AN ABSTRACT OF THE THESIS OF
for the degree of
Stephen D. Scarborough
August 26, 1982
presented on
Department of Mathematics
in
Doctor of Philosophy
A Moment Rate Characterization for Stochastic Integrals
Title:
Signature redacted for privacy.
Abstract approved:
David S. Carter
D. S. Carter has described the following model for security
prices.Letz(t)=(.
zi(t)),
(t
0), (i=1,...,n)
ous stochastic process on the probability space
(6,!A7.
t
0)
t
'
be a nondecreasing family of sub
be a continu-
(0,4X,P).
Let
a-fields of
is a vector of security prices
to which
z
and
contains the information known up to and including time
jet
is adapted.
Here
z(t)
Suppose there are stochastic processes
8(t) =
(8ij(t)),
a(t) = (ai(t))
0), (i,j=1,...,n)
(t
and
such that the following
moment rate conditions hold:
lim
h4-0
lim
it
Erz.(t+11)-z.(01 &J
hLi
1
a(t) as.
,
EPz.(t+h)-z.(t))(z.(t+h)-z.(t))1
j
5] =B(t)
t
114,0
(8)
lim
h+0
for some in
1
m
E[ n
k=1
3.
z.
lk
(t+h)-z; (t)
Ik
I
7]
= 0
a.s.
ij
a.s.
t.
The question arose as to how this compares with models in
which prices are given by stochastic integrals.
This thesis dis-
cusses the relationship between stochastic integrals and processes
satisfying conditions (1), (2) and (3) for various values of
It is shown that for each value of
m
a broad class of Ito
stochastic integrals satisfy (1), (2) and (3).
Furthermore, under
reasonable restrictions on the processes involved, a process
that satisfies (1), (2) and (3) with
m.
z
m = 3 is representable as an
Ito stochastic integral.
A preliminary result, of interest in itself, is that sufficient conditions are obtained for the conditional expectations of a
continuous stochastic process to have continuous versions.
A Moment Rate Characteeization for
Stochastic Integrals
by
Stephen D. Scarborough
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Completed August 26, 1982
Commencement June 1983
APPROVED:
Signature redacted for privacy.
Professor of Mathematics in charge of major
Signature redacted for privacy.
Chairman of Department of Mathematics
Signature redacted for privacy.
Dean of Graduate1Shoo1
Date thesis is presented
August 26, 1982
Typed by Donna Lee Norvell-Race for
Stephen D. Scarborough
ACKNO'4LEDGMENTS
wish to give my thanks to Dr. David S. Carter for his
patience and encouragement during the research phase of this
thesis.
I would like to give special thanks for his assistance
turning a rough draft into an end-product.
I also wish to thank Kennan Smith, Don Solmon, Ed Waymire,
and most notably Robert Burton, for several fruitful
conversations.
Finally, I thank the people in the Mathematics Department,
especially the office staff, who have been very kind and helpful
to me during my time at Oregon State University as a graduate
student.
TABLE OF CONTENTS
Page
I
II
INTRODUCTION
........
MATHEMATICAL PRELIMINARIES
2.1
III
.....
Probability Spaces and Conditional
Expectation
1
6
6
2.2
Independence
10
2.3
Stochastic Processes
12
2.4
Separability
14
2.5
Integration of Sample Paths
17
2.6
Sample Path Continuity
21
2.7
Brownian Motion and Martingales
30
2.8
The Ito Integral
33
2.9
Miscellaneous Results
43
THE MOMENT RATE CHARACTERIZATION
47
3.1
The Spaces MP and QP
47
3.2
Moment Rate Properties of Certain
Stochastic Integrals
51
Representation Theorem
63
3.3
BIBLIOGRAPHY
75
INDEX
76
A Moment Rate Characterization for
Stochastic Integrals
INTRODUCTION
I
Motivated by a search for a model of security prices, D. S.
Carter suggested a stochastic model as described below.
Carter was
interested in generalizing the standard models in which security
prices are represented by diffusion processes, in such a way as to
avoid the Markov and independence assumptions of diffusion processes
while retaining a characterization in terms of drift and covariance
functions.
To describe Carter's model let
z(t)=(z1(t)),(t>0),(i=l,...,n)
be a continuous stochastic process on the probability space
Let ('
t
(97',1)).
f7r
a-fields of
t
'to which
be an increasing family of sub
0)
is adapted.
z
of the security prices at time
and
t
tion known up to and including time
a(t) = (a(t))
processes
and
t.
z(t)
Here
is a vector
contains the informa-
Then there are stochastic
B(t) = (Bij(t)),(t>0),(i,j=1,...,n)
such that
lim
h40
E z
h
i(t)1
5-6(t]
= a(t)
a.s. (i=1,...,n),
1
'rim
h40
EL-(z.(t+h)-z1(0)(z .(t+h)-z.(t))!
t
(i,j=1,...n) and
= B(t)
ij
a.s.
2
(3) h4,0
UrnE[
n
k=1
for some
m
1
ft);
z. (t+h
ik
3, where
dices from {1,...,n}.
0
:
-
any sequence of
k=1,...,rn
drift
is the
Here
a.s.
0
vector and
B
m
in-
is the
covariance matrix.
The defining equations for a diffusion process are similar
to (1), (2) and (3).
(t) =
if
E
(ci(t)),
(t
Specifically, a stochastic process
0), (i=1,...,n)
is a diffusion process
is a continuous Markov process and there exist functions
Rn
x[0,co)
R
such that for all
t
0
and
(i,j=1,...,n)
lim
h0 "
1im
h+0
1
h
lim
114
for some
EE(.(t+h)-i(t))R.(t+h)-yt))1 c(t)=x]
E[11
(S
pi(x,t)
I E(t)=x]
ErE.1(t+11)_ci(t)
(t+h)-E,(01124-6
I
*(t)=x
(I
> 0, where
= vij(x,t)
= 0
x.2)
i=1
1
There is a close relationship between diffusion processes and
stochastic integrals.
Let
p = (pi)
a matrix, of functions from Rnx [0,.)
integral equation
be a vector, and
R.
a = (aij)
Consider the stochastic
3
,t
F,(t) =
(s),$) ds +
(0) +
where
w
!
a((s),$) dw(s)
'0
0
is an n-dimensiorpal Brownian motion.
Ito (1951) has shown,
under suitable continuity and growth aonditions on
i
and a, that
there is a unique solution to (7) satisfying
T
E[
f
i'(t) dt ] <
0
for all
T
0
i=1,...,n.
and
Furthermore, this solution is a
diffusion process with
Vij =
kll
Gik Gjk
in correspondence with the notation in (5), [see Friedman (1975,
Ch. 5) for an account more in line with our notation].
With this in mind, the question arose as to how Carter's model
compares with models in which prices are represented by stochastic
integrals.
The main result of this thesis is to show, under
reasonably broad restrictions on the stochastic processes involved,
that these two approaches are mathematically equivalent.
In
section 3.2 we give conditions under which an Ito stochastic
integral
ft
ft
a(s) ds +
z(t) = z(0) +
0
b(s) dw(s)
0
4
m
satisfies equations (1), (2) and (3) for fixed values of
B.. =
1J
with
k1 bik bjk
Section 3.3 is devoted to showing under suitable hypotheses -a = (ai) and
including the assumptions that
are bounded and that
B
B = (Bij), (i,j=1,...,n)
is strictly positive definite -- that a
process satisfying (1), (2) and (3) with
sented by a stochastic integral.
m = 3
can be repre-
Roughly speaking, we use (1), (2)
and (3) to integrate
lim
1
-
.
f(z(t))I,,rt]
h-1,0
with respect to
f
:
Rn
-4-
t
for sufficiently smooth functions
R, to obtain a form of Ito's formula.
This allows us
to apply a result of Stroock and Varadhan (1979), which states
under certain restrictions that if a stochastic process
z
satis-
and
B
bounded, then there exists
an n-dimensional Brownian motion
w
such that (8) holds with
fies
Ito's formula with
bbT = B.
When
B
a
is only nonnegative definite, we show that
there is a stochastic integral (possibly on a different probability
space) that has the same finite dimensional probability distributions as
z.
5
In Chapter II the necessary background .material in probability
theory, continuous parameter stochastic processes and Ito integrals is briefly sketched with references given in lieu of proofs.
Little effort has been made to reference the original sources.
It
is assumed that the reader has a basic knowledge of measure theory.
Chapter II contains one noteworthy result which we were unable
to find in the literature.
In section 2.6 the theory of Banach
space-valued random variables is used to establish a sufficient
condition for the conditional expectations of a continuous stochastic process to have continuous versions.
6
II
MATHEMATICAL PRELIMINARIES
Probability Spaces and Conditional Expectation
2.1
,P) be a measure space where
Let
a-field of subsets of
defined on TX
2
and
P
2
is a set, fr is a
is a positive CT-additive measure
If P(2) = I then (2, f;;;- ,P) is a probability space.
.
When dealing with probability spaces, the term "almost surely (a.s.)"
On a
is used interchangeably with "almost everywhere (a.e.)".
probability space convergence in measure is called convergence in
probability.
A real valued measurable function on a probability space is
If a random variable
called a random variable.
f dP
then we write E[f] for
f
is integrable
and we call this integral the expec-
2
tation of
f.
An n-dimensional random variable is an
Rn- valued
Most of the definitions
measurable function on a probability space.
and results for random variables can be extended to n-dimensional
random variables by dealing with their components individually.
Let
e
variable.
fX
and
f
an integrable random
The conditional expectation of
f
with respect to
be a sub CT-field of
is defined to be any e'-measurable random variable, E[f
such that for all
f dP =
c
CE er
E[f
IJ
dP.
I
e
er
],
7
The Radon-Nikodym Theorem gives existence and uniqueness a.s. of
e
E[f
Following is a list of properties of conditional expectation.
