Mathematical Modeling in Finance with Stochastic Processes

5.7. QUADRATIC VARIATION OF THE WIENER PROCESS 187

Section Starter Question

What is an example of a function that “varies a lot”? What is an example of a
function that does not “vary a lot”? How would you measure the “variation”
of a function?

Key Concepts

1. The total quadratic variation of Brownian motion ist.


2. This fact has profound consequences for dealing with Brownian motion
   analytically and ultimately will lead to Itˆo’s formula.

Vocabulary

1. A functionf(t) is said to havebounded variationif, over the closed
   interval [a,b], there exists anMsuch that

|f(t 1 )−f(a)|+|f(t 2 )−f(t 1 )|+···+|f(b)−f(tn)|≤M

for all partitionsa=t 0 < t 1 < t 2 < ... < tn< tn+1=bof the interval.

2. A functionf(t) is said to havequadratic variationif, over the closed
   interval [a,b], there exists anMsuch that

(f(t 1 )−f(a))^2 + (f(t 2 