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di¬erence martingale, and the corresponding σ 2 is identically 1. Therefore
we can construct stochastic integrals with respect to w.
Formally, dw(t) = σ ’1 (t) dy(t), so that dy(t) = σ(t) dw(t). Let us
prove that, in fact,
y(b) ’ y(a) = σ(s) dw(s).

A simple calculation shows that if f is in H0 then

f (s)σ ’1 (s) dy(s).
f (s) dw(s) =

Consequently, the same holds for any f in H . Therefore,
b b
σ(s)σ ’1 (s) dy(s)
σ(s) dw(s) =
a a
= y(b) ’ y(a) = y(a, b).

The theorem follows from the de¬nition (11.5) of y. QED.

It is possible that the theorem remains true without the regularity
assumption (R3) provided that one is allowed to enlarge the underlying
probability space and the σ-algebras Pt .
A fundamental aspect of motion has been neglected in the discus-
sion so far; to wit, the continuity of motion. We shall assume from now

on that (with probability one) the sample functions of x are continuous.
By (11.5), this means that the same functions of y are continuous. (Use
ω to denote a point in the underlying probability space. We can choose
a version Dx(s, ω) of the stochastic process Dx that is jointly measur-
able in s and ω, since s ’ Dx(s) is continuous in L 1 [15, §6, p. 60 ¬].
Then a Dx(s, ω) ds is in fact absolutely continuous as b varies, so that
y(b) ’ y(a) has continuous sample paths as b varies.) Next we show (fol-
lowing Doob [15, p. 446]) that this implies that ω has continuous sample

THEOREM 11.7 Let y be an (R2) di¬erence martingale whose sample
functions are continuous with probability one, and let f be in H . let
z(b) ’ z(a) = f (s) dy(s).

Then z is a di¬erence martingale whose sample paths are continuous with
probability one.

Proof. If f in in H0 , this is evident. If f is in H , let fn be in H0
with f ’ fn ¤ 1/n2 , where the norm is given by (11.10). Let
zn (b) ’ zn (a) = fn (s) dy(s).

Then z ’ zn is a di¬erence martingale. (We already observed in the proof
of Theorem 11.6 that z is a di¬erence martingale”only the continuity of
sample functions is at issue.)
By the Kolmogorov inequality for martingales (Doob [15, p. 105]), if
S is any ¬nite subset of [a, b],

1 1 1
· n2 = 2 .
Pr sup |z(s) ’ zn (s)| > ¤
n n

Since S is arbitrary, we have

1 1
sup |z(s) ’ zn (s)| > ¤
Pr .

(This requires a word concerning interpretation, since the supremum is
over an uncountable set. We can either assume that z ’ zn is separable in

the sense of Doob or take the product space representation as in [25, §9]
of the pair z, zn .) By the Borel-Cantelli lemma, z converges uniformly
on [a, b] to z. QED.

Notice that we only need f to be locally in H ; i.e., we only need
f χ[a,b] to be in H for [a, b] any compact subinterval of I. In particular, if
y is an (R3) process the result above applies to each component of σ ’1 ,
so that w has continuous sample paths if y does.
Now we shall study the di¬erence martingale w (with σ 2 identically 1)
under the assumption that w has continuous sample paths.

THEOREM 11.8 Let w be a di¬erence martingale in satisfying

E{[w(b) ’ w(a)]2 | Pa } = b ’ a

whenever a ¤ b, a ∈ I, b ∈ I, and having continuous sample paths with
probability one. Then w is a Wiener process.

Proof. We need only show that the w(b) ’ w(a) are Gaussian. There
is no loss of generality in assuming that a = 0 and b = 1. First we assume
that = 1.
Let ∆t be the reciprocal of a strictly positive integer and let ∆w(t) =
w(t + ∆t) ’ w(t). Then

[w(1) ’ w(0)]n = ∆w(t1 ) . . . ∆w(tn ),

where the sum is over all t1 , . . . , tn ranging over 0, ∆t, 2∆t, . . . , 1 ’ ∆t.
We write the sum as = + , where is the sum of all terms
in which no three of the ti are equal.
Let B(K) be the set such that |w(1) ’ w(0)| ¤ K. Then

lim Pr B(K) = 1.

Let “(µ, δ) be the set such that |w(t) ’ w(s)| ¤ µ whenever |t ’ s| ¤ δ,
for 0 ¤ t, s ¤ 1. Since w has continuous sample paths with probability

lim Pr(“ µ, δ) = 1

for each µ > 0.

