3.4 Quadratic Variation 105We write informally
dW (t) dW (t) = dt, (3.4.10)
but this should not be interpreted to mean either (3.4.8) or (3.4.9). It is only
when we sum both sides of (3.4.9) and call upon the Law of Large Numbers
to cancel errors that we get a correct statement. The statement is that on an
interval [0, T], Brownian motion accumulates T units of quadratic variation.
If we compute the quadratic variation of Brownian motion over the time
interval [0, TI), we get [W, W] (T1) = T 1. If we compute the quadratic variation
over [0, T 2 ], where^0 < T 1 < T 2 , we get [W, W](T 2 ) = T 2. Therefore, if we
partition the interval [T 1 1 T 2 ], square the increments of Brownian motion for
each of the subintervals in the partition, sum the squared increments, and
take the limit as the maximal step size approaches zero, we will get the limit
[W, W](T 2 )-[W, W](TI) = T 2 - T 1. Brownian motion accumulates T 2 - T 1
units of quadratic variation over the interval [T 1 , T 2 ]. Since this is true for
every interval of time, we conclude that
Brownian motion accumulates quadratic variation at rate one per unit
time.
We write (3.4.10) to record this fact. In particular, the dt on the right-hand
side of (3.4.10) is multiplied by an understood 1.
As mentioned earlier, the quadratic variation of Brownian motion is the
source of volatility in asset prices driven by Brownian motion. We shall even
tually scale Brownian motion, sometimes in time-and path-dependent ways,
in order to vary the rate at which volatility enters these asset prices. 0Remark 3.4.5. Let II= {to,t 1 , ... ,tn} be a partition of [O,T) (i.e., 0 =to<
t1 < · · · < tn = T). In addition to computing the quadratic variation of
Brownian motion(3.4.11)we can compute the cross variation of W(t) with t and the quadratic variation
of t with itself, which aren-^1
11 N 1 �^0 �
(W(ti+1)-W(ti))(ti+1-ti) = o,
J=O
n- 1
lim ""(t·+l-t·)^2 = 0.
111711-tO � J J
J=OTo see that 0 is the limit in (3.4.12), we observe that
(3.4.12)(3.4.13)