THE PETTIS-STIELTJES (STOCHASTIC) INTEGRAL
by
Richard H. Shachtman
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 593
September 1968
This research was supported by the National Science Foundation under Grant
CU-2061, at the University of Maryland, and Grant GU-2059, at the University
of North Carolina at Chapel Hill.
TABLE OF CONTENTS
SECTION
PAGE
l.
INTRODUCTION
1
2.
VECTOR MEASURES
2
3.
THE PETTIS-STIELTJES INTEGRAL
6
4.
THE PS-STOCHASTIC INTEGRAL: A REPRESENTATION
.13
Figure 1
• 23
5.
OTHER PROPERTIES •
• 26
6.
EXAMPLES
29
APPENDIX
· 32
REFERENCES
· 34
INTRODUCTION
1.
Various definitions for integrals of functions with values in an arbitrary
Banach space
topologies:
X (vector-valued integrals) have been given in at least three
X and pointwise in the scalar
The weak and strong topologies on
field topology.
See, for example, [1], [4] and [5].
In each case conditions
X.
must be determined so that the integral exists as a well-defined vector in
The purpose of this paper is to define and exhibit some of the properties
of a Stieltjes integral for vector-valued measures (functions) in the weak
topology on
X
As the definition is motivated by B. J. Pettis' definition
of a Lebesgue integral in
X,
the integral will be called the Pettis-Stieltjes
integral.
In Section 2, some properties of vector measures used in the analysis are
mentioned.
The basic definition of the Pettis-Stieltjes integral and a listing
of some of the commonly indicated integral properties comprise Section 3.
The
main result of Section 4 is a representation for the Pettis-Stieltjes stochastic
integral in the form of a generalized integration by parts formula.
The latter
entails the use of a duality formula and an unsYmmetric Fubini theorem.
In
Section 5, a comparison with other stochastic integrals is noted as well as
another condition for existence via the definition of a modified Stieltjes
X.
integral in the strong topology on
Examples are given in Section 6 and there is an
appe~dix
erties of the scalar valued modified Stieltjes integral.
in the real line,
R
take
-
00
<
a
=
inf t
tET
<
sup t
tET
=
cataloging prop-
For the interval
b
<
00
•
This research was supported by the National Science Foundation under Grant
GU-206l, at the University of Maryland, and Grant GU-2059, at the University
of North Carolina at Chapel Hill.
T
2
VECTOR MEASURES
2.
This section contains some preliminary data on vector-valued measures,
definitions of their variations and comparisons.
(T, Q), where
Consider the measure space
Q
reals, and
(measure) on
is a sigma field of subsets of
Q to the Banach space
DEFINITION 2. 1.
for
{A}
n
disjoint in
~:
Q -+ X
T
T
is an interval in
Let
~
R,
the
be a set function
X over R.
is strongly (weakly) countably additive if
Q
~
=
( u A)
n
n
2: ~A
n
n
where convergence is in the norm (weak) topology.
Three definitions for a function
x: T
-+
X to be of bounded variation
are given in Hille-Phillips, [4, p.60), and two are shown to be equivalent.
Below, variation functions for the measure
are introduced and equivalence
~
is extended to yet another definition from Dunford-Schwartz, [1, p.320).
AE Q
supremum is taken over all finite partitions of
The
unless otherwise
noted.
X*, X**
denote the first and second dual spaces of
X.
DEFINITION 2.2.
i)
The weak variation of
~
A
on
sup
n
2:
k=l
E
Q
with respect to
Ix*(~~) I •
x*
E
X*
is
3
ii)
~
The (total) weak variation of
iii)
~
The semi-variation of
on
on
A
A
Q
E
E
Q
is
is
n
11\.111 (A)
supll L: a k ~~II
k=l
-
where the supremum is also taken over all
iv)
The variation of
\.I
on
A
Q
E
is
n
sup II L: \.I~ II
k=l
v)
~
The strong variation of
-
on
sup
n
L:
k=l
A
II
E
Q
is
\.I~ II
PROPOSITION 2.3.
i)
s.J
E
[1, 2].
See, also, Section 6.
n
Proof:
II x* II :;.
Let
-
sup L:
k=l
lak
1 , then
and adapting the argument in [4], get
x*(\.I~)
I
over all
4
WA(~) ~ 2VA(~)
as well as
COROLLARY 2.4.
~
i)
.
The remainder of the inequalities are evident.
The following are equivalent.
has finite (total) weak variation (is of weak bounded
variation) on
A.
ii)
~
has finite semi-variation on
iii)
~
has finite variation on
A.
A.
Moreover, the finiteness of any of the variation functions
I I~I
I(T)
WT(~)
or
additivity when
Q
is equivalent to
~
PROPOSITION 2.5.
.
The statements below are equivalent for
i)
~
is strongly countably additive on
ii)
~
is weakly countably additive on
§(T)
iii)
~
is of weak bounded variation on
T,
m(') = x*[~(')] ,
(ii) => (iii):
for
~
X:
finite.
Q.
x*
X* •
E
the scalar case result is known (Hahn decomposition)
this follows from the correspondence between Lebesque-
Stieltjes measures and non-decreasing, bounded functions on
Note.
+
m(') .
(iii) => (ii):
f(')
§(T)
(i) <=> (ii) in [4].