Let ± > 0. Choose K ≥ 1 so that Pr B(K) ≥ 1 ’ ±. Given n,
choose µ so small that nK n µ ¤ ± and then choose δ so small that
Pr(“ µ, δ) ≥ 1 ’ ±. Now the sum can be written
+··· +
+ n’3 ,
0 1

where ν means that exactly ν of the ti are distinct and some three of
the ti are equal. Then ν has a factor [w(1) ’ w(0)]ν times a sum of
terms in which all ti that occur, occur at least twice, and in which at least
one ti occurs at least thrice. Therefore, if ∆t ¤ δ,

d Pr ¤ K ν µ ∆w(t1 )2 . . . ∆w(tj )2 d Pr ¤ K ν µ,

where the t1 , . . . , tj are distinct. Therefore

d Pr ¤ nK n µ ¤ ±.

Those terms in in which one or more of the ti occurs only once
have expectation 0, so

d Pr = µn ,

where µn = 0 if n is odd and µn = (n ’ 1)(n ’ 3) . . . 5 · 3 · 1 if n is even,
since this is the number of ways of dividing n objects into distinct pairs.
Consequently, the integral of [w(1) ’ w(0)]n over a set of arbitrarily
large measure is arbitrarily close to µn . If n is even, the integrand
[w(1) ’ w(0)]n is positive, so this shows that [w(1) ’ w(0)]n is integrable
for all even n and hence for all n. Therefore,
E[w(1) ’ w(0)]n = µn
for all n. But the µn are the moments of the Gaussian measure with mean
0 and variance 1, and they increase slowly enough for uniqueness to hold
in the moment problem. In fact,


[w(1) ’ w(0)]n
Ee [w(1) ’ w(0)] = E

(i»)n »2
µn = e’ 2 ,

so that w(1) ’ w(0) is Gaussian.
The proof for > 1 goes the same way, except that all products are
tensor products. For example, (n ’ 1)(n ’ 3) . . . 3 · 1 is replaced by

(n ’ 1)δi1 i2 (n ’ 3)δi3 i4 . . . 3δin’3 in’2 1δin’1 in .


We summarize the results obtained so far in the following theorem.

THEOREM 11.9 Let I be an interval open on the right, Pt (for t ∈ I) an
increasing family of σ-algebras of measurable sets on a probability space, x
a stochastic process on ‚ having continuous sample paths with probability
one such that each x(t) is Pt -measurable and such that

x(∆t + t) ’ x(t)
Dx(t) = lim E

[x(∆t + t) ’ x(t)]2
σ (t) = lim E

exist in L 1 and are L 1 continuous in t, and such that σ 2 (t) is a.e.
invertible for a.e. t. Then there is a Wiener process w on ‚ such that
each w(t) ’ w(s) is Pmax(t,s) -measurable, and
b b
x(b) ’ x(a) = Dx(s) ds + σ(s) dw(s)
a a

for all a and b in I.

— — — — —

So far we have been adopting the standard viewpoint of the theory of
stochastic processes, that the past is known and that the future develops
from the past according to certain probabilistic laws. Nature, however,
operates on a di¬erent scheme in which the past and the future are on
an equal footing. Consequently it is important to give a treatment of
stochastic motion in which a complete symmetry between past and future
is maintained.

Let I be an open interval, let x be an ‚ -valued stochastic process
indexed by I, let Pt for t in I be an increasing family of σ-algebras
such that each x(t) is Pt -measurable, and let Ft be a decreasing family
of σ-algebras such that each x(t) is Ft -measurable. (Pt represents the
past, Ft the future.) The following regularity conditions make the con-
ditions (R1), (R2), and (R3) symmetric with respect to past and future.
The condition (R0) is already symmetric.

(S1). The condition (R1) holds and, for each t in I,
x(t) ’ x(t ’ ∆t)
D— x(t) = lim E

exists as a limit in L 1 , and t ’ D— x(t) is continuous from I into L 1 .

Notice that the notation is chosen so that if t ’ x(t) is strongly dif-
ferentiable in L 1 then Dx(t) = D— x(t) = dx(t)/dt. The random variable
D— x(t) is called the mean backward derivative or mean backward velocity,
and is in general di¬erent from Dx(t).
We de¬ne y— (a, b) = y— (b) ’ y— (a) by
x(b) ’ x(a) = D— x(s) ds + y— (b) ’ y— (a).

It is a di¬erence martingale relative to the Ft with the direction of time

(S2). The conditions (R2) and (S1) hold and, for each t in I,
[y(t) ’ y(t ’ ∆t)]2
σ— (t) = lim E

exists as a limit in L 1 and t ’ σ— (t) is continuous from I into L 1 .


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