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is arbitrary, (11.1) holds. QED.
68 CHAPTER 11

Theorem 11.1 and its proof remain valid without the assumption that
x(t) is Pt -measurable.

THEOREM 11.2 An (R1) process is a martingale if and only if Dx(t) =
0, t ā I. It is a submartingale if and only if Dx(t) ā„ 0, t ā I and a
supermartingale if and only if Dx(t) ā¤ 0, t ā I.

We mean, of course, martingale, etc., relative to the Pt . This theo-
rem is an immediate consequence of Theorem 11.1 and the deļ¬nitions (see
Doob [15, p. 294]). Note that in the older terminology, āsemimartingaleā
means submartingale and ālower semimartingaleā means supermartin-
gale.
Given an (R1) process x, deļ¬ne the random variable y(a, b), for all a
and b in I, by
b
x(b) ā’ x(a) = Dx(s) ds + y(a, b). (11.5)
a

We always have y(b, a) = ā’y(a, b), y(a, b) + y(b, c) = y(a, c), and y(a, b) is
Pmax(a,b) - measurable, for all a, b, and c in I. We call a stochastic process
indexed by I Ć—I that has these three properties a diļ¬erence process. If y is
a diļ¬erence process, we can choose a point a0 in I, deļ¬ne y(a0 ) arbitrarily
(say y(a0 ) = 0), and deļ¬ne y(b) for all b in I by y(b) = y(a0 , b). Then
y(a, b) = y(b) ā’ y(a) for all a and b in I. The only trouble is that y(b) will
not in general be Pb -measurable for b < a0 . If I has a left end-point, we
can choose a0 to be it and then y(b) will always be Pb -measurable. By
Theorem 11.1, E{y(b) ā’ y(a)|Pa } = 0 whenever a ā¤ b, so that when a0 is
the left end-point of I, y(b) is a martingale relative to the Pb . In the gen-
eral case, we call a diļ¬erence process y(a, b) such that E{y(a, b) | Pa } = 0
whenever a ā¤ b a diļ¬erence martingale. The following is an immediate
consequence of Theorem 11.1.

THEOREM 11.3 Let x be an (R1) process, and deļ¬ne y by (11.5). Then
y is a diļ¬erence martingale.

From now on we shall write y(b) ā’ y(a) instead of y(a, b) when y is a
diļ¬erence process.
We introduce another regularity condition, denoted by (R2). It is
a regularity condition on a diļ¬erence martingale y. If it holds, we say
that y is an (R2) diļ¬erence martingale, and if in addition y is deļ¬ned in
KINEMATICS OF STOCHASTIC MOTION 69

terms of an (R1) process x by (11.5) then we say that x is an (R2) process.

(R2). For each a and b in I, y(b) ā’ y(a) is in L 2 . For each t in I,

[y(t + āt) ā’ y(t)]2
Pt
2
Ļ (t) = lim E (11.6)
āt
ātā’0+

exists in L 1 , and t ā’ Ļ 2 (t) is continuous from I into L 1 .

The process y has values in Ā‚ . In case > 1, we understand the
expression [y(t + āt) ā’ y(t)]2 to mean [y(t + āt) ā’ y(t)] ā— [y(t + āt) ā’ y(t)],
and Ļ 2 (t) is a matrix of positive type.
Observe that āt occurs to the ļ¬rst power in (11.6) while the term
[y(t + āt) ā’ y(t)] occurs to the second power.

THEOREM 11.4 Let y be an (R2) diļ¬erence martingale, and let a ā¤ b,
a ā I, b ā I. Then
b
E{[y(b) ā’ y(a)] | Pa } = E Ļ 2 (s) ds Pa .
2
(11.7)
a

The proof is so similar to the proof of Theorem 11.1 that it will be
omitted.
Next we shall discuss the ItĖ-Doob stochastic integral, which is a gen-
o
eralization of the Wiener integral. The new feature is that the integrand
is a random variable depending on the past history of the process.
Let y be an (R2) diļ¬erence martingale. Let H0 be the set of functions
of the form
n
f= fi Ļ[ai ,bi ] , (11.8)
i=1

where the interval [ai , bi ] are non-overlapping intervals in I and each fi
is a real-valued Pai -measurable random variable in L 2 . (The symbol Ļ
denotes the characteristic function.) Thus each f in H0 is a stochastic
process indexed by I. For each f given by (11.8) we deļ¬ne the stochastic
integral
n
fi [y(bi ) ā’ y(ai )].
f (t) dy(t) =
i=1
70 CHAPTER 11

This is a random variable.
For f in H0 ,
n
2
E fi [y(bi ) ā’ y(ai )] fj [y(bj ) ā’ y(aj )].
E f (t) dy(t) = (11.9)
i,j=1

