Mathematical Modeling in Finance with Stochastic Processes

5.7. QUADRATIC VARIATION OF THE WIENER PROCESS 191

Now let

Znk=

(

W

(kt

n

)

−W

(

(k−1)t

n

))

√

t/n

Then for eachn, the sequenceZnkis a sequence of independent, identically
distributed standard normalN(0,1) random variables. Now we can write
the quadratic variation as:

∑n

k=1

∆^2 nk=

∑n

k=1

t

n

Znk^2 =t

(

1

n

∑n

k=1

Znk^2

)

But notice that the expectationE(Znk^2 ) of each term is the same as calcu-
lating the variance of a standard normalN(0,1) which is of course 1. Then
the last term in parentheses above converges by the weak law of large num-
bers to 1! Therefore the quadratic variation of Brownian motion converges
tot. This little proof is in itself not sufficient to prove the theorem above
because it relies on the weak law of large of numbers. Hence the theorem
establishes convergence in distribution only while for the theorem above we
want convergence almost surely.

Remark.Starting from

lim

n→∞

∑^2 n

n=1

[

W

(

k

2 n

t

)

−W

(

k− 1

2 n

t

)] 2

=t

and without thinking too carefully about what it might mean, we can imagine
an elementary calculus limit to the left side and write the formula:

∫t

0

[dW(τ)]^2 =t=

∫t

0

dτ.

In fact, with more advanced mathematics this can be made sensible ad math-
ematically sound. Now from this relation, we could write the integral equality
in differential form:
dW(τ)^2 =dτ.

The important thing to remember here is that the formula suggests that
Brownian motion has differentials that cannot be ignored in second (or
squared, or quadratic) order. Brownian motion “wiggles” so much that even
the total of the squared differences add up! In retrospect, this is not so