Page 208 - Elements of Distribution Theory

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194 Stochastic Processes

Consider
W t+h − W t
(6.9)
h
for small h;of course, the derivative of W t is simply the limit of this ratio as h → 0. Since
√
W t+h − W t has variance h, the difference W t+h − W t tends to be of the same order as h;
for instance,
1
E(|W t+h − W t |) = (2h/π) 2 .
1
Thus, (6.9) tends to be of order h − 2 , which diverges as h → 0.
Let f :[0, ∞) → R denote a continuous function of bounded variation and consider the
quantity
n
2
Q n ( f ) = [ f ( j/n) − f (( j − 1)/n)] .
j=1
This is a measure of the variation of f over the interval [0, 1]. Note that
n

Q n ( f ) ≤ max | f ( j/n) − f (( j − 1)/n)| | f ( j/n) − f (( j − 1)/n)|.
1≤ j≤n
j=1
Since f is continuous on [0, ∞)itis uniformly continuous on [0, 1] and, hence,
lim max | f ( j/n) − f (( j − 1)/n)|= 0.
n→∞ 1≤ j≤n
Since f is of bounded variation,
n

| f ( j/n) − f (( j − 1)/n)|
j=1
is bounded in n. Hence, Q n ( f ) approaches 0 as n →∞.
Now consider Q n as applied to {W t : t ≥ 0}.By properties (W2) and (W3), W j/n −
W ( j−1)/n , j = 1,..., n, are independent, identically distributed random variables, each
with a normal distribution with mean 0 and variance 1/n. Hence, for all n = 1, 2,...,
n

2
E (W j/n − W ( j−1)/n) ) = 1;
j=1
that is, E[Q n (W t )] = 1 for all n = 1, 2,.... This suggests that the paths of a Wiener process
are of unbounded variation, which is in fact true; see, for example, Billingsley (1968,
Section 9).

The Wiener process as a martingale
Since, for any s > t, W s − W t and W t are independent, it follows that
E[W s |W t ] = E[W s − W t |W t ] + E[W t |W t ] = W t ,

a property similar to that of martingales. In fact, a Wiener process is a (continuous time)
martingale; see, for example, Freedman (1971). Although a treatment of continuous time
martingales is beyond the scope of this book, it is not difficult to construct a discrete time
martingale from a Wiener process. Let {W t : t ≥ 0} denote a Wiener process and let 0 ≤
t 1 < t 2 < t 3 < ··· denote an increasing sequence in [0, ∞). For each n = 1, 2,..., define