If i < j then fi [y(bi ) ā’ y(ai )] fj is Paj -measurable, and

E{y(bj ) ā’ y(aj ) | Paj } = 0

since y is a diļ¬erence martingale. Therefore the terms with i < j in (11.9)
are 0, and similarly for the terms with i > j. The terms with i = j are
bi bi
Ļ (s) ds Pai
Efi2 2
Efi2 Ļ 2 (s) ds
E =
ai ai

by (11.7). Therefore
2
Ef 2 (t)Ļ 2 (t) dt.
E f (t) dy(t) =
I

This is a matrix of positive type. If we give H0 the norm

2
Ef 2 (t)Ļ 2 (t) dt
f = tr (11.10)
I

then H0 is a pre-Hilbert space, and the mapping f ā’ f (t) dy(t) is iso-
metric from H0 into the real Hilbert space of square-integrable Ā‚ -valued
random variables, which will be denoted by L 2 I; Ā‚ .
Let H be the completion of H0 . The mapping f ā’ f (t) dy(t)
extends uniquely to be unitary from H into L 2 I; Ā‚ . Our problem
now is to describe H in concrete terms.
Let Ļ(t) be the positive square root of Ļ 2 (t). If f is in H0 then f Ļ is
square-integrable. If fj is a Cauchy sequence in H0 then fj Ļ converges
in the L 2 norm, so that a subsequence, again denoted by fj , converges
a.e. to a square-integrable matrix-valued function g on I. Therefore fj
converges for a.e. t such that Ļ(t) = 0. Let us deļ¬ne f (t) = lim fj (t)
when the limit exists and deļ¬ne f arbitrarily to be 0 when the limit
does not exist. Then fj ā’ f ā’ 0, and f (t)Ļ(t) is a Pt -measurable
square-integrable random variable for a.e. t. By deļ¬nition of strong
measurability [14, Ā§5], f Ļ is strongly measurable. Let K be the set of
KINEMATICS OF STOCHASTIC MOTION 71

all functions f , deļ¬ned a.e. on I, such that f Ļ is a strongly measurable
square-integrable function with f (t) Pt -measurable for a.e. t. We have
seen that every element of H can be identiļ¬ed with an element of K ,
uniquely deļ¬ned except on sets of measure 0.
Conversely, let f be in K . We wish to show that it can be approxi-
mated arbitrarily closely in the norm (11.10) by an element of H0 . Firstly,
f can by approximated arbitrarily closely by an element of K with sup-
port contained in a compact interval I0 in I, so we may as well assume
that f has support in I0 . Let
ļ£±
ļ£“ k, f (t) > k,
ļ£²
|f (t)| ā¤ k,
fk (t) = f (t),
ļ£“
ā’k, f (t) < ā’k.
ļ£³

Then fk ā’ f ā’ 0 as k ā’ ā, so we may as well assume that f is
uniformly bounded (and consequently has uniformly bounded L 2 norm).
Divide I0 into n equal parts, and let fn be the function that on each
subinterval is the average (Bochner integral [14, Ā§5]) of f on the preceding
subinterval (and let fn be 0 on the ļ¬rst subinterval). Then fn is in H0
and fk ā’ f ā’ 0.
With the usual identiļ¬cation of functions equal a.e., we can identify
H and K . We have proved the following theorem.

THEOREM 11.5 Let H be the Hilbert space of functions f deļ¬ned a.e.
on I such that f Ļ is strongly measurable and square-integrable and such
that f (t) is Pt -measurable for a.e. t, with the norm (11.10). There is a
unique unitary mapping f ā’ f (y) dy(t) from H into L 2 I; Ā‚ such
that if f = f0 Ļ[a,b] where a ā¤ b, a ā I, b ā I, f0 ā L 2 , f0 Pa -measurable,
then

f (y) dy(t) = f0 [y(b) ā’ y(a)].

We now introduce our last regularity hypothesis.

(R3). For a.e. t in I, det Ļ 2 (t) > 0 a.e.

An (R2) diļ¬erence martingale for which this holds will be called an
(R3) diļ¬erence martingale. An (R2) process x for which the associated
diļ¬erence martingale y satisļ¬es this will be called an (R3) process.
72 CHAPTER 11

Let Ļ ā’1 (t) = Ļ(t)ā’1 , where Ļ(t) is the positive square root of Ļ 2 (t).

THEOREM 11.6 Let x be an (R3) process. Then there is a diļ¬erence
martingale w such that

E{[w(b) ā’ w(a)]2 Pa } = b ā’ a

and
b b
x(b) ā’ x(a) = Dx(s) ds + Ļ(s) dw(s)
a a

whenever a ā¤ b, a ā I, b ā I.

Proof. Let
b
Ļ ā’1 (s) dy(s).
w(a, b) =
a

This is well deļ¬ned, since each component of Ļ ā’1 Ļ[a,b] is in H . If f
b
is in H0 , a simple computation shows that a f (s) dy(s) is a diļ¬erence
martingale, so the same is true if f is in H or if each f Ļ[a,b] is in H .
Therefore, w is a diļ¬erence martingale, and we will write w(b) ā’ w(a) for
w(a, b).
If f is in H0 , given by (11.8), and if f (t) = 0 for all t < a, then

2
Pa
E f (t) dy(t) =

fi2 [y(bi ) ā’ y(ai )]2 Pa
E =
i
bi
Ļ 2 (s) ds Pai Pa
fi2
EE =
ai
i
bi
Ļ 2 (s) ds Pa
fi2
E =
ai
i

f 2 (s)Ļ 2 (s) ds Pa .
E
KINEMATICS OF STOCHASTIC MOTION 73

By continuity,
2
b
Pa
E f (t) dy(t) =
a
b
f 2 (s)Ļ 2 (s) ds Pa
E
a

for all f in H . If we apply this to the components of Ļ ā’1 Ļ[a,b] we ļ¬nd

E{[w(b) ā’ w(a)]2 | Pa } =
b
Ļ ā’1 (s)Ļ 2 (s)Ļ ā’1 (s) ds Pa
E =
a
b ā’ a,

whenever a ā¤ b, a ā I, b ā I. Consequently, w is an (R2) in fact, (R3)
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