Let
for
~:
§(T)
Statements (i) and (ii) are equivalent for arbitrary
Proof:
,
having either type of countable
~(T)
is the Borel field,
VT(~)
= m( (-00, .] ) ,
then
f
T
is the difference of two such functions.
There is no confusion about
~
being of finite weak variation or
finite (total) weak variation since they are, also, equivalent:
(iii) => (i) =>
II ~ II (T)
<
00 ,
Let
[DS, p.320],
=>
W (~)
T
<
00 •
5
Let
BV(T)
be the space of functions of bounded variation on
T
under
the supremum norm
II f(.) I u
sup
tET
If( t) I
In Section 4, we will be interested in the case where the vector measure
~
is induced by
a
function
x: T
~
X .
x
~
defines
on the field of
half-open intervals by
~(
and when
x
(a, t] )
is of weak bounded variation on
x* [x ( . ) ]
for all
x(t) - x(a)
x*
E
X*,
then
and extends uniquely to
g*(')
=
6g*(')
dg*(')
x*[~]
x*(dx)
=
E
BV(T)
is countably additive on the field
countably additive on
spondence mentioned in the above proof.
defined by the values
T , i.e., the scalar function
§(T)
Q = ~(T)
n = 1, 2, ...
VA(x) ,
{(skn' t
kn
)},
sup
x .
is an interval with endpoints
X=
{t
x: T
kn
~
X ,
we will
}, k =0, "', n;
for all finite collections of non-
n
l:
!i1x*[x(t
k=l
~
is
For example,
WA(x, x*)
x: T
dx
All of the above remarks are valid; in particular, for
use partitions
A
=
dg* .
and take the usual partitions
overlapping intervals.
When
~
So the vector measure
When dealing with the measure induced by a function
always use
by the corre-
c
and
kn
d,
)]I
.
write
W
cd
or
V
'
cd
If
R , all of the above reduce to the ordinary variation of the function
6
THE PETTIS-STIELTJES INTEGRAL
3.
The basic definition is motivated by the definition of the Pettis integral,
see Hille-Phillips [4, p.77], and is made possible by the following proposition,
which is similar to the ordinary case.
(T,
Q)
and we say that
f(·) E Ll(T, Q,
f
Let
f
be measurable with respect to
is weakly integrable with respect to
x*[~(·)])
for all
w
x* EX, where
W if
is (weakly) countably
Q.
additive on
PROPOSITION 3.1.
Let
W be countably additive on
Then there exists
~
integrable with respect to
x**
Q
E
and
X**
f
weakly
such that
x**(x*)
for all
x*
E
X* •
The scalar-valued integral on the right side is the ordinary Lebesgue
integral.
Proof.
Let
IT
I(x*)
f(t)
x*[~(dt)]
x*
for
E
X*.
I
is obviously
linear and
wat (w,
for some
S
E
[1, 2] by Proposition 2.3.
II(x*) I S
Therefore
I
x*) s sl~~lv
IT
!f(t)
IdWat(~'
at
(w)
So
x*) s Ilx*11 S
is bounded and, hence, in
X** .
IT
!f(t) IdVat(~)
.
7
With the above as a justification,
f
T
X
f(t)~(dt)
E
may be set equal to the symbol
X** .
x **
f , there is an
More precisely, for each
in the image of
x**
X**
E
f
which mayor may not be
under the natural mapping
i: X ~ i(X)
c
X** .
If this correspondence obtains, we may define the Pettis-Stieltjes integral.
DEFINITION 3.2.
The scalar-valued function
~:
Stieltjes (PS-) integrable with respect to
all
A
E
Q,
there exists
x
A
E
x*
E
X* .
f
E
Q~ X ,
Q)
(T,
is Pettis-
a Banach space, if for
f(t)
x*[~(dt)]
f(t)
~(dt)
By definition
x
and write
on
X such that
fA
for all
f
PS(~)
A
ps-f
A
E
X
.
For notational purposes, we will often write
ps-f
f(t)
A
Let
f
E
PS(~)
always for
X
be reflexive.
if and only if
X=
L
P
Then if
f
~
~(dt)
is weakly countably additive,
is weakly integrable with respect to
over an arbitrary measure space,
I
<
P
<
00
~
hence,
8
COROLLARIES 3.3.
i)
The PS-integral is uniquely defined.
~
ii)
iii)
must be weakly countably additive for the definition.
E PS(~)
f , f
l
2
f
iv)
PS(
E
l-J..),
=> alf
PS(~2)
~>A
and
<f,
If
x=R
+ a2f 2
l
=> f
E PS(~)
PS(SI~1
E
+
,
a.
J
S2~2)
is a bilinear function
then
<f,
~>
E
R,
,
for fixed
A
E
Q •
reduces to the ordinary Lebesgue
integral.
Proof.
Use the definition and the fact that
implies
x
=
x*(x)
x*(y)
for all
x*
X*
E
y .
PROPOSITION 3.4.
Let
f
PS(~)
E
and
<f, ~>A ' then
v(A)
Q~
v:
X
is (strongly) countably additive.
Let
Proof.
{A}
n
x*[v(uA )]
n
So
v
Then
Q.