In this list
and
f
probability space
and
,!;
(2,
,
P),
[Ash (1972, Ch. 6)] (a)
a.s. then E[f
e;
]
'
I
f
If
E[g
1
are real numbers,
]
I
IcY:]
E[f
I
e
= f
C
]
= a E[f
tr ] +
I
a.s.
= E[f]
a.s.
15 then
I
]
]
E[f.
I
]
]
= E[f
In particular
I e ]]
is equal to a constant
1e]
E[lfl
<
1
f0,2} ]
6'
f
a.s.
e
bg
E[f
If
a.s. then
and
E[E[f
b
a.s.
k
qf 1 6 ]
If
E[E[f
]
g
E[af +
E[E[f
and
a
are sub a-fields of
2.1.1
E[f
are integrable random variables on a
g
= E[f]
a.s.
]
I]
a.s.
a.s.
a.s.
b E[gle]
a.s.
k
8
If
(h)
E[fg
is tr-measurable and
g
1e]=
g E[f
Ie]
is integrable then
fg
a.s.
We also have the Dominated Convergence Theorem for conditional
expectations:
2.1.2
[Ash (1972, p. 257)]
Let
fm
be a sequence of random vari-
ables such that
lim
fm = f
a.s.
If there is an integrable random variable
I
then
fm
t]
E[fm
lim
such that
g
I
E[f
g
exists and
I
t; ]
= E[f1
6]
a.s.
m÷.
Since conditional expectation is order preserving (2.1.1 (b)),
it is a simple exercise to show that Wilder's inequality is true
for conditional expectations.
2.1.3
Let
p
l/p +1/q = 1.
integrable and
and
If
q
In')
be real numbers greater than
and
10
1
are integrable then
such that
Ifg1
is
9
EUfgl
]
(E[FfIP
e'
])11P(E[IgIcil &])11c1
a.s.
A useful consequence of FOlder's inequality occurs when g = 1:
2.1.4
If
p
E[Ifl
1
I
and
]/)
ifiP
is integrable then
E[IfIP
]
a.s.
10
Independence
2.2
Let (2, 507,P) be a probability space.
(i = 1,...,m) of sub o-fields of
C.
fzr
A finite family
is independent if for all
1,...,m)
E
II cm)
p(cirl
p(cm)
P(C1)
E I) of sub a-fields of
A familyi'
.
fr
is independent if
every finite subcollection is independent.
a(f
(i
fi,
If
E J) is a collection of random variables then
J) will denote the smallest a-field with respect to
i
which each
E
A collection of random variables
is measurable.
f.
J) is independent if the collection of a-fields
E
J) is independent.
Often, for convenience, we will
say that two a-fields, or two random variables, are independent
when the collection consisting of the two objects is independent.
When we say that a random variable
we mean that
2.2.1
a(g)
and
[Loeve (1978, p. 15)]
and the 0-field
E[f
6
g
is independent of a a-field
are independent.
If the integrable random variable
are independent, then
] = E[f]
a.s.
f
11
2.2.2
[Ash (1972, P. 227)]
If the integrable random variables
fi, (i = 1,...,m) are independent, then
E[fl
fm] = E[fl]
E[fm]
.
12
Stochastic Processes
2.3.
An n-dimensional stochastic process on a probability space
f
54-,P) is an R'1-valued function of two variables
t C
where I, the index set, is any set such that for each
2
function f(t, ) :
f(t, ) by
f(t) = (fi(t)), (t E
is the
ith
component.
We will refer to this process as
In our applications,
f.
For a fixed
usually be an interval of real numbers.
path of
.,w)
:
I
We
I), (i = 1,...,n) to denote an n-
f(t)
f(
the
I
We will use the
f(t).
dimensional stochastic process where
function
Rn
Ix 2
Rn is an n-dimensional random variable.
-4-
will denote the random variable
notation
:
Rn
-0-
wE
I. will
2
the
is called a sample function or sample
f.
Acollection.0
J) of stochastic processes, with a
fi'
common index set
t E I),
I, is independent if the collection of a-fields
(if 3) is independent.
Given a one-dimensional stochastic process
f(t), (tE I) , let
D = (ti,...,tr) be a finite sequence of distinct elements of I.
let fD = (f(t1),f(tr)) and notice that
random variable.
(Rr,
We use
fD
Then
is an r-dimensional
fD
to induce a probability measureD on
csr
Lig ) by the formula
(E) =
Let L.D
P(fD-1
(E))
for
E
C,S r
.
be the collection of all finite sequences of distinct ele-
ments of I.
Then
{,2D
D
'
C(201
is called the system of finite
13
dimensional probability distributions determined by
If
and
f(t)
E
g(t),
1)
are two stochastic processes
on a probability space, we say that
equivalent if for every
g
is a version of
f.
tE
f
and
are stochastically
g
We then say that
1, f(t) = g(t) a.s.
Often, given a process
find another process which is a version of
conditions.
f.
f
we will need to
and satisfies certain
f
When the original process is no longer referred to we
will use the same symbol for the version of
f
as for
f.
The
next result shows that two versions of the same process are equivalent for empirical purposes.
2.3.1
[Yeh (1973, P. 3)]
Two versions of the same stochastic
process have the same system of finite dimensional probability
distributions.
14
Separability
2.4
f(t), (tEI) be a one-dimensional stochastic process where
Let
I
Unfortunately functions such as
is an interval of real numbers.
g = sup (f(t)
or
h = lim f(t)
:
t
C I) may or may not be random
t-4-c
As an example consider a set of the form
variables.
A = fw
:
g(w) > dI =
1.1
{03
f(t,w) > d}
:
.
tEI
The union on the right involves an uncountable number of sets.
Even in the cases where
A
is not in general
is measurable, P(A)
determined by the finite dimensional probability distributions of
f
(for an example, see Ash and Gardner (1975, p. 161)).
The notion
of separable stochastic processes was introduced to avoid these
difficulties.
Suppose that
f(t), (t C
I
A stochastic process
is an interval.
I) on a probability space (Q, jr,P) is separable if
there is a countable dense subset
set, and a set
N Cf.;" with
Io of
I, called the separating
P(N) = 0, called the negligible set,
such that the following condition holds for
f(t,w)
t
is a closed interval and
A
If
e
A
for all
t
E Ion J,
J
then
w
fi(
N
is an open interval and
f(t,w) C A
for all
El rbi.
2.4.1
[Doob (1953, P. 55)]
Let
f(t), (t
stochastic process with separating set
10
E I) be a separable
and negligible set N.
15
Suppose
(1)% N, to
e
f(t.1)
lim
oE
,tT.
I, and
o
exists (and is thus a random variable due to the countability of
I0).
Then
lim
f(t,w)
t+to,tEI
exists and the two limits are equal.
An extended random variable is a measurable function from a
probability space to the extended real line.
process is a function
f
:
An extended stochastic
such that for each
14'2
is an extended random variable.
t E I, f(t, -)
The concepts and results discussed
so far for random variables and stochastic processes are also valid
in the extended case.
2.4.2
[Doob (1953, p. 53)]
If
stochastic process, then sup (f(t)
f(t), (t E
:
t
I)
is a separable
E I) is an extended random
variable.
Statements (2.4.1) and (2.4.2) will allow us to study some
sample path properties of separable processes.
existence of separable versions.
We still need the
16
(Separability Theorem), [Doob (1953, p. 57)]
2.4.3
f(t), (t E I) be a stochastic process where
I
Let
is an interval.
Then there exists a separable extendec real valued stochastic process
g
that is a versio5 of
f.
Notice that by (2.3.1), replacing a process by any of its
separable versions does not change the finite dimensional
probability distributions.
17
Integration of Sample Paths
2.5
f(t), (t E
A stochastic process
(2,
5re ,P) is measurable if
f
and
I42
:
I
1)
on a probability space
is a Borel measurable subset of
R
R is a measurable function on the product space
IxQ.
The following result, which is an application of Fubini's
Theorem, uses measurability to justify the integration of sample
functions.
Let
[Doob (1953, p. 62)]
2.5.1
f(t), (t E I) be a measurable
Then
stochastic process.
f
almost all sample functions of
of
are measurable functions
t.
if
able function of
if
t E
exists for all
E[f(t)]
I
then
E[f(t)] is a measur-
t.
DI is Borel measurable and
En-qt)1]
tegrable over D, then almost all sample functions of
integrable over
D
and
f
f(t) dt
is Lebesgue inf
are Lebesgue
is a random variable.
To establish conditions under which we can interchange an integral and a conditional expectation, we need the following result:
2.5.2
Lemma
If
h
and
on the probability space
k
are
(2,17,P)
&-measurable random variables
where
is a sub a-field of
18
e-measurable random
04 and E[nh] = E[nk] for every bounded
variable
n
a.s.
h = k
then
Let
Proof:
if
h(w) -
if
h(w) - k(w) < 0
k(w)
C
n(w)=
Since
E[n(h-k)] = 0.
2.5.3
e -measurable random variable, we have
is a bounded
n
Hence E[
h-k
I
f(t),
Let
Theorem
I
]
= 0, and h = k
C I) be a measurable stochastic
a sub a-field
process on the probability space (0, 54-,P) and
of
X
0
a.s.
Suppose that E[ If(t)I] is Lebesgue measurable and
.
JE[If(t)I] dt <
If there exists a measurable version of
.
the process E[f(t)ltp° ], (teI), then for any such version
I
Proof:
E[f(01
e]
If(t) dt
dt = E[
]
I
-measurable random variable.
Let n be any bounded
In
view of the properties of conditional expectations (2.1.1) we see
that,
E[
In E [f(t)
]
I
]
dt
E[E[ In f(t)Ilg ]
] dt
19
=
I]
In f(t)
E
j
since
dt
is bounded.
n
Using this and the hypothesis that
E[If(t)I] is Lebesgue inte-
grable, we may apply Fubini's Theorem twice to obtain
E[f(t),
E[ n
E[f(t)
E[
=
dt ]
e; ] ]
1
dt
EE n f(t)] dt
j
=
e]
f(t) dt ]
= E[ n
.
