ńņš. 11 |

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, āsemimartingaleā

means submartingale and ālower 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 + āt) ā’ y(t)]2

Pt

2

Ļ (t) = lim E (11.6)

āt

ātā’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 + āt) ā’ y(t)]2 to mean [y(t + āt) ā’ y(t)] ā— [y(t + āt) ā’ y(t)],

and Ļ 2 (t) is a matrix of positive type.

Observe that āt occurs to the ļ¬rst power in (11.6) while the term

[y(t + āt) 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)

ńņš. 11 |