Then for
<f, x*(~»UA
X*
x* E
2: <f, x*(~»A
n
n
2: x*[v(A )]
n
.
is countably additive by Proposition 2.5.
Call
REMARK.
all
disjoint in
¢: Q ~ X continuous with respect to
> 0 , there exists
E:
v
<5
> 0
is continuous with respect to
x*(~)
respect to
Let
[4, p.76].
x*
E
X*
for any
If
x*
x*[~A]
E
II¢ (A) I
such that
~
<
; in fact,
E:
v
~:
Q ~ X,
whenever
if for
II~(A)
II
<
is continuous with
X*:
=
0
then
vA
o
and the result follows by
<5
9
PROPOSITION 3.5.
f
E PS(~)
, then
L be a bounded, linear operator on X.
Let
E PS(L~)
f
and for
~>
L <f,
Proof.
y*
E
X*
given by
y* =
L.
y*(~)
L*x* and
Q
E
A
L* be the adjoint of
Let
A
If
For
=
x*
x*(L~)
E
X* , there is a unique
is countably additive.
Thus there exists
y*(~»
<f,
Hence,
<f,
For
f
E
PS(lJ) ,
v
=
<f, lJ>
has finite variation on
does not necessarily have finite strong variation.
Let
f
i)
~
in weak variation then
If
lJ
n
- lJ
0
E
PS(lJ ) , n
n
Proof.
and
f
n
~
m, n
E
PS(lJ)
and for
is countab1y additive since
PS(lJ) •
~
as
f
in weak variation in (i) is not sufficient.
lJ
i)
E
lJ
~
00
,
< If
Q
but
See Section 6.
(weak) •
=
lJ
•
= 1, 2,
PROPOSITION 3.6.
ii)
~»
L<f, lJ> .
LlJ>
REMARK.
x*(L<f,
y*«f, lJ»
I, Wat(lJm
so there exists a weak limit of
- lJ , x*) >A ~ 0
n
<f, lJ
n
>
and
A
E
Q
10
x*«f, fl »
<f, X*(fl »
n
The left side converges to
Let
ii)
+0
2W(f.i, x*)
fl
n
and
-fl
==
•
n
x*(lim <f, fl »
n
Let
~
W(fl , x*)
n
<f, X*(fl n -
PROPOSITION 3.7.
n
f
n
<f, X*(fl»
.
and the right side is
W(fl, x*)
but
W(fl
n
- fl
'
x*)
x*«f, fl».
=
+--0
fl»
E
~
, n
PS(fl)
1, 2, ... , and either of the
following hold:
fn ~ f
i)
ii)
f
Then for
A
E
n
in
~ f
II' I~ .
pointwise,
If
n
I
~ g
weakly integrable with respect to
Q
lim <f , fl>
n
n
A
Proof.
f, f
n
fl
(weak) •
are bounded by an integrable function; the argument in 3.6.
applies and the inequality
is evident.
Note that it is sufficient for the convergences and inequality in (i) and
(ii) to be (weak)
fl -
a.e., of course.
An acceptable strong definition implies the weak definition in much the
same way that existence of the Bochner (ordinary) integral implies existence
and equality of the Pettis integral.
For example, using a definition in
Dunford-Schwartz [1, p.323],
DEFINITION 3.8.
A scalar-valued measurable function
integrable if there exists a sequence of simple functions
f
is said to be
{f }
n
such that
•
11
f
i)
n
{f
ii)
~
A
fA f
w-
f
a.e.
f (t) w(dt)}
and
converges in norm for
A
n
(t) w(dt)
and
Q,
E
where
partitions
A.
n
The limit in (ii) is defined to be
Ds-f
f(t) w(dt) .
A
REMARK.
The existence of the strong integral
Ds-f
f(t) W(dt)
implies
A
the existence of the (weak) Pettis-Stieltjes integral
PS-<f, w>A
and the
integrals coincide:
the left side converges to
x*(DS-f
f dw)
and the right side is
A
PROPOSITION 3.9.
v(A)
= <g, w>A.
Then
Let
f
f, g
be scalar-valued functions on
PS(v)
E
if and only if
fg
PS(w)
E
(T,
Q)
and for
and
A
E
Q
<fg, P>A .
Proof.
{f}
n
Let
f
E
PS(v)
simple such that
and
f
n
x*
~
E
f,
consider the non-negative parts of
X* .
Then there exists
x*(p) - a.e.
f, g
and
<f
Hence
h*
E
{fng}
L (T)
l
is Cauchy in
such that
f g
n
LI(T)
~
h* .
m
and in
x*(w)
<f, x*(v»
LI(T)
A
and
(If necessary,
and use the linearity.)
- f , v>A .
n
with respect to
Consequently,
x*(w)
h*
=
and there exists
fg a.e.
and
12
<f, x*(v» A
If
f
n
and
fg E
-+
PS(~)
f a.e.
, there exists
So
f E PS(v) .
{f}
n
<fg,
x*(~»A .
simple such that
and
13
4.
THE PS-STOCHASTIC INTEGRAL: A REPRESENTATION
In this section, consider the vector measure
x: T
function
X , where
7
X=
variables over a probability space.