Also by (2.1.1),
f(t) dt
E[ n E[
J
I
e; ] ]
f(t) dt ]
E[
.
This establishes the formula
E[f(t)I g ] dt ]
E[ n
1
= E[ n E[
for all bounded
j
f(t) dt
I g J]
-measurable random variables
now follows by Lemma 2.5.2.
n.
The theorem
0
20
Let
where
be a stochastic process on
f(t), (t C I)
is an interval,and
I
of sub a-fields of
5c 5.
is adapted to
F; i.e.,
f(t)
If
t
fFt,
(
t
(
E I) a nondecreasing family
s,t E.
if
and
I
s < t
c'97-measurab1e for each
is
(2, 5F.,P),
then
t C I
t C I).
We will generally need a condition stronger than adapted.
each
t E I
a-field on It.
tE
(
f
ni
It =
let
Then
It
is
and let
S(It)
For
be the Borel
is progressively measurable with respect to
f
I) if for all
restricted to
f
then
45
t
C
(It)
I
the stochastic process given by
xPf-measurable.
When it is clear
which family of sub a-fields we are referring to, we simply say
that
f
is progressively measurable.
In the next section we will
state a result which gives sufficient conditions for the existence
of a progressively measurable version of a process.
21
Sample Path Continuity
2.6
Given a stochastic process
f(t), (t E I) on a probability
I, we say that
space (2, 5,17(-,P) and a real interval
continuous if almost every sample function of
ous on
I.
f
f
is right
is right continu-
(Although "a.s. right continuity" would be appropriate
here, it is common practice to omit "a.s." in this context.)
is continuous then we say that
f
almost every sample path of
If
f
is continuous.
The following is a list of results relating continuity with
separability and progressive measurability.
2.6.1
[Ash and Gardner (1975, p. 163)]
continuous stochastic process, then
f
If
f(t), (tE 1) is a
is separable and the
separating set Io can be taken as any countable dense subset of
2.6.2
[Ash and Gardner (1975, p. 163)]
right continuous, then
right endpoint
f
f(t), (te 1)
Further if
I
is
has a
y, then the separating set Io can be taken as any
countable dense subset of
2.6.3
is separable.
If
[Yeh (1973, p. 57)]
I
containing
If
y.
f(t), (te I) is separable and is
stochastically equivalent to a continuous stochastic process then
f
I.
is itself continuous.
22
If the continuous stochas-
[Ash and Gardner (1975, p. 170)]
2.6.4
tic process
f(t), (tE I) is adpated to a nondecreasing family of
tEI) thenfhasaversion which is both progres-
a-fields (5"(
t'
sively measurable with respect to
t
(
I) and separable.
Next we state a theorem due to Kolmogorov that gives a sufficient condition for a process to be continuous.
2.6.5
Let
[Ash and Gardner (1975, p. 164)]
separable stochastic process.
f(t), (t E I) be a
If there are positive constants,
a, b, and C such that
C It-
f(t)-f(s)I b]
for all s,t E I, then
2.6.6
Corollary
(Q, X,P)
e
is continuous.
f
f(t), (tE I) is a stochastic process on
satisfying the hypotheses in (2.6.5) with b
is a sub a-field of
process E[f(t)
Proof:
If
sil+a
Ie],
(t
5
1, and
, then there is a version of the
E I) which is continuous.
We will show that every version of E[f(t)
e]
satisfies
the hypotheses of (2.6.5) with the same a, b, and C as for
Hence, any separable version of E[f(t)
use of (2.1.1) and (2.1.5) we see that
1
e]
is continuous.
f.
Making
23
E[I
E[f(t)!
EHEE
E[f(s)
]
f(t)
E[E[If(t)
-
-
f(s)1
f(s)i b
f(t) - f(s)
= E[
]
!
]
t;' Db]
r]-_!
b]
0
C1t-sli+a
Next we wish to give another condition under which continuity
of a process
implies that the process
f
E[f(t)
]
I
has a con-
To do this we make use of the theory of Banach
tinuous version.
space valued random variables and Bochner integration.
The source
used for this material was Diestel and Uhl (1977, pp. 41-50, 121-123).
Let
with norm
f
0, and
II
be a finite measure space, X
a Banach space
4
A function
15
(A,
the Borel a-field of
X.
: A÷X is a simple function if there exists xiEX and EE (5
= 1,...,n) such that
f =x. xE.
i=1
where
XE.
1
1
is the characteristic function of the set
E..
A
1
function
f: A -> X is
(i) strongly measurable if there is a sequence of simple functions
fn
with
lim LI
n÷00
fn - f
Ii
= 0
a.e.
24
measurable if the inverse image of a measurable set is
measurable
weakly measurable if, for each bounded linear functional
x
R, the real valued composite function
X
:
x of
is
measurable.
First we will establish the relationships between these different concepts of measurability.
(Diestel and Uhl (1977) only
define strong and weak measurability.)
(Pettis' Theorem)
2.6.7
A function
f
:
is strongly
A -> X
measurable if and only if
5
there exists NE
with
p(N) = 0
and such that
f(A \ N)
is a separable subset of X, and
f
If
is weakly measurable.
f
is measurable then
functional x
.
x of
is measurable for each linear
Thus, measurability implies weak measurability.
This plus Pettis' Theorem gives us the next result:
2.6.8
f
:
Let X be a separable Banach space.
Then a function
A -> X is strongly measurable if and only if
2.6.9
Lemma
If
f
:
A -> X
II fH :A->Ris measurable.
f
is measurable.
is strongly measurable then
25
Let
Proof:
lim
be a sequence of simple functions such that
fn
a.s.
fn - f P = 0
H
n-+.0
Then
fn
is a sequence of simple functions from
to R and
A
since
Ofn II
f
-
II
II
fn -.111
we have
limllifnR-ilf01 =
0f U
Thus
0
is strongly measurable.
a.e.
Since
R
is separable,
11f
0
is measurable by (2.6.8).
A function
f
:
A
is Bochner integrable if
X
f
measurable and if there is a sequence of simple functions
is strongly
fn such
that
lim
I
ii fn - f
II
dp
0
.
A
If
f
is Bochner integrable, then for each
define
dp
f dp = lim
In
E
EE
5
we
26
where
If dp =
x. 1.1 (E.)
1
i=1
n
E
for
f
x.
=
x
En
.
1=1
Standard techniques are used to show that this definition is
independent of the sequence of simple functions chosen to
approximate
2.6.10
f.
(Bochner)
A strongly measurable function
f
:
A
is
X
Bochner integrable if and only if
I
2.6.11
HfH dp
T
Let
00
be a bounded linear operator on a Banach space
into a Banach space
Tf : A
Y.
If
Tf dp
dp\
function
G
C
.
G
(jf
T
Let
is Bochner integrable then
f : A 4- X
is Bochner integrable and for each
Y
X
(Q,
fX,P)
f
Q 4- X is called an X-valued random variable.
:
be a sub o-field of
be a probability space.
Y
and
f
A strongly measurable
Let
a Bochner integrable X-valued
er
27
respect to
variable
e
t
[f
]
I
such that for all
t; ]
with
-strongly measurable X-valued random
to be any
t[f
j
f
We define the conditional expectation of
random variable.
f dp
dp =
C
e
.
J C
Now if
] exists it is uniquely defined
;.7,[f
a.s.
The usual
existence proof for conditional expectations does not apply since
the Radon-Nikodym Theorem does not generally hold for Banach spaces.
However, by defining
e]
t[fl
for simple functions and using an
t[fl
extension argument, the existence of
grable
f
]
for Bochner inte-
can be established (cf. Diestel and Uhl (1977, p. 123)).
We will use as our Banach space the space
continuous function on a finite closed interval
C(I)
I, with the sup norm.
be the (7-field generated by the open sets of
Let
of real valued
C(I).
Since
is separable, our three concepts of measurability coin-
(C(I),
cide by (2.6.7) and (2.6.8).
f(t), (t E
Let
be a continuous stochastic process on the
I)
probability space
(0, 5X-,P).
P(20)
w E.
in
=
1
and if
20
Let
20 E X
be such that
f(.
then the sample function
,
w) is
C(I).
2.6.12
[Freedman (1971, p. 50)]
The function
by
f(.,w)
if
0
otherwise
w
E Q0
F(w) =
F
:
0 -4- C(I) given
98
is
43c-measurab1e.
f(t), it E
Let
Theorem
2.6.13
(2, 5d,P)
process on the probability space
of
fX
be a continuous stochastic
I)
e
and
a sub
a-field
If
.
I
< c°
f(t)
tE. I
E
[sup
E[f(t)
then any separable version of the process
1
t; ],
(t
E
1)
is continuous.
Let
Proof:
strongly measurable by (2.6.9).
II
Since
C(I)
is separable, F
Next for
a.e.
6.)C 2
be as in (2.6.12).
F
F(w)
II
=
sup
1
is
f(t,w)I
tE I
so
so
<
E[1, Fl ]
e[F
e]
co
.
Thus by (2.6.10)
exists.
For
t
E
I
by formula= g(t), and notice
't
F
is Bochner integrable,
define the function
that r
:
C(I)
is a bounded linear
't
operator.
We will apply (2.6.11) and the definitions of conditional expectation to establish the identity
t[F
D = E[f(t)
a.s.
R
29
t; ]
Since tr_F
of
E[f(t)
of
E[f(t)
Let
this gives us a continuous version
Then (2.6.3) shows that any separable version
].
1
C(I)
is in
tr ]
is continuous.
and make the calculation
C E
E[f(t)
1 e]
dP
= Ic
f(t) dP
(F) dP
=
C
t
F dP
=
\ C
\Jcd,
[F
]
dP)
C
I
C
\
CF I 6 3
which gives the desired result.
dP
0
30
Brownian Mction and Martingales
2.7
A Brownian motion is a stochastic process
w(t), (t
0)
satisfying
w(0) = 0
a.s.
the random variables
< tm
for any 0 < to < t/ <
w(tk) - w(tk_1), (k = 1,...,m) are independent.
if 0 < s < t
then for every Borel set A
r
/
1
p(w(t) - w(s)E A) = JA
V2(t-s)
w
\
2(t-s)2/
t-s.
