192 CHAPTER 5. BROWNIAN MOTION
surprising given the law of the iterated logarithm. We know that in any
neighborhood [t,t+dt] to the right oft, Brownian motion must come close
to
√
2 tlog logt. That is, intuitively,W(t+dt)−W(t) must be about√
2 dt
in magnitude, so we would guessdW^2 ≈ 2 dtThe theorem makes it precise.
Remark.This theorem can be nicely summarized in the following way: Let
dW(t) = W(t+dt)−W(t). Let dW(t)^2 = (W(t+dt)−W(t))^2. Then
(although mathematically not rigorously) we can say:
dW(t) ∼ N(0,dt)
(dW(t))^2 ∼ N(dt,0).Theorem 25.
lim
n→∞∑^2 nn=1∣
∣
∣
∣W
(
k
2 nt)
−W
(
k− 1
2 nt)∣
∣
∣
∣=∞
In other words, the total variation of a Brownian path is infinite, with prob-
ability 1.
Proof.
∑^2 nn=1∣
∣
∣
∣W
(
k
2 nt)
−W
(
k− 1
2 nt)∣
∣
∣
∣≥
∑ 2 n
n=1∣
∣W(k
2 nt)
−W
(k− 1
2 nt)∣∣ 2
maxj=1,..., 2 n∣
∣W
(k
2 nt)
−W
(k− 1
2 nt)∣
∣
The numerator on the right converges tot, while the denominator goes to 0
because Brownian paths are continuous, therefore uniformly continuous on
bounded intervals. Therefore the faction on the right goes to infinity.
Sources
The theorem in this section is drawn from A First Course in Stochastic
Processesby S. Karlin, and H. Taylor, Academic Press, 1975. The heuristic
proof using the weak law was taken fromFinancial Calculus: An introduction
to derivative pricing by M Baxter, and A. Rennie, Cambridge University
Press, 1996, page 59. The mnemonic statement of the quadratic variation in
differential form is derived from Steele’s text.