Elx(t)1
<
for all
00
given
t
x(t)
E
E
T
L
l
E, E)
X(~,
X
If
induced by a random
~
is a Banach space of random
L
l
(~)
, then there exists
Recall the representation for the dual space
(~)
and
x*
E
Ll*(~)
~* E
, there exists
L
00
(~)
such that
E[x(t)~*]
x*[x(t)]
.
The general definition for the PS-integral becomes in this case,
DEFINITION 4.1.
The scalar-valued function
f
PS-integrable with respect to the random function
Ll(~)
values in
, if for all
A
Q,
E
there exists
fA f(t)
for all
~*
E
Loo(~).
x
Q)
on
(T,
=
{x(t): t
x
A
E Ll(~)
is
E
T}
with
such that
dE[x(t)~*]
By definition
=
To have the induced measure
restrict the scalar function
~* E
L
00
(~)
.
~
g*(.) -
be countably additive, it suffices to
E[x(·)~*]
to be in
BV(T) , for all
Make this assumption in the sequel.
The above is a definition of a stochastic integral in the weak topology
of
X=
Ll(~)
.
We will also need another stochastic integral defined point-
wise in the scalar topology.
14
Let
x
and
y
be random functions on
DEFINITION 4.2.
to
x
over
A
E
Q
(T,
Q)
to
X(~,
The sample path stochastic integral of
f,
E) .
y
with respect
exists and is denoted by
Sp-fA yet, .) dx(t, .) ,
if the scalar-valued integrals
fA
(type) -
w
exists for almost all
~
E
,
Yet, w) dx(t, w)
where the type may be one of the Riemann-
Stieltjes (RS-) integral definitions or the Lebesgue-Stieltjes (LS-) integral.
Conditions for existence and properties of the SP-integral are
[5].
discussed in
defined on
~
In both of the above definitions, the integrals are
a.s. with respect to
Assume that
~(T»
x: (T,
measurable random function and
THEOREM 4.4.
For
f
E
E
~ Ll(~"
T
BV(T)
I, E)
is a non-trivial product
is a finite interval.
and
x
of weak bounded variation on
the Pettis-Stieltjes integral exists and has the representation
ps-
where
on
m (·)
f
§(T)
E
fA
CA(T)
f(t) dx(tl
the countably additive scalar-valued set functions
T ,
15
The integral on the right side is the sample path Lebesgue-Stieltjes
type.
By the remark following Definition 3.8, the formula is true for strong
integrals.
COROLLARY 4.5.
When it exists,
Ds-IT
f(t) dx(t}
The proof of the theorem requires a few preliminary results.
Let
BDl(T)
be the space of bounded, scalar-valued functions on
with at most discontinuities of the first kind.
g
E
BV(T),
defined;
the modified Stieltjes integral
For
f
E
BDl(T)
Ms-fT f(t) dg(t)
T
and
may be
it is a generalization of the Riemann-Stieltjes integral and is
discussed in the appendix.
DEFINITION 4.6.
Let
£(T)
open intervals and points in
£(T)
be the field of finite unions of (finite)
T.
For
f
E
BDl(T), define
m (·)
f
on
by
Let
BA(T)
set functions on
be the space of bounded, finitely additive, scalar-valued
T.
LEMMA 4.7.
(i)
When
f
E
=
BDl(T),
m
f
E
BA(T)
f(c+) - f(d-),
on
mf({c})
£(T)
=
and
f(c-) - f(c+) .
16
(ii)
Proof:
When
f
E
BV(T) ,
m
f
E
CA(T)
m
C= u (c., d.)
J
J
j=l
Let
u
~(T)
on
n
u
k=l
and is bounded.
{e } , disjoint.
k
(i) follows
from Proposition A.6 (appendix), Definition 4.9 and the form of
n
m
Z If(c.+) - f(d.-)I + Z If(e -) - f(e +)I ~ VT(f)
k
k
J
J
j=l
k=l
hence,
sup {lmf(C)
increasing,
where
C.
k
I :
C
= 1, 2.
E
~(T)}
is finite.
Also
f = f
l
<
00
- f , f
2
k
Therefore,
m
f
E CA(T).
As a result, m
may be (uniquely) extended to the
f
k
sigma field generated by £(T), by the Caratheodory Extension theorem, but
this is
~(T).
PROPOSITION 4.8.
T,
For
f
BV(T)
E
and
x
of weak bounded variation on
a duality formula is obtained:
~* E Loo(~)'
for all
Proof:
where
m
f
E
CA(T)
and
g*(')
E[x(·)
~*]
.
We refer to Proposition A.8 in the appendix for an integration by
parts result for modified Stieltjes integrals.
MS-f.
+
Z
x
f dg
- MS-JT gdf +
b
[fg]a
[f(x-){g(x)-g(x-)} - f(x){g(x+)-g(x-)} + f(x+){g(x+)-g(x)}]
where the sum is over the (common) discontinuities of
f
and
g.
17
Letting
IT
g
get) dmf(t)
be
g*,
our desired result obtains if we can show that
expands to become the right hand side of the above equation.
