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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 ∈ I. It is a submartingale if and only if Dx(t) ≥ 0, t ∈ I and a
supermartingale if and only if Dx(t) ¤ 0, t ∈ I.

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

We always have y(b, a) = ’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 —I that has these three properties a di¬erence process. If y is
a di¬erence process, we can choose a point a0 in I, de¬ne y(a0 ) arbitrarily
(say y(a0 ) = 0), and de¬ne y(b) for all b in I by y(b) = y(a0 , b). Then
y(a, b) = y(b) ’ 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) ’ y(a)|Pa } = 0 whenever a ¤ 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 di¬erence process y(a, b) such that E{y(a, b) | Pa } = 0
whenever a ¤ b a di¬erence martingale. The following is an immediate
consequence of Theorem 11.1.

THEOREM 11.3 Let x be an (R1) process, and de¬ne y by (11.5). Then
y is a di¬erence martingale.

From now on we shall write y(b) ’ y(a) instead of y(a, b) when y is a
di¬erence process.
We introduce another regularity condition, denoted by (R2). It is
a regularity condition on a di¬erence martingale y. If it holds, we say
that y is an (R2) di¬erence martingale, and if in addition y is de¬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) ’ y(a) is in L 2 . For each t in I,

[y(t + ∆t) ’ y(t)]2
Pt
2
σ (t) = lim E (11.6)
∆t
∆t’0+


exists in L 1 , and t ’ σ 2 (t) is continuous from I into L 1 .

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

THEOREM 11.4 Let y be an (R2) di¬erence martingale, and let a ¤ b,
a ∈ I, b ∈ I. Then
b
E{[y(b) ’ y(a)] | Pa } = E σ 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 Itˆ-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) di¬erence martingale. Let H0 be the set of functions
of the form
n
f= fi χ[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 χ
denotes the characteristic function.) Thus each f in H0 is a stochastic
process indexed by I. For each f given by (11.8) we de¬ne the stochastic
integral
n
fi [y(bi ) ’ 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 ) ’ y(ai )] fj [y(bj ) ’ y(aj )].
E f (t) dy(t) = (11.9)
i,j=1


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

E{y(bj ) ’ y(aj ) | Paj } = 0

since y is a di¬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
σ (s) ds Pai
Efi2 2
Efi2 σ 2 (s) ds
E =
ai ai

by (11.7). Therefore
2
Ef 2 (t)σ 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)σ 2 (t) dt
f = tr (11.10)
I

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

all functions f , de¬ned a.e. on I, such that f σ 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 identi¬ed with an element of K ,
uniquely de¬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
±
 k, f (t) > k,

|f (t)| ¤ k,
fk (t) = f (t),

’k, f (t) < ’k.


Then fk ’ f ’ 0 as k ’ ∞, 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, §5]) of f on the preceding
subinterval (and let fn be 0 on the ¬rst subinterval). Then fn is in H0
and fk ’ f ’ 0.
With the usual identi¬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 de¬ned a.e.
on I such that f σ 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 ’ f (y) dy(t) from H into L 2 I; ‚ such
that if f = f0 χ[a,b] where a ¤ b, a ∈ I, b ∈ I, f0 ∈ L 2 , f0 Pa -measurable,
then

f (y) dy(t) = f0 [y(b) ’ y(a)].


We now introduce our last regularity hypothesis.

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

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

Let σ ’1 (t) = σ(t)’1 , where σ(t) is the positive square root of σ 2 (t).

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

E{[w(b) ’ w(a)]2 Pa } = b ’ a

and
b b
x(b) ’ x(a) = Dx(s) ds + σ(s) dw(s)
a a

whenever a ¤ b, a ∈ I, b ∈ I.

Proof. Let
b
σ ’1 (s) dy(s).
w(a, b) =
a


This is well de¬ned, since each component of σ ’1 χ[a,b] is in H . If f
b
is in H0 , a simple computation shows that a f (s) dy(s) is a di¬erence
martingale, so the same is true if f is in H or if each f χ[a,b] is in H .
Therefore, w is a di¬erence martingale, and we will write w(b) ’ 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 ) ’ y(ai )]2 Pa
E =
i
bi
σ 2 (s) ds Pai Pa
fi2
EE =
ai
i
bi
σ 2 (s) ds Pa
fi2
E =
ai
i

f 2 (s)σ 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)σ 2 (s) ds Pa
E
a

for all f in H . If we apply this to the components of σ ’1 χ[a,b] we ¬nd

E{[w(b) ’ w(a)]2 | Pa } =
b
σ ’1 (s)σ 2 (s)σ ’1 (s) ds Pa
E =
a
b ’ a,

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

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