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6.6 Wiener Processes 193

Theorem 6.12. Let {W t : t ≥ 0} denote a Wiener process.
(i) For a given h > 0, let Y t = W h+t − W h . Then {Y t : t ≥ 0} is a Wiener process.
(ii) Let K(t, s) = Cov(W t , W s ). Then

K(t, s) = min(t, s).

Proof. Toverifypart(i)itisenoughtoshowthattheprocess{Y t : t ≥ 0}satisfiesconditions
(W1)–(W4). Clearly, Y 0 = 0 and since
,
Y t 1 − Y t 0 = W h+t 1 − W h+t 0
{Y t : t ≥ 0} has independent increments. Continuity of the sample paths of Y t follows imme-
diately from the continuity of Z t . Hence, it suffices to show that (W3) holds.
For 0 ≤ t 1 < t 2 ,

Y t 2 − Y t 1 = W h+t 2 − W h+t 1
has a normal distribution with mean 0 and variance (h + t 2 ) − (h + t 1 ) = t 2 − t 1 ,verifying
(W3). Hence, {Y t : t ≥ 0} is a Wiener process.
Suppose t < s. Then

K(t, s) = Cov(W t , W s ) = Cov(W t , W t + W s − W t )
= Cov(W t , W t ) + Cov(W t , W s − W t ).
By (W2), W t and W s − W t are independent. Hence, K(t, s) = t; the result follows.

Example 6.16 (A transformation of a Wiener process). Let {W t : t ≥ 0} denote a Wiener
process and for some c > 0 define Z t = W c t /c, t ≥ 0. Note that Z 0 = W 0 so that Pr(W 0 =
2
1). Let 0 ≤ t 0 ≤ t 1 ≤· · · ≤ t m ; then
2 /c, j = 1,..., m.
Z t j − Z t j−1 = W c t j
2 /c − W c t j−1
has a normal
Hence, {Z t : t ≥ 0} has independent increments. Furthermore, Z t j − Z t j−1
distribution with mean 0 and variance
2 2
c t j − c t j−1
= t j − t j−1 .
c 2
Finally, continuity of Z t follows from continuity of W t ; hence, Z t is a continuous function.
It follows that {Z t : t ≥ 0} is also a Wiener process.

Rigorous analysis of Wiener process requires advanced results of probability theory and
analysis that are beyond the scope of this book. Hence, in this section, we give an informal
description of some of the properties of Wiener processes.

Irregularity of the sample paths of a Wiener process
By definition, the paths of a Wiener process are continuous; however, they are otherwise
highly irregular. For instance, it may be shown that, with probability 1, a Wiener process
{W t : t ≥ 0} is nowhere differentiable. Although a formal proof of this fact is quite difficult
(see, for example, Billingsley 1995, Section 37), it is not hard to see that it is unlikely that
derivatives exist.