Mathematical Modeling in Finance with Stochastic Processes

5.7. QUADRATIC VARIATION OF THE WIENER PROCESS 189

Quadratic Variation of the Wiener Process

We can guess that the Wiener Process might have quadratic variation by
considering the quadratic variation of our coin-flipping fortune record first.
Consider the function piecewise linear functionWˆ(t) defined by the sequence
of sumsTn=Y 1 +···+Ynfrom the Bernoulli random variablesYi= +1 with
probabilityp= 1/2 andYi=−1 with probabilityq= 1−p= 1/2. With some
analysis, it is possible to show that we need only consider the quadratic vari-
ation at points 1, 2 , 3 ,...,nThen each term (Wˆ(i+ 1)−Wˆ(i))^2 =Yi^2 +1= 1.
Therefore, the quadratic variation is the total number of steps,Q=n. Now
remember the Wiener Process is approximated byWn(t) = (1/

√

n)Wˆ(nt).
Each step is size 1/

√

n, then the quadratic variation of the step is 1/n
and there arensteps on [0,1]. The total quadratic variation ofWn(t) =
(1/

√

n)Wˆ(nt) on [0,1] is 1.
We will not completely rigorously prove that the total quadratic variation
of the Wiener Process ist, as claimed, but we will prove a theorem close to
the general definition of quadratic variation.

Theorem 24.LetW(t)be standard Brownian motion. For every fixedt > 0

lim

n→∞

∑^2 n

n=1

[

W

(

k

2 n

t

)

−W

(

k− 1

2 n

t

)] 2

=t

with probability 1 (that is, almost surely).

Proof. Introduce some briefer notation for the proof, let:

∆nk=W

(

k

2 n

t

)

−W

(

k− 1

2 n

t

)

k= 1,..., 2 n

and
Wnk= ∆^2 nk−t/ 2 n k= 1,..., 2 n.

We want to show that

∑ 2 n

k=1∆

2

nk → t or equivalently:

∑ 2 n
k=1Wnk → 0.
For eachn, the random variablesWnk,k = 1,..., 2 nare independent and
identically distributed by properties 1 and 2 of the definition of standard
Brownian motion. Furthermore,

E[Wnk] =E

[

∆^2 nk

]

−t/ 2 n= 0