Mathematical Modeling in Finance with Stochastic Processes

190 CHAPTER 5. BROWNIAN MOTION

by property 1 of the definition of standard Brownian motion.
A routine (but omitted) computation of the fourth moment of the normal
distribution shows that
E

[

Wnk^2

]

= 2t^2 / 4 n.

Finally, by property 2 of the definition of standard Brownian motion

E[WnkWnj] = 0,k 6 =j.

Now, expanding the square of the sum, and applying all of these computa-
tions

E





{ 2 n

∑

k=1

Wnk

} 2 

=

∑^2 n

k=1

E

[

Wnk^2

]

= 2n+1t^2 / 4 n= 2t^2 / 2 n.

Now apply Chebyshev’s Inequality to see:

P

[∣

∣

∣

∣

∣

∑^2 n

k=1

Wnk

∣

∣

∣

∣

∣

> 

]

≤

2 t^2

^2

(

1

2

)n

.

Now since

∑

(1/2)nis a convergent series, the Borel-Cantelli lemma implies
that the event ∣
∣
∣
∣
∣

∑^2 n

k=1

Wnk

∣

∣

∣

∣

∣

> 

can occur for only finitely manyn. That is, for any >0, there is anN,
such that forn > N ∣
∣
∣
∣
∣

∑^2 n

k=1

Wnk

∣

∣

∣

∣

∣

< .

Therefore we must have that limn→∞

∑ 2 n
k=1Wnk= 0, and we have established
what we wished to show.

Remark.Here’s a less rigorous and somewhat different explanation of why
the squared variation of Brownian motion may be guessed to bet, see [5].
Consider
∑n

k=1

(

W

(

kt

n

)

−W

(

(k−1)t

n

)) 2

.