[Friedman (1975, Ch. 3)]
2.7.1
dx
exp
is normally distributed with
i.e., the random variable w(t) - w(s)
mean 0 and variance
x2
If
w
has a version which is continuous
almost all the sample paths of
w
w
almost all the sample paths of
is a Brownian motion, then
.
are nowhere differentiable.
have infinite variation
on any finite interval.
Further, if the Brownian motion
ing family of a-fields
see that
w
(fc
w
is adapted to a nondecreas-
t0) then by (2.6.4) and (2.6.3) we
has a version that is continuous and progressively mea-
surable with respect to
(, t
0).
From now on, in discussing
Brownian motion, we will assume that such a version has been chosen.
(1953,
Next we give a characterization of Brownian motion due to Doob
p. 384).
31
2.7.2
f(t), (t
Let
be a continuous stochastic process
0)
adapted to a nondecreasing family of
f(0) = 0
If
a.s.
E[(f(t)-f(s))21
ffs]
and
a.s.
f;'s] = 0
E[f(t) - f(s)
= t - s
a.s.
is a Brownian motion.
f
An n-dimensional stochastic process
(t
0).
0 < s < t
for all
then
0-fields ( 5,7;, t
0), (i=1,...,n)
w(t) = (wi(t)),
is an n-dimensional Brownian motion if each wi
is a Brownian motion and the collection of processes
wi, (i=1,...,n)
is independent.
The proof for the next result, which is an extension of (2.7.2),
may be found in Friedman (1975, p. 51).
2.7.3
0), (i=1,...,n)
f(t) = (fi(t)), (t
Let
be a continuous
n-dimensional stochastic process adapted to the nondecreasing
family of
0-fields
f(0) = 0
(Grt, t
0).
If for all
0
s < t
a.s.
E[f(t) - f(s)
5A-s]
= 0
as.
and
ENfi(t)-fi(s))(fi(t)-fi(s))17s]=dij(t-s) a.s. (i,j=1,...,n),
where
13
is the Kronecker delta, then
Brownian motion.
f
is an n-dimensional
39
be a probability space and
Let (Q, .57- ,P)
a nondecreasing family of sub o-fields of
interval.
A stochastic process
martingale with respect to
t
(i)
'
(ii)
E[
1
f(s)
I
]
<
co
E[f(t)17-s] = f(s)
f(t), (tE I)
0---
( okt, t
s
and
a.s.
X,
El)
(
where
(
--t,t
I
E
I)
be
is an
is called a
if for all s, t E
I
with
33
The Ito integral
2.8
Let
(0 < a
f(t), (tE7[(1,13]),
and w(t), (t
547-,P).
<0
be a stochastic process
a Brownian motion on a probability space
0)
In this section we will give Ito's definition (cf. Ito
(1944)) of the integral
J13f(t) dw(t)
a
and state those properties which will be used later in this disserThe proofs for the statements made in this section can be
tation.
found in Friedman (1975, Ch. 4).
Since the sample functions of
are of unbounded variation
w
(cf. 2.7.1) integration with respect to Brownian motion cannot be
defined in the usual Lebesgue-Stieltjes sense.
An integration by
f
parts formula was used by Weiner; however, this restricts
to
absolutely continuous processes.
Let (
of
54--
5,
such that
each t
0
0) be a nondecreasing family of sub (7-fields
t
w
is adapted to
a(w(x+t) - w(t), x
assume that a version of
(
5,
processes
1, we define
f(t), (t
has been chosen so that
w
KID[a]
E [..a,])
f is separable,
0) and that for
0) is independent of
gressively measurable with respect to
Next for p
t
(
5prt, t
o),
ti4t.
w
We
is pro-
(cf. 2.6.5).
to be the set of all stochastic
such that
34
is progressively measurable with respect to
f
(,7'
t
0)
and
1
a
f(t)
IP dt
Notice that if
a.s.
<
f
E KP[a,]
1 < q < p, then by Holder's
and
inequality
f(t)
I
<
dt
1
f
f E Kg [a,13].
E e[ct,]
f(t)
1
I
dt
P1(13-c1) (1 6a
a
so that
q/P
N-a)
Ito's integral will be defined for
and is thus defined for
We define
f
E Kqa,13], p
MP[a,(3] to be the subset of
2.
KP[a,] such that
n
E[ J
f(t)
dt] <
00
.
a
We say that a matrix of processes is in
each element is in
K1[a,(3]
A stochastic process
MP[a,],
or
f(t), (t
KP[a,]
or
M[a,13]
if
respectively.
E [a,(3])
is a step function
if there is a partition
a = t0 < t1 <
< tm =
(3
of
[cl,]
such that for each
E
f(t,w) =
f(ti,w)
for
tE[ti,ti.41), (i = 0,...,m-1).
35
2.8.1
fk
fEK2[a,(3]
If
in
then there is a sequence of step functions
such that
K2[ct,]
2
lim
I
locc,
If
f(t) - fk(t)
dt = 0
I
is a step function in
g
a.s.
do,
a = to < tl <
< tm =
,
spect to the Brownian motion
K2[(1,]
with partition
then the Ito integral of
w
g
with re-
is defined to be the random variable
m-1
.1
g(ti) [w(ti+l) - w(t)]
1=0
and is denoted by
9(0 dw(t).
f'a
To define the Ito integral for any process
f
fk be a sequence of step functions which approaches
in
K2[a,], let
f
in the sense
It can be shown that the sequence
of (2.8.1).
'
f
dw(t)
f
is convergent in probability and that the limit is independent
of the particular sequence
integral of
f
fk
chosen.
a.s.
This limit is called the Ito
with respect to the Brownian motion
w
and is written
36
f(t) dw(t)
Ja
a
to be zero.
f(t) dw(t)
We define
a
2.8.2
K2[a,f3] and Ai
and f2 be processes in
Let fl
Then
real numbers.
is in
Xlfl + X2f2
rExifi(t) + X2f2(t)]
K2[a,6]
and A2
be
and
dw(t)
a
= X1
+ A9
dw(t)
fI
'
f2(t) dw(t)
For the next result notice that if
[a,131
to
and fE K2[a,f3]
is in
[-y,6]
2.8.3
=
I
Ja
or
restricted
f(t) dw(t)
f(t) dw(t)
+
K2[ot,13]
or
M2[a,], respectively, and
t;f(t) dw(t)
a.s.
a
a
1:21a,]
f
then
jZ7.-measurable random variable then
a bounded
of
and YE Ca,
is an element of
f
is a subinterval of
then the process given by
If f E K2[a,(3]
If
[y,(S]
K2[-y,6].
rf(t) dw(t)
a
2.8.4
a.s.
a
Ef
f(t) dw(t)
a
a.s.
M2[(1,]
f
and
E
is an element
is
37
2.8.5
If
E
f C M2Ea,], then
[f
f(t)
dw(t)1
0
=
a
rr
Li a
E
[(
f(t) dw(t)
j13'
f (t)
9-0]
I
dw(t) )2
a.s.
=
E[Jf2(t) dt]
a
E
2
[(1
f(t) dw(t)
I
a
j
=
2.8.6
(i)
fE
Let
f2(t) dt
a
(c70]
M2m[a,f3]
where m is a positive integer.
Then
2m1
EDf
f(t) dw(t) )
[m(2m-1)]m (-a) m-1
ii
a.s.
E[rSa
TaarTia
sup
,t3]
(
f(t) dw(t)
Err
Li2maf
(t)
dt]
)2m]
[4m3/(2m-1)]m N-a)m-1
E[r f2m(t)
a
Friedman gives a proof for (2.8.6) in the case where a
0.
It is clear in the proof where to make the appropriate changes
to obtain the result stated here.
expectation version of (2.8.6(i)).
We will need the conditional
To obtain this we make use of
38
the following lemma (notice its similarity to (2.5.2)):
2.8.7
Lemma
and
f
If
are
g
E[f]
with finite expectation and
E[g]
for every bounded, nonthen
if-measurable random variable
negative,
Proof:
;--measurable random variables
f < g
a.s.
Let
(f-g)(w) > 0
1
if
0
elsewhere
=
(0.))
-measurable
is bounded, nonnegative and
Since
However, due to the way
E[ E(f-g)] < 0.
(f-g)
E(f-g) = 0
Thus
0.
0
2.8.8
Theorem
was defined
f - g = 0
The set on which
a.s.
= 0
= 0; hence
included in the set on which
g - f
E
a.s.
is
Thus
0
a.s.
If
f
E M2m[a]
m
where
is a positive integer,
then
E
r(rf(t)
dw(t)
)2m
I
-ct
L
<
Proof:
Let
variable.
we obtain
E
[m(2m-1)]m (-a)m-'
LJ a
f'9 m
be a bounded, nonnegative,
(t) dt
I
Ye-a]
a.s.
fiX'-measurable random
Making use of properties (2.1.1), (2.8.4) and (2.8.6 (i))
39
E
R2m E[(i
f(t) dw(t)
y7cil
a
.
E
r-E r (''''rf(t)
L_ .c:
=
E
[ (j:f(t) dw(t) '1,2111.j
dw(t)
,
< [m(2m-l)]m
(5-a)11171
E
'',2m
/
I
511
]
a
a.s.
1
a.s.
n2m f2m
(t) dt
1
a.s.
a
= E Ec-,2m [m(2m-1)]m (ft-a)m-1 EL .rf2m (t)
dt
I
a
a
0
Then by Lemma (2.8.7) we obtain the desired result.
[A proof similar to that above can be used to prove the conditional expectation version of
2.8.9
If
is in
f
(2.3.6 (ii)).]