From the definition,
n
lim ~ g(tiJ Limf(t )
k
D k=l
with partitions from
For
t'
For
t~
E
k
Hence
£(T)
and the
t'
are arbitrary interior points.
k
(t k - l , t )
Limf(t )
mf«t - l , t ))
k
k
f(t k- 1+)
f(\-)
t
Limf(t k )
IDf({t })
f(t -)
f(t +)
k
k
:
k
k
k
k
g(t~) Lim (t k )
f
~
~
intervals
in D
g(tk')[f(t _ +) - f(t -)] + ~ get )[f(t -) - f(t +)]
k 1
k
k
k
k
and writing the first square bracket as
we get the following sums
are non-zero only for the discontinuities of
8 , 8
2
3
set.
f, {d.},
J
a countable
Therefore, including these in a sequence of increasing partitions and
taking the limit,
8
1
-r
- IT
get) df(t)
, (+[fg]ba
for
f
continuous at endpoints)
()()
8
8
-r
2
3
-r
~
j=l
[g(d.-){f(d.) - f(d.-)} + g(d.+){f(d.+) - f(d.)} J
J
J
J
J
J
J
()()
. ~l g(d.)[f(d.-) - f(d.+)]
J=
J
J
J
,
18
If the endpoints appear in
S3'
we get
L g(t )[f(t -) - f(t +)]
k
k
b
g(a)[-f(a)] + g(b)[f(b)]
[fg]a .
k
Adding, we get
- fT
g(t) df(t) +
[fg]~
00
+
L:
j=l
[f(d.-){g(d.)-g(d.-)} - f(d.){g(d.+)-g(d.-)} + f(d.+){g(d.+)-g(d.)}] .
J
J
J
J
J
J
J
J
J
LEMMA 4.9.
For
f
E
BV(T) ,
the Lebesgue integral
SP-JT x(t, .) dmf(t)
exists.
Proof.
Use the Fubini theorem and the fact that
uniformly in
(T
x ~,
t
(x
~
.§(T)
£,
is bounded,
is product measurable with respect to
A ~
PROPOSITION 4.10.
(i)
Ex(·)
E) ,
Let
f
where
E
is Lebesgue measure.)
A
PS(x), then
An unsymmetric Fubini theorem obtains,
fT
f(t) dEx(t)
in fact,
E[fT
for all
F,*
E
L
00
(~)
E[fT
for all
F,*
E
L
00
(~)
f(t) dE[x(t)
,*]
.
Moreover, i f
(ii)
,*] " fT
f(t) dx(t)
x(t,
.
.)
f
E
BV(T) ,
drof(t)
,*j
the above also coincide with
19
Proof.
The equality in (i) follows since
f
Lemma 4.9 and the ordinary Fubini theorem.
PS(x)
E
and that in (ii) from
Using the duality formula in
Proposition 4.8,
i
~*]
f(t) dE[x(t)
T
<==>
there exists
lim E[
D
~
m
f(t~)
llx(t.)
J
j=l
J
where the limit exists as a MS-integral.
lim
lim
L
D
j=l
f(t~)
llE[x(t.)
J
J
~*]
Thus
E[YD~*]
E[y~*]
D
for all
~*
E
Loo(~)
; i.e.,
x*(y)
lim x*(YD)
for all
x*
E
X*.
Hence
D
exists and is
the weak limit of
weak lim L f (t ~) IIx(t.)
J
J
D
Denoting the left limit by
L
f
E[xT~*]
DEFINITION 4.11.
X
y ;
=
T
x(t, .) dmf(t) E
iT f(t)dx(t),
T
dE[x~*]
Define
i
topology, of the (linear) span of
(~)
we get the desired relation
iT E[x<*] dm f
Ll(x)
L
l
E[f T xdmf <*] .
to be the closure, in the norm
{x(t) : t
E
T} .
Proof of Theorem 4.4.
x = {x(t) : t
E
T}
+{OJ
=>
dim Ll(x) ~ 1 => Ll(x)
is a non-trivial
20
Banach space in
Ll(~)
is total.
=>
is assured; see, for example, Corollary 5.5.
x*<f
for all
x*
E
f
f
dx)
X* .
Ll(x) ,
dm f
=
x*<J
x dm f l
Consequently,
ps-J
in
PS- <f, x>
By Proposition 4.10,
J x*<x)
=
f dx*<x)
Existence of the
sp-f
f dx
x dm f
which means a.s.
In effect, the Pettis-Stieltjes stochastic integral of
to the random function
x
f
with respect
is represented by an integration by parts formula.
The following discussion motivates Theorem 4.4, indicates the origin of
Definition 4.6 and outlines the argument supporting the view of the formula
as a representation.
By the definition of the PS- integral, existence requires
Identifying the respective dual spaces,
L
00
(~) -+
L
00
*(~)
where the equivalences are isometric isomorphisms.
is in the curve
x
= {x(t): t
E
BA(~)
•
Our interest, however,
T}
lying in the space
~*]
,
Ll(~)
and we want
to characterize the functionals
E[·
which act on
we look at
x(t),
E['
~*]
t
E
T.
~* E L 00 (~)
Rather than considering functionals on
on the space generated by the curve
x,
Ll(x) .
Ll(~)'
21
Ll(x)
is a closed, linear subspace of
Ll(~)
Ll(~)
Banach space under the relative topology of the
functionals on
L
.L
l
(x)
look like
Ll(x)
is the annihilator of
this is just
E[x(')s*]
g*
Ll(x)
which is in
Since the operator topology on
topology on
(BV(T),
Loo(~) ,
I,. I 1)*
u
where
Ll*(~)'
in
consequently, is a
norm,
EI·
I .
