g >_
(3.6), then by assumption (iii), G()
1.
[xl![u] x(r(k))G(x)mw(dX)
Thus
(3.6) reduces
to
G(u((k)))
(3.7)
In view of (1.9) and (3.5), we conclude that
I xG(x)mw(dX)
G(u) for almost all
u in
C[O,T],
x_<u
and so
(3.4)
is established.
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SMOLOWITZ, L.H. Wiener Derivatives and Corresponding Fundamental Theorem
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PARK, C. and SKOUG, D.L. A Simple Formula for Conditional Wiener Integrals
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CLARKSON, J.A. and ADAMS, C.R. On Definitions of Bounded Variation for
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