K2[a,3], then the process
z(t), (tE [a,])
given by
rt
z(t) =
f(s) dw(s)
I
Jot
has a continuous version.
The process
integral of
f.
z
defined above is called the indefinite Ito
Henceforth, when we speak of an indefinite Ito
integral, we shall always mean a continuous version.
We next define the n-dimensional Ito integral.
Let
40
motion and
that
0) a nondecreasing family of
( OZ7' t
t
a-fields such
is progressively measurable with respect to(
w
and that for each
t
a(w(X+t) -
0
If b(t) = (bij(t)), (t
tc.;.
be an n-dimensional Brownian
0), (i=1,...,n)
w(t) = (w.(t)),(t
K2[a,6]
,
A
C [0.:,])
then the Ito integral of
t'
0)
t
is independent of
0)
is an m x n matrix in
with respect to
b
f.,71r
w
is de-
fined to be the m-dimensional random variable
b(t) dw(t) =b..(t) dw.d
a(t) = (ai(t))
Next let
,
(i=1,...,m)
.
IJ
j=1 Ja
and
b(t) = (bij(t)), (t
CoL,(i1),
be a stochastic processes satisfying
(i=1,...,m), (j=1,...,n)
It
a
for each
t
C [a,].
a version of
z
z
a.s.
a
By (2.8.9), (2.6.4) and (2.6.3) we can find
that is continuous and progressively measurable
with respect to ('
t
t
say that
b(s) dw(s)
a(s) ds +
z(t) = z(0) +
C).
If
z
is such a version then we
is a stochastic integral.
Our next result is the chain rule for the Ito calculus.
It
and several other identities derived from it are all known as Ito's
formula.
Examination of this equation highlights several of the
major differences between Ito integrals and ordinary integrals.
41
2.8.10
u(x,t), (x E Rm, t E [a,])
Let
ut, ux., ux.x.
with continuous partial derivatives
(zi(t)),
z(t) =
Let
.
Ea,Cj
Rmx
valued function in
be a continuous real
(t
E [a,]),
(i=1,...,m) be
j
a stochastic integral
ft
z(t) =
z(a) +
a(t) dt +
j
Kl[a,]
and
a.s.,
0
0
wherea=(a.)and
b(t) dw(t)
b = (b..)' (i=1,...,m), (j=1,...,n) belong to
ij
K2[a,13], respectively.
Then
u(z(t), t)
is the
stochastic integral
u(z(t), t) =
u(z(a), a)
m
ft
+
a
i=1
+
n
I
xix'
m
(z(s),
1
a k=1
i=1
xi
(z(s), s) bik(s) bik(s)] ds
u
i,j=1
k=1
(z(s), s) a(s)
u
(z(s), s) +
[ut
s) bik(s) dwk(s)
a.s.
Ai
The next result is obtained by applying (2.8.5) to the previous
result.
It is this that we will call Ito's formula:
2.8.11
Let
u
:
Rm
R
be a continuous function with continuous
first and second derivatives.
Assume further that
satisfy the same hypotheses as in (2.8.10).
Let
z, a, and b
42
b. (t) b. (t)
1
=
B.
ij
.
jk
ik.
k=1
If the process
m
y(t) =
u
m
ux.(z(t)) bik(t)
k=1
is in
ux
.
x
.
(z(t))B(..
i,j=1
and the process
M1 [a,B,]
n
2
xi
i=1
is in
(z(t))ai(t)+1
i=1
,
(t
E [a,])
1
M2 Ca,], then for each
E[u(z(t)) - u(z(a))
t
E [ot,]
5t]
a
y(s) ds
= E
a
tore
a _1
a.s.
t),
(tE[a,$]).
43
2.9
Miscellaneous Results
First we quote a well-known inequality often used to prove
Minkowski's inequality.
[Ash (1972, p. 85)]
2.9.1
c,d
If
and
0
m
1, then
(c + d)m < 2m-' (cm + dm)
We need the following generalization of this result:
2.9.2
m
Lemma
If ci, (i=1,...,n)
are nonnegative real numbers and
is a positive integer then
m
n
C.
(i=1
Proof:
.
< 2(n-1)(m-1)
n
,
c.m
i=1
11
The proof for n
2
is by induction on
For the induction step from
(2.9.2) is (2.9.1).
use (2.10.1) to obtain:
r11
c.
1)
m
-
/ c
I
k\.
n
+
n-1
1
_m
rn-1
il
1
c.
1
n.
n-1
For n = 2,
to
n
we
44
m
I
< 2m-1
n
+ 2
(n-2)(m-1)
n-1
m
ci )
1=1
p.
< 2
=
9(n-2)(m-1
m-1
\_
i=1
Lemma
2.9.3
0
c.m
2(n-1)(m-1)
1
For n = 2,3,4,
n-1
/2n
n
(
j=0 \ji
/2n)
n
j=0
"
2'1+1)
n-1
(2n-2j) =
j=0
j=0 \4J
2n
2j+1
-2j-1)
n
j=0
Proof:
()`j (2n-2j
1
)(2n-2j-1) =
)(2n-2j-1)(2n-2j-2)
9;'
j=0
By the Binomial Theorem,
2n
(x-.)2n
j=0
/ \
2p
(-.03
\i/
2n)
=
Evaluation at
x =
./
j=0
1
x, then evaluation at
x2n-j
(2n
2n-25
2j
x
-/
gives (i).
j=0
2j+1
A
Differentiation with respect to
x = 1, gives (ii).
Equation (iii) is obtained
by a second differentiation with evaluation at x = 1.
0
45
Lemma
2.9.4
If
f
[a,b)
:
tE [a,b)
on [a,b), then for each
UrnT;
1
rtjrh
is integrable and right continuous
R
f(s) ds = f(t)
.
h4O
Proof:
if
s
Given
6 > 0, use right continuity to choose
E [t, t+d), then
ii
1
f(s) - f(t)
1
< E.
d > 0
Then for h
so that
E (0,)
ds - f(t)
(f(s) - f(t)) ds
1
it+h
f(s) - f(t)
ds
<C
2.9.5
0
[Saks (1937, p. 204)]
right differentiable on
continuous at a point
[a,b)
If
g
[a,b)
:
R
is continuous and
and the right derivative of
tE-[a,b) then
g
g
is differentiable at
is
t.
Combining this result with the Fundamental Theorem of Calculus
we obtain
46
2.9.6
If
Lemma
g
:
[a,b)
continuous right derivative
R
is cntinuous on
g.
[a,b), then for each
fc
(x) dx = g(c) - g(a)
g
a
r
[a,b)
.
and has a
cE[a,b]
47
THE MOMENT RATE CHARACTERIZATION
The Spaces MP and QP
3.1
Given a probability space
(2, Zje.
,P), let
We define
nondecreasing family of sub a-fields of
the set of all stochastic processes
f(t), (t
0) be a
57;:t, t
(
on
0)
to be
MP
(2, 5747:,P)
such that
is separable,
f
to( 9rt'
f is progressively measurable with respect
T
for all
and
0,
fT
Mt)
0)
t
P
dt
I
<
I
0
is in
iff for all
MP
Notice that
f
given by
restricted to the interval
f
We define
QP to be the subset of
sup
tlE[O,T]
E [ess
a <
0
[ct,]
MP
B
<
co
is in
the process
MP[a,f3]
such that for all
.
T > 0
f(t)
I
the set of right continuous processes in
M.
We denote by
DM P
The set
consists of those continuous processes that are in
M.
CM P
The sets
DOP
and
CQP
are similarly defined.
The following results are consequences of Holder's inequality
and the fact that the essential supremum of a product of positive
48
functions is less than or equal to the product of the essential
suprema of these functions.
q
If
1
3.1.2
If
f
3.1.3
If fE
3.1.4
If
f,
3.1.5
If
f E Q2P
E M2P
M4P
gE
MPC:Mq
then
p
3.1.1
and
and
gE
M2
then
and
g
E
m4
then
fg E Q2
then
Q4
and
g
E Q2
QPC10.
fPg E ml
fPg E m2
.
.
.
then
fPg E
Ql
.
For notational convenience, we will say that a matrix of processes is in one of the spaces
MP, DM, CM, QP,
DQP
or
CQP
if
all of its components are in that space.
Lemma
3.1.6
If
f
E DQP, where
p
is a positive integer, then
for any separable versions of the processes
t+h
P
f(s) ds)
0.71-
owt
,
(h
0)
and
rt+h
fP(s) ds
it
where
t
is fixed and
h
It
,
(ho)
is the parameter, we have
49
-1
N
0) lim
E
h+0
(1 ft+h f(s) ds,1 P
[...Ti:
t
(ii) lim
11,-
if
E
"
h+0
1
7
f"(s) ds
f(t)
=
a.s.
_1
,
F t+h
,
cdk
I
t
I
f(t)
1=
a.s.
Lt
r , t+h
Proof:
Let
be a separating set for
Ho
El
Lit
f(s) ds )
p
I
jet]
.
To justify the use of the Dominated Convergence Theorem (2.1.2) notice
0< h1
that for
P
t+hf(s)
ds
<
II
=
sup
sa0,t+1]
f
t+h
\ t
sup
slE[0,t+1]
f(s) ds) P
f(s)
and by hypothesis
f(s)
sup
E
LsE[0,t+1]
Then using (2.1.2) and (2.9.4) we obtain
lim
E
(1 It+h f(s)
-fr
114-0,helo
=E
Lh4-01,1hEn1H0
=E
[fP(t)
I
\
1
-h-
Ft_il
a.s.
d's)
I
I
r
= f(t)
it
it+h
J
t
a.s.
f(s) ds) P
a.s.