The
= E[x(.)s*]
g*(')
[1, p.72].
On
and
Ll(X) ,
BV(T)
Ll*(x)
II· I lu
consider the
(Note:
+ Ll .L (x),
and,
I 1'11
is the induced
topology on
BV(T)
00
and identify
This is not necessarily an exact isometric
isomorphism; the latter holds only in a special case.)
We recall the definition of
dual space by Hildebrandt,
PROPOSITION 4.12.
in
BA(T)
BDl(T)
[2, p.873].
(BDl(T),
and the characterization of its
(The result allows
II· IIu )*
are defined on the field,
£(T),
~ BA(T) ,
T
to be
R.)
where the set functions
of subsets of
T
and the
correspondence is an isometric isomorphism.
BV(T)
C
BDl(T),
so the characterization may be used for our purposes;
in fact, there is an improvement due to the form of the functionals
x **(.)
f
on
(BV(T), 11'11)
IT
f(t) d(o)
.
Now we see exactly the motivation of Definition 4.6 since
mf(C)
x **(1 )
f
C
IT
f(t) dIC(t)
.
Combining this last result with the duality formula, the following framework
is established for the correspondences between the various spaces used in the
analysis:
i)
=
E ~
x(·)
~(T,
L1(~)
with
where
~* E
L1(~»'
Ex(·)
L
00
(~)
E
the space of random functions on
g*(.) = E[x(.)
BV(T).
=
~*)
T
~*[x(·»)
E
to
BV(T),
.
~ ]. BV(T) :
ii)
n[x(.») = g*(')
~ ~ BV(T)
iii)
g~*
by
n[x(t»)
g*(t)
~* [x(t»)
CA(T).
(g**
.
R:
g**(') = IT (.) dm f
E
~
BV*(T)
0
n)[x(.»)
= g**{n[x('»)}:g**[g*('») = IT g*(t) dmf(t) = JT
Thus
g**
n:
0
~ ~
Rand
functional on
iv)
::: ..1i- L
00
[ L
~ ~
00
~*,
~**
~
(~)
~**(.) = IT
=
0
n
E
n
0
is a bounded, linear
:::*
(~):
s[x(·») =
v)
g**
g**
~*[x(t)dmf(t).
~**
fd(')
0
functional on
~*
defines
g* .
R:
~**{s[X('»)}
Hence
where
s:
=
E
Loo**(~)
~**[~*(.»)
~ ~
R and
-
BA(~)
= IT
~**
.
f(t)
0
s
(~**
0
s)
d~*[x(t»)
[x(·»)
= IT f(t)dg*(t).
is a bounded, linear
- ,
Comparing the above under the appropriate conditions, we get the represen tat ion
f
0
~*
= mf
~**
0
0
~*
s = g**
0
n
or, symbolically,. the duality formula from 4.8,
The relationship is illustrated in Figure 1.
23
LQ) (Q)
s-** ".
BA(S?)
24
We record the following from [4, p.77].
DEFINITION 4.13.
is
The function
x
A
Pettis-integrable if for all
E
on
Q,
x*
E
to the Banach space
there exists
JA x*[x(t)]
for all
Q)
(T,
x
A
E
X
X such that
dA(t)
X* , where the integral on the right is the scalar-valued
Lebesgue integral.
By definition
xA
p-fAX(t) dA(t) .
The result of Theorem 4.4 holds for the Pettis integral.
COROLLARY 4.14.
f
E
PS(x) .
Suppose
is of weak bounded variation and
Then
fA
ps-
Proof.
x
f(t) dx(t)
PS-<f, x> .
Let
t
Then for
t
fdx*(x)
by the duality formula.
x*
E
X*
x*(x) dmf
(Existence of the Pettis integral is insured by
existence of the sample path integral, Lemma 4.9, and the usual Fubini theorem.)
REMARK..
Let
[0, 1]
T
and
f
E
BD (T),
I
but not in
BV(T) ,
be
defined by
00
f(t)
=
(_l)n+l n -1 I
l:
n=l
where
A
n
(
1
n + 1
, 1:.]
n
n
VT(f) >
l:
k=l
A
(t)
t
n
+0;
0
f(O)
Then
If(~)
1
- f(k+l)
n
I
l:
k=l
1
("k
1
+ k+l) -+
n
00
•
.
25
Therefore, even for a random function as simple as
IT x dm f
not in
=
In fact, should we restrict
00
BV(T) , we can't define
sp-IT x(t,
+0
,
to be continuous on
f
.)
=K
x(t)
for, say,
dmf(t)
x
T
but
being
the Brownian motion process, since the paths are a.s. not of bounded variation.
As a result,
f
E
BV(T)
is a necessary hypothesis for the representation.
Although the representation for arbitrary continuous functions cannot
be obtained in general, a strong approximation is available.
PROPOSITION 4.15.
exists a polynomial
IIIT
Proof.
p
Let
on
f
be continuous on
T
T.
For
such that
IT
f(t) dx(t) -
x(t, .) dm ( t)
P
II
<
E
p
with
I If - pi lu
E/4V (x) .