50
Applying (2.4.1) establishes (i).
by noticing that if
in (i).
f E DQP
then
Equation (ii) follows from (i)
fP
DQI and taking
p =
1
0
51
Moment Rate Properties of Certain
Stochastic Integrals
3.2
be an n-dimensional
0), (i=1,...,n)
w(t) = (w(t)), (t
Let
Brownian motion on the probability space
(2,
Tier,P)
and
c-(
okt' t
( 57'v t
is adapted to
w
that
and
a(t) = (ai(t))
processes
(i,j = 1,...,n), where
a
(w(X+t)-w(t), X
and
0)
for each
is independent of
0.
t
Given the stochastic
0),
b(t) = (bii(t)), (t
E M1
and
b
0)
E M2, let
z = (zi),
be a stochastic integral given by
(i=1,...,n)
a(s) ds +
z(t) = z(0)
Recall that since
b(s)
f
is a stochastic integral, it is continuous and
z
progressively measurable with respect to
adapted to
(i=1,...,n)
(fXt'
t
0).
(A.-'
t
t
n-dimensional stochastic process
Given an
(3.2.1)
dw(s)
0
0
f
such
0) a nondecreasing family of sub 0-fields of
f(t) = fi(t), (t
0), a moment rate function for
is a process given by a limit of the form
E
h
h)
where
(ik
:
II
(f. (t+h) -
k=1
k=1,...,n)
5
fik(t))1
/k
is a sequence of indices in {1,...,n}
and a separable version has been chosen for the process
-m
E
II
k=1
(f. (t+h)(t))1
1k
lk
0),
52
For the remainder of this section the symbol
a stochastic integral with
will be placed on
and
a
and
a
b
of moment rate functions for
b
z
as in (3.2.1).
will denote
Conditions
that will guarantee the existence
and allow us to obtain explicit
z
expressions for them.
3.2.2
Lemma
If
a
E M1
and
b
E M2
then
Frt+h
E[z(t+h)
-
z(t)1
a.s.
a(s) dsl
9rt]= E
it
Proof:
We use (2.8.3) and (2.8.5) to see that
E[z(t+h)
-
r= E
I
z(t)I
5z:t]
t+h
clrl
t+h
a(s) ds +
b(s)dw(s)
j
I
a.s.
t
= E
F t+h
I
a(s) dsl
O4t
0
a.s.
L.
3.2.3
Theorem
If
a E
version of the process
DQ1
and
E M2
then for any separable
E[zi(t+h) - z(t)1
lim- E[z.(t+h) - z(t)1
h
b
ge-]
t
= a(t)
(h
0) we have
a.s.
h4,0
Proof:
This follows from Lemmas 3.2.2 and 3.1.6.
0
53
For higher moments we require
Lemma
3.2.4
If
and
a
(i=1,...,n), where
are in
b
E[z2p. (0)i< c°
is a positive integer, then
p
Proof:
By (2.9.2) there is a constant
and
such that
p
i
and
M-9p
Cn,p
Q2p
is in
z
depending only on
n
/rt
Izi(t) 12P
C
z.(0
L
n,p
)I
2P +
1
1
ds) 2P
i
2p
it
dw.(s)
b..(s)
1J
j
la (s)
\JO
It suffices to show that the essential supremum of each term individually in the above equation has finite expectation.
(ft
ess sup
L tE[O,T;
ai(s)1
1
\
/
1.-.
00
< E FT
2p-1
J
\ 2p]
/P rT
1
0
ds
\\JO
r"
= E
1
T
For the second, we have for
term is covered by hypothesis.
21ElF
The first
1
a.(s)I ds
/
1
T
2
a.-P(s) ds]
i
f0
CO
where Holder's inequality was used on the integral
I
a(s)
1
ds
.
0
To handle the Ito integrals we make use of (2.8.6) to see that
for some constant K. depending only on p,
54
Eless sup
rtI
tE0 TJ
Jo
L'
P-1
< K T,
<
3.2.5
Ir
Lio
j
3
13(s)
El
P
dw.(s)11
b..
213
li 2P(s) ds,j
b.
00
Theorem
If
a,b
E Derlm4
then for any separable version
of the process
E[(zi(t+h) - zi(t))(zj(t+h) - zj(t))I
fFt-t],
(h
0)
we have
Er(zi(t+h) - zi(t))(zi(t+h) - z(t)fl 5rt]
lim
=
nb.lk (0
k=1
(t)
a.s.
Since we are dealing with differences we can assume without
Proof:
loss of generality that
(
bjk
t
z(0) = 0.
First, since z is adapted to
0),
E[(zi(t+h) - zi(t))(zj(t+h) - zj(t))I
5AI]
zi(t)zj(t)i
f.-Xt]
= E[zi(t+h)zj(t+h) -
z(t) E[zj(t+h) - Zj(t)I
6PYt]
z(t) E[zi(t+h) - zi(t)I
570
a.s.
(3.2.6)
55
u(x) = xixj
Next we will apply Ito's formula (2.8.11) with
to the first term on the right side of the above equation to
obtain
E[zi(t+h)zj(t+h) - zi(t)zj(t)I
f;Tt]
t+h
a1(s) + z1(s) ad(s)
j.(s)
t
b.k(s) bjk (s)] dsl
+
=
k1
Crti
(3.2.7)
a.s.
1
To show that the hypotheses of (2.8.11) are satisfied, first notice
that by Lemma 3.2.4
are in
zjbjk
M2.
are in Q1
bikbjk
z
is in
Q4; thus by (3.1.2)
From (3.1.5) we observe that
and hence in
M1
z.jbik
and
and
zjai' ziaj
.
The above paragraph also shows that the integrand in (3.2.7) is
in
Q
1
;
so we can apply Lemma 3.1.6 to obtain
lim
1
1.7
EEzi(t+h)z.(t+h) - z-(t)z.(t)i
5.t]
11+0
= z(t)a1(t) + zi(t)aj(t) + kribik(t) bjk(t)
a.s.
Combining this with Theorem 3.2.3 and equation 3.2.6, we obtain
the desired result.
0
56
3.2.8
m
If for an integer
Lemma
lim 1
h40
"
11M
T;
h+0
"
2
z(t) Im
E[1z.(t+h)
= 0
(j=1,...,n)
a.s.
then
m
E[
E
1z4
k=1
kt+h) -
lk
(ik
for any sequence of indices
Proof:
z. (t)1t
,
:
= 0
k=1,...,m)
a.s.
{1,...,n}
in
.
This follows by applying Holder's inequality (2.1.3)
m-1
times in the following manner:
E [H Iz.
k=1.
1t]
(t+h) - z4
lk
lk
l/m
(Ellz. (t+h) - z. (t) 1 m1
11
-LP ])
11
(m--1)
x (E[
E
z. (t+h) - z. (t)
I
k=2
m
2
E (El
1
z4 (t+h) - z,
ii
j=1
x (El R lz. (t+h)
k=3
i
(t)
I
tftt])
m/(m-2)(m-2)/m
't])
z. (t)1
-
l/m
I
k=1
a.s.
1
1k
m
E
'let])
I
lk
1k
a.s.
1
(Ell z. (t+h)
lk
-
zi
(t)
1
1/M
1
..)kt])
a.s.
o
57
IZ.
ET E
where
z.lk(t)11 Ft],
0)
(h
is an integer greater than or equal to
p
k=1,...,p)
:
(t+h) -
lk
k=1
(ik
Consider a process given by
Theorem
3.2.9
is a sequence of indices in
and
3
{1,...,n}
.
If
any one of the four conditions below holds, then every separable
version of this process satisfies
1
Urn-17
11+0
"
P
E[ E
4L]
(t+h) -
k=1
is even and
p
is odd, a E DQP
p
is even, a E Derlm2P-4
p
is odd, a E Derlm2P-2 and
and
b
C
;
b E
and
b
1
h0
E 0041-1M2P-2
m
;
.
P
ETI zi(t+h)
-
z(t)
=
I
Suppose condition (1) holds.
Write
and Lemma 2.9.2 we have, for some constant
on
DQ4()M2P-4
By Lemma 3.2.8 we need only show that
lim
Case 1:
a.s.
0
a,b E DQP
p
Proof:
=
zik(t)11
lk
and
n,
0
p = 2m.
Cm,n
a.s.
By (2.8.3)
depending only
58
E[(zi(t+h) - zi(t)
r-
t
(
tJ1
21111
2m
h
a(s) ds)
< Cm,n
,t
\2m
(1
7
o4t
b,j(s) dw(s))
Now we use Lemma 3.1.6 to see that
2m
t+h
1
a.(s) ds
E
11+0
"
I
t
\2m
ft+h
= lim h2m-1
h+0
=0
a,(s) ds
E
I
t
a.s.
For the remaining terms we apply (2.8.6) to obtain, for some
m,
constant Km depending on
/
E
2m
t+h
jb ..(s)
'
dw(s))
J't
I
t+h
b.."(s) dsl
< Km hm-1
(3.2.10)
a.s.
.111
Then noticing that
m
2
and applying Lemma 3.1.6, we find that
t+h
lim
h4
m-1
1
-F
Kmh
.1
bij2m(s) dsl
E
t
Or`t
=
0
a.s.
59
Case 2:
Write
Assume condition (2).
p = 2m+1.
By Holder's
inequality (2.1.3), we have
E[
zi(t+h) - z(t)
2m+1
t-
1
Liet ]
(E[(zi(t+h) - zi(t))2m1
)1/2
'])1/2
a.s.
x (E[(zi(t+h) - zi(t))2m4-21
This reduces Case 2 to Case 1.
Case 3:
Suppose condition (3) hold and write
p = 2m.
By the
Binomial Theorem
E[(zi(t+h) -
2m
= E[
zi(t))2m1
(
`'.")
ct74-.0
.
(-1)J
i
J(t+h)
j(t)I
5c]
a.s.
j=0 ,j
= E[ 1
2)
j=0
m-1
- Er
L,00
z42m-23(t+h)
zi2jWI
ji
/
z.2m-2j-1 (t+h) z 2j+
\23+1
Now making use of Lemma 2.9.3(i), we subtract
m\
z.2m(t)
i=0
t#-t]
a
1(t)I
60
from the first term and add the equal expression
z.