T
<
exist and the representation holds for
Since
ps-I
p dx.
p
E
Let
T
=
•
The Weierstrass approximation theorem insures the existence of a
polynomial
y
E > 0 , there
IT f dx - IT P dx,
then using
Ix*(y) I ::.
which means that
II y II < E .
L
Proposition 2.3
If - pldw
at
(x, x*)
< E /2
BV(T) , the integrals
I /x*1 I ~
land
26
5.
OTHER PROPERTIES
Restricting
x
to be of weak bounded variation on
T
is, in a sense,
a weakest possible assumption, since we shall usually want to integrate, at
the least, all continuous functions.
PROPOSITION 5.1.
T,
then
and
f(t) dx*[x(t)]
f(t) dx*[x(t)]
[r, t]
PS-JT f(t) dx(t)
for
f
is the same.
in
L (rt,
2
E,
f(t) dx(t)
i)
ii)
Proof.
When
=
X*
E
and equals
continuous, since the oscillation over
So
x*[x(·)]
E
For
(r, t)
RS-integra1s, the propo-
BV(T) ,
for all
x*
E
X* .
be the well-known Ito-Doob stochastic integral
When the following stochastic integrals exist, the
exists and coincides with them:
sp-It
f(t) dx(t,
.) ,
ID-f
f(t) dx(t)
,
spIT
exists,
y
f bounded,
where
Ex(·)
LS-type in
E
L (rt)
1
.
.
BV(T)
T
=
bounded variation on
E[y ,*1
continuous on
f)
PROPOSITION 5.2.
PS-fT
x*
(See the appendix.)
ID-fT f(t) dx(t)
f
T .
exists for all
sition is known, [3, p.271].
Let
exists for all
is of weak bounded variation on
MS-fT
Proof.
RS-JT
x
If
f dx
T
,
hence
is measurable and
f
Ex(· )
JT Ht) dE[x(t) ,*J • hence
E
BV(T).
y
See [5].
= PS-JT
f
dx.
x
Let
is a.s. of
~* E
L00 (rt)
The argument
for the ID-integra1 is similar to the remark following Definition 3.8.
,
27
Returning to the strong topology on
X
,the standard definition of a
Riemann-Stieltjes integral in a Banach space, [4, p.62], may be slightly
generalized.
n
DEFINITION 5.3.
t~
with
If
f(t~)
lim E
D k=l
(t _ , t )
k l
k
arbitrary in
~x(tk)
exists in the norm topology
and partitions are successively finer,
then denote the limit by
MS-f
f(t) dx(t) •
T
PROPOSITION 5.4.
Let
weak bounded variation on
f
T.
E
BDI(T)
Then
and
Ms-f
x: T
~
f(t) dx(t)
X such that x is of
exists (in the norm
T
topology).
For
Proof.
E
>
select a partition
0,
D
over any open subinterval of any
W(x, x*) ~ 2MII x*11
that
for
x*
::>
X*
E
D of
E
D
E
,
T
such that
where
M = VT(x)
D, D'
and let
::>
osc (f)
<
00
osc
is over
E/4M
Recall
D
E
E
osc(f) I ~x*[x(t ok)]
DuD'j,k
J
where
.
<
I
<
IIx* I E/ 2
and
So
II E
f ~x
-
D
E
f ~xll < E
D'
and the limit exists.
COROLLARY 5.5.
T,
f
E
PS(x)
For
f
E
BDI(T)
and
x
of weak bounded variation on
and
pS-f
f(t) dx(t)
T
=
Ms-f
f(t) dx(t) .
T
28
Proof.
IT
x*[MS-I f(t) dx(t)]
T
When the p-th moments of
f(t) dx*[x(t)] .
x(t)
exist,
t
E
T,
the computation of the
p-th moments of the PS-integral falls out from the definition.
fA
EXA
EX
2
A
f(t) dEx(t).
=
fA fA
r ("
where we assume
.)
E
Let
2
BV (T ),
f(s) f(t) E[dx(s) dx(t)]
r (s,
t)
00,
take
E[x(s) x(t)] .
If
x
has
orthogonal increments
In general, if
since
X = Lp(~)'
Elx~-llq = ElxAI P
EX~
For
p
=
<
fA
I
00
~ p
<
r
E
f(s) E[dx(s)
BV(T P ) ,
xr-
l
E
Lq(~) , q = p/(p-l),
and
{fA
an integer,
again assuming
~*
r(sl' " ' , sp)
f(t) dx(t)}p-l] .
29
6.
EXAMPLES
6.1.
To illustrate Theorem 4.4, the representation for the PS-integral,
consider a Poisson process on
is finite, hence the
T
= [0, b]
PS -integral may be
A > O.
with parameter
defi~ed
VT(x)
(see below).
For this process, almost all sample paths are increasing step functions,
integer-valued with jumps of magnitude one and continuous from the left.
Also, there are only a finite number of discontinuities in any finite interval.
Let
f
be continuous on
T.
Then the SP-integral exists and is
N
fT
where
of
f
k
k=l
N = NT
{d (')},
k
E f(d ) [x(d +) - x(d )]
f(t) dx(t)
=
k
k
is a random variable representing the number of discontinuities,
of the sample path functions on
T.
So the (stochastic) integral
with respect to the Poisson process is a random sum of random variables.
such that
For
max
1 ::; j
lit
::; m
j
max
<
1 ::; k ::; N(w)
lId (w)
k
and
m
m
x(t~, w) lImf(t.)
E
j=l
where
t'
j
E
subinterval
J
J
(t. l' t.)
JJ
and
But
lId (w)
k
the telescoping
E
=
E'
j=2
E'
is the sum of differences
x(', w)
reduces to
(j-l) {E' [f(t _ ) - f(t )]}
j 1
j
f[dk_l(w)] - f[dk(w)]
ps-f
f dx
T
and
N(w)
E x(tJ~' w) lImf(t .)
that is,
f
on each
is constant on these subintervals, so
m
j=l
lIm
J
=
E
k=l
f [d (w)]
k
•
30
Here, the SP-,
6.2. Define a process
6x (t)
PS-
and
x
on
y
IDT
=
integrals all exist and coincide.
by letting the increments
[0, b]
be independent and normally distributed with mean
y
variance
2
a 6t;
y, a
>
x
0
y
ya6t
and
is a shift of the Brownian motion for
is finite.
y > 0
Here the PS-
and
ID-
integrals may be defined, but not the SP -integral.
Using the above three examples and others, a table may be set up
displaying all possible combinations of existence for the sample path, PettisStieltjes and Ito-Doob integrals.
REMARK.
The Brownian motion shift is an example of a process which
~(T)
induces a measure of finite variation on
(countably additive) but not
of finite strong variation.
where
x
is (zero-mean) Brownian motion.
Hence,
and
2:
~n
a'; "- /
7f
L
k=l
16 t kn
-+
00
•
31
Recalling that
WT(~)
=
SVT(~)
where
~:
(T,
Q)
~
X and 1
~
~
S
2 ,
the Poisson process provides an example for which the equality is obtained
(S=l);
let
Ilx*ll~l,
n
sup
L
k=l
E[~x(tkn)]
=
Ab •
But
=
where
NT
is defined in 6.1.
Ex(b)
Therefore
WT(x)
Ab ,
~
VT(x) ,
so
S
1.
This
may also be seen from the fact that
Ab ,
so
VT(x)
in
Ll(~)
= WT(x) = ST(x)
.
In fact, these equalities obtain for any process
with nondecreasing (or nonincreasing) sample paths.
ACKNOWLEDGEMENT.
This research constituted a part of my doctoral
dissertation and I wish to express my gratitude to my advisor, Professor
Ryszard Syski, for his valuable advice and helpful guidance.
]2
APPENDIX
The properties of the scalar-valued modified Stieltjes integral are
summarized here for reference.
The integral is due to
results are listed in [3, p.273].
DEFINITION A.I.
Let
f
B. Dushnik and some
Proofs are omitted.
and
g
be real-valued functions on
T.
The
modified Stie1tjes integral is defined as
n
MS-f
f(t) dg(t)
lim I f(t
D k=l
T
k) ~g(tk)
where the limit is taken over successively finer partitions
t
k
are arbitrary interior points of the subintervals
Note that the
D of
T.
The
(t _ , t ) .
k l
k
MS-integra1 is more general than the usual Riemann-
Stie1tjes (RS-) integral.
DEFINITION A.2.
interval
T
and equals
LEMMA A.3.
partition
osc(f)
D
E
of
sup {If(s) - f(t)
Let
T
is the oscillation of
f
E
osc(f)
When
f
E
BD (T)
1
over a prescribed
T}
E
E >
a , there exists a
over any open subinterval of
< E
and, hence, over any open subinterval of
Ms-f
s,t
BD! (T) , then for every
such that
PROPOSITION A.4.
I:
f
D
~
D
E
D
E
and
g
E
BV(T) ,
there exists
f(t) dg(t) .
T
PROPOSITION A.5.
Let
f
E
BD (T)
1
and
g
continuous in
there exists
RS-f
f(t) dg(t)
T
MS-f
f(t) dg(t)
T
BV(T).
Then
33
PROPOSITION A.6.
Let
f
E
BD (T)
l
and
g
E
BV(T) .
Then
MS-f fdg
RS-f fdg c +
~
{f(t+) [g(t+) - get)] + f(t-)[g(t) - g(t-)]}
is the continuous part of
where
PROPOSITION A.7.
For
f
E
exhibits the discontinuities.
BDl(T)
and
g
E
BV(T) ,
the Lebesgue-
Stieltjes (LS-) integral exists and
LS-f fdg
PROPOSITION A.8.
When both
MS-f fdg .
f
and
g
are in
BV(T) ,
a type
of integration by parts theorem holds.
Jfd& + J&df
+
~
x
= [ g f]ba
[f(x-){g(x)-g(x-)} - f(x){g(x+)-g(x-)} + f(x+){g(x+)-g(x)}]
where the sum is taken over the (common) discontinuities of
f
and
g.
34
REF ERE N C E S
[1]
Dunford, N.
General Theory,
[2]
Schwartz, J. T.
and
Linear Operators, Part
1:
Interscience, New York.
Hildebrandt, T. H.
(1934)
Trans. Amer. Math. Soc.,
[3]
(1958)
Hildebrandt, T. H.
"On Bounded Linear Functional Operations",
868-875.
~,
(1938)
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