2mto
kt./
(2m)
n.,,
L
`37-1
j=0
to the last term, to obtain
E[(zi(t+h)
=
m-1
zi2i (t) E[zi
2217.1
(
j=0
-
Z(t)2m 191-0
-
J
f)44.1
2m-23 -1 (t+to
"I
21:1 z7J l(t) E[zi
)
j=0
z2.m-2j-1
(01
t j
(3.2.11)
a.s.
1
We will apply Ito's formula (2.8.11) with
q = 2,3,...,2m
zi2m-2j(01 ?-7-.t3
2m-2j(tol)
u(x) = xiq
for
to obtain
E[zig(t+h)
-
ziq401
64.74]
t+h
I
= E[
{ q a(s) z.q-1
(s)
it
} ds
2
a.s.
(3.2.12)
11
where
B(s) =
.
k=1
bik2(s)
First, however, the hypotheses in (2.8.11) must be verified.
By
61
Lemma 3.2.4 we see that qz.cFrom (3.1.4) B
so by (3.1.5) Bii zic1-2 and
(3.1.2)
zic1-1
b.:1K
ai
zq-1
are in
01.
E Q2,
Finally by
E M2.
By taking the last term to be 0, equation (3.2.12) also holds
for
q = 1
by Lemma 3.2.1.
all terms equal to
It holds trivially for
q = 0, with
0.
The next step is to obtain the limits
1
UrnErz ci(t+h) - z.cl(t)1
t
i
h4,0
q a(t)
z1(t)
a.s.
+ q(c1-1) B. (t) z.c1-2( t)
2
This is done by applying Lemma 3.1.6 to the right side of equation
3.2.12.
ai
and
The hypothesis that the integrand is in
are right continuous,
bij
z2 and
above,
a.
z.c1-1
12101
holds since
is continuous and, as shown
zi
are in
Q.
Then applying this to equation 3.2.11, we obtain
E[(zi(t+h) - zi(t)
2m
orwt]
h4,0
,.
=
(2m-2j) a(t) zi2m-1(t)
j0
=
(7
1
+
(-2m\I
1
(2m-2j)(2m-2j-1) B(t) z.2m-2(t)
11
j=02j)
m-1
2m
m-2j-1) a(t)
j=0
z.
-.(t)
62
1
m7:1
(i 2m
'2j+1
(2m-2j-1)(2m-2j-2) B(t) zi2m-2(t)
a.s.
i=0
Reference to Lemma 2.9.3 snows that the above expression is equal to
a.s.
Case 4:
Assume condition (4) holds.
The technique used to reduce
Case 2 to Case 1 also serves to reduce Case 4 to Case 3.
0
63
Representation Theorem
3.3
Let
(Q,
57-,P)
ia-fields of
nondecreasing family of sub
(Bij(t)),
B(t) =
and
a(t) = (a(t))
(5rt,
be a probability space and
fX.
t
0)
a
Suppose that
0), (i,j=1,...,n) are
(t
stochastic processes which are progressively measurable with respect to
(L9rt, t
function
f
(Lt
:
For any twice continuously differentiable
Rn
0).
-4-
f)(x) = -12-
i,J=1
(Lt
tic process, then
f, (t
0)
(x) +
B. .(t) f
xixj
13
1=1
z(t), (t
Notice that if
Lt
define
R
0)
by
a.(t) f
1
(x)
(3.3.1)
xi
is a progressively measurable stochas-
f)(z(t)), (t
0)
defines another progres-
sively measurable stochastic process.
Rn
possessing continuous derivatives of all orders and having comFollowing Stroock and Varadhan (1979) we say that a
pact support.
stochastic process
B
the space of real valued functions on
C; (Rn)
We denote by
and drive
a
z(t), (t
if for every
metric nonnegative definite and
0)
is an Ito process with covariance
t
0
z
t
measurable with respect to fX
('
t
and AE2,
B(t,(0)
is sym-
is continuous, progressively
0)
and the process
ft
(Lf)(z(u)) du, (t
f(z(t)) 0
0)
u
is a martingale with respect to (9;..t, t
0)
for all
-HEX; (Rn).
64
(This generalizes Stroock and Varadhan's definition by omitting
their requirement that
and
a
are bounded.)
B
A more convenient formulation of the martingale property for
Ito processes is
E[f(z(t))-f(z(s))1F]=E[
f (Lu
S
a.s.,
f)(z(u)) du154;]
S
(0
(3.3.2)
s
From this we see that Ito's formula (2.8.11), with
u6C°90 (Rn),
holds for Ito processes.
Stroock and Varadhan (1979, p. 108) have proved the following
theorem:
Suppose that the stochastic processes
3.3.3
B(t) =
sively measurable with respect to
that
matrix for each
bbT = B.
B
If
and drift
motion
w
(fFt,
t
0).
Suppose further
is a strictly positive definite n x n-symmetric
B(t,w)
(i,j=1,...,n)
are bounded and progres-
0), (i,j=1,...,n)
.(t)), (t
(B.1 J
a(t) = (ai(t)) and
t
0
and
wE Q.
Let
b(t) =
(bij(t)),
(t
be a progressively measurable process such that
z(t), (t > 0)
a.
is an Ito process with covariance
Then there exists an n-dimensional Brownian
adapted to
(frt,
t > 0)
for which
0),
65
rt
Pt
a(s) ds +
z(t) = z(0) +
b(s) dw(s)
!
do
0
The above result needs the existence of a progressively
measurable
b
such that
bbT = B.
In our application
B
is a
continuous process.
Under this condition we can establish the
existence of such a
b.
We will use the next result to accomplish
this.
[Friedman (1975, p. 128)]
3.3.4
xe G
For each
n x
n-matrix.
root of
order
D(x).
be an open set in
d(x) = (d..(x))
be the positive definite square
1J
i
and
i.e., has continuous derivatives up to
j, then for all
positive square root is symmetric.
B
for
b.
By (3.3.4), b
(2.6.4) can be chosen
to
and
i
To apply this result, notice that since
of
R.
be a strictly positive definite
If D Cm,
m, for all
G
(Dii(x))
let D(x) =
Let
Let
B
j
d..ELCm
1J
is symmetric its
Thus we can use the square root
is a continuous
process which by
be progressively measurable.
In the following theorem we establish sufficient conditions
on a process and its moment rates to insure that it is an Ito
process.
3.3.5
Theorem
Let
z(t) =
(zi(t)),
(t
0), (i=1,...,n)
be a
continuous stochastic process that is progressively measurable with
66
(57e7t, t
respect to
0).
a(t) = (a(t))
Let
0), (i,j=1,...,n)
B(t) = (B. .(t)), (t
13
in CQ1 such that for each
and
be stochastic processes
and t > 0, B(t,w)
wE,C2
is a symmetric
Suppose that whenever separable
nonnegative definite matrix.
versions of the processes given by the conditional expectations
below are taken, the following conditions hold for
u
and
0
i,j=1,...,n:
(i) there is a
numberu >
and a nonnegative random variable
0
with finite expectation such that for every
Mu
1
E[zi(u+h)-zi(u)1 °--,74tu] <Mu
I
i)
1 im
E[z. (u+h)-z (u)
1-
h
(0,5u)
a.s.,
= a . (u)
a.s.,
u
1-14,0
lim
1
a.s. and
E[(z4(u+h)-zi
u(u))(z.(u+h)-z.(u))1X]=B..(u)
'
114,0
lim
Enz1(u+h)-zi(u)13
127"-u] = 0
a.s.
h4,0
Then
z
Proof:
is an Ito process with covariance
B
and drift
a.
We take separable versions of all processes occurring through-
out the proof.
All limits taken here are then justified in the
sense of (2.4.1).
First we note that by Lemma 3.2.8 and condition (iv) we have,
for
1 < i,j,k < n,
1(3.3.6)
67
II
m=i,j,k
E[
Urn-F
114
Now let
for
1
be in
f
z m (u+h\-z m(u'.
-,,
/
C
(Rn).
Ill]
= 0
We will show that (3.3.2) holds
0,
By Taylor's formula, we have for h
z.
(zi(u+h)-zi(u)) fx
f(z(u+h))-f(z(u)) =
a.s.
(z(u))
1=1
n
1
+
(zi(u+h)-zi(u))(zi(u+h)-zi(u)) fx.x. (z(u))
1
2-
J
i ,j=1
1
+
[
-6-
II
i,j,k=i m=i ,j,k
where
is
Since
x.x.x
j
a
(z(u))
f
x.
point on the line connecting
and
]
(zm(u+h)-zm(u)) f x.xx.
j k
f
(z(u))
are
and
z(u)
z(u+h).
-measurable and
xxij
are bounded functions we see that by (ii), (iii)
and (3.3.6)
k
1
lim
E[f(z(u+h))-f(z(u))
5;-]=
(Lf)(z(u))
u
u
a.s.
Thus for 0
u,
1-14-0
where
E[lim
h+0
Luf
is as defined in (3.3.1).
s
E[f(z(u+h))-f(z(u))11n= E[(Luf)(z(u))1
"
s]
a.s.
(3.3.7)
68
be a separating set for the process
Let
Ho
E[f(z(u+h)) - f(z(u))Ifc, (h
the fact that
f
has compact support,
If(z(u+h)) - f(z(u))
for some constant
By the Mean Value Theorem and
0).
C.
C iz(u+h) -
z(u)1
Using (i) with the above equation shows that
we can apply the Dominated Convergence Theorem (2.1.2) and use
(2.1.1g) and (2.4.1) to obtain
E[f(z(u+h)) - f(z(u))
E[lim
15ru] 15rs]
114
E[f(z(u+h) - f(z(u))
lim
= EE
1:Xu] IfXs]
a.s.
h4-0,hcHo
lim
=
.
-
f(z(u))
IF's]
a.s.
h4-0,hfH0
1
.
= lim
E[f(z(u+h)) - f(z(u)) Is
a.s.
-F
h4-0
Thus,
.
lim
11+0
1
E[f(z(u+h))-f(z(u))1
]=EHLu f)(z(u));]
a.s. (3.3.8)
69
Since
f
xi
and
(z(t))
processes,
f
x.
f
1
1
and
f
x.x
1
1
j
j
(z(t)), (t > 0)
are bounded functions and
CQ1, we see that the process
are in
are continuous
Cu
(L
a
and
B
is in
> 0)
CQ1.
Then by Theorem 2.6.13 we may choose a version of the process
E[(Lu
f)(z(u))1j4rs], (u
0)
which is continuous.
Using (2.6.3)
and (2.6.4) we make this choice so that the process is both continuous and progressively measurable with respect to
(fXt,
t
0).
In accordance with (3.3.8), this provides a continuous and progressively measurable version of
.
lim
h+0
1
F[f(z(u+h)) - f(z(u))ljes], (u
h
0).
Thus we can integrate (3.3.8) to obtain
I
1
E[f(z(u+h)) - f(z(u))
lim
s h+0
=EEL_ f)(z(u)) If;;] du
s
] du
a.s.
"
We now apply Lemma 2.9.6 to the left side and Theorem 2.5.3 to the
right side of the above equation to get (3.3.2).
0
Combining (3.3.3) and Theorem 3.3.5, one obtains the following
representation theorem, which characterizes certain Ito integrals
in terms of moment rate properties:
70
Suppose that
Theorem
3.3.9
bounded and that for each
w
adapted to
coE Q
and
0
t
,
and
a
are
B
is strictly
B(t,w)
bbT = B, there is an n-dimensional Brownian motion
(frt.,
t
0)
for which
t
z(t) = z(0) +
f
t
a(s)ds +
b(s)dw(s)
i
0
0
We next consider the case when
finite.
satisfy the hypo-
B
Then for any progressively measurable process
positive definite.
such that
and
a
Suppose further that
thesis of Theorem 3.3.5.
b
z,
B
is only nonnegative de-
The difficulty is that we have no guarantee in this case
that the probability space (0,P) is "large enough" to have a
Brownian motion.
To handle this type of situation, Doob (1953,
p. 287) had the idea of adjoining a probability space, containing
a Brownian motion independent of the process in question, to the
original space.
Quoting Stroock and Varadhan (1979, p. 108), "We
then hold this new independent Brownian motion in reserve and only
call on it to fill the gaps created by lapses in 'randomness' of
the original Ito process."
Using this technique, Stroock and
Varadhan (1979, p. 108) obtained:
3.3.10
Let
z(t) = (zi(t)), (t
process with covariance
bounded.
Let
B
and drift
b(t) = (bij(t)), (t
be an Ito
0),(i=1,...,n)
a, where
a
and
0), (i,j=1,...,n)
B
be a
are
71
process which is progressively measurable with respect to
t
t
0)
such that
Then there is a probability
bbT = B.
'
(A0.1,Q), a nondecreasing family of sub
space
ost' t
0)
a, B, 6
stochastic processes
of
an n-dimensional Brownian motion
(A,c.15,Q)
on
W
gressively measurable with respect to
a-fields
and
and
that are pro-
(5t' t>0), such that
a, B, b and z
have the same finite dimensional probability dis-
tributions as
a, B, b and z, respectively,
and
6(s)dW(s) a.s.[Q]
i(t) = i(0) +-i(s)ds +
0
.
0
Now combining (3.3.10) and Theorem 3.3.5 we obtain
3.3.11
Theorem
Suppose that
in Theorem 3.3.5 and suppose further that
Let
b
satisfy the hypotheses
z, a and B
a
and
B
are bounded.
be a progressively measurable process such that
Then the conclusion of
bbT = B.
(3.3.10) holds.
We will end this section with a theorem which explains in
what sense mathematical models described by a stochastic integral
are mathematically equivalent to models described by Carter's conditions, as given in the introduction.
1emma.
First we need the following
72
and
be stochastic processes in
(i,j=1,...n)
b(t) =
CQ2
(bii(t)),
r1m4
0), (i=1,...,n)
z(t) = (zi(t)), (t
If
tively.
a(t) = (a(t))
Let
Lemma
3.3.12
and
0),
(t
CQ4, respec-
is a stochastic
integral given by
rt
a(s)ds +
z(t) = z(0) +
a.s.,
is an n-dimensional Brownian motion, and
w
where
b(s)dw(s)
1
JO
0
is a process given by
0), (i,j=1,...,n)
B(t) = (B1.(t)), (t
J
z, a and B
B = bbT, then
wEQ
That for each
Proof:
satisfy the hypotheses of Theorem 3.3.5
and
t
0, B(t,w) is a symmetric non-
negative definite matrix follows from the form of
Since
CQ4
is in
b
hence that
B
we see by (3.1.5) that
is in
bbT
B, i.e., bbT.
is in
CQ2
and
CQ1.
Conditions (ii), (iii) and (iv) in Theorem 3.3.5 follow from
Theorems 3.2.3, 3.2.5 and 3.2.9(4), respectively.
To show that
z
satisfies condition (i), we have the follow-
ing estimates, where the first step is justified by Lemma 3.2.2.
u+h
1
.11
1E[zi(u+h)-zi(u)1 rit]1 =
a4(s)ds1fXLI]l
111-1E[f
u
Eu+h
[f la(s)
<
<
E[
sup
tE[0,u+1]
1
ds
a.(t)
1
1,cru]
a.s.
1Y]
a.s.
a.s.
73
Since
a c Q1,
E[E[
we have
sup
te[0,u+1]
1
a . (t)1
1
tE
1
This establishes (i) with du = 1
Mu = E[
sup
tE [0,u+1]
sup
[0,u+1]
Fu]] = E[
a(t)
1
ai (t)
1
1
]<
.
and
]
1
0
.
1
Making the observation that a bounded, progressively measurable
stochastic process is in
for all
QP
p, we combine Lemma 3.3.12,
Theorem 3.3.9 and Theorem 3.2.9 to obtain
3.3.13
Theorem
Let
a(t) = (a(t))
and
B(t) = (Bii(t)), (t > 0),
be bounded, progressively measurable, continuous
(i,j=1,...,n)
stochastic processes, such that for each
t > 0
is symmetric strictly positive definite.
Let
(i=1,...,n)
and
wE
z(t) = (zi(t)), (t >0),
be a progressively measurable, continuous stochastic
Then conditions 1 and 2 below are equivalent.
process.
Condition 1
Whenever separable versions of the processes given by
the conditional expectations below are taken, then for all
n, m > 3 and all finite sequences, (ik
i, j
1
1,...,n
:
u > 0,
k=1,...m) from
.
(i) there is a
Mu
B(t,w)
numberu >
0
and a nonnegative random variable
with finite expectation such that for every
hfa(0,6u)
74
IiiE[zi(u+h) - z(u) 1074-u]
.
.
lim
1
1
< Mu
E[(z.(u+h) - zi(u))!f]
-F
a.s.
a(u)
=
a.s.
h4-0
1
TEEz4(u+h)-z4W(z.(li+h)-z
.(u) ) 1 f2r] = B. .(u)
urn
liM
1
E[1z4
M
k=1
11+0
Condition 2
trix
1J
u
a.s.
"
h4,0
5ril]
(U+h)-Zil(U)1
= 0
a.s.
'k
For each progressively measurable n x n stochastic ma-
b(t), (t
Brownian motion
0)
w
that satisfies bbT=B there is an n-dimensional
adapted to (! t,
t
0)
such that
ft
a(u)du +b(u)dw(u)
z(t) = z(0) +
0
a.s.
0
A corresponding result is obtained for the case in which
B
nonnegative definite by replacing Condition 2 above by the conclusion of (3.3.10).
is
75
BIBLIOGRAPHY
1972.
Ash, R. B.
New York.
Real Analysis and Probability.
Academic Press,
Topics in Stochastic Pro1975.
Ash, R. B. and Gardner, M. F.
cesses. Academic Press, New York.
Doob, J. L.
1953.
Stochastic Processes.
Wiley, New York.
Diestel, J. and Uhl, J. J. 1977. Vector Measures.
matical Society, Providence, Rhode Island.
1971.
Freedman, D.
San Francisco.
Brownian Motion and Diffusion.
American Mathe-
Holden Day,
Stochastic Differential Equations, Vol. 1.
1975.
Friedman, A.
Academic Press, New York.
Stochastic Integral, Proc. Imp. Acad. Tokyo (Nihon
Gakvshiin) 20, 519-524.
Ito, K.
1944.
1951.
Ito, K.
Soc. 4.
On Stochastic Differential Equations, Mem. Am. Math.
Probability Theory, Vol. 2, 4th Ed.
1978.
Loeve, M.
Verlag, New York.
Springer-
Theory of the Integral, Second Revised Edition.
Saks, S. 1937.
Hefner, New York.
1979.
Multidimensional
Stroock, D. W. and Varadhan, S. R. S.
Springer-Verlag,
New
York.
Diffusion Processes.
Stochastic Processes and the Wiener Integral.
Dekker, New York.
Yeh, J.
1973.
76
INDEX
Page
20
Adapted
Almost surely
(a.s.)
6
48
CM
Conditional expectation
Continuous stochastic process
Convergence in probability
6
21
6
CQ
48
DM
48
DQ
48
E[
E[
I]
]
6
6
27
Expectation
6
Finite dimensional probability distribution
12
Ito's formula
41
Ito integral
35
Ito process
62
e[a,(23]
33
Measurable stochastic process
17
Moment rate function
51
Mip[a,]
34
MP
47
Nondecreasing family of G-fields
20
Progressively measurable
20
QP
47
Right continuous stochastic process
21
Sample function (path)
12
Separable stochastic process
14
Separating set
14
Step function-stochastic process
34
Stochastic integral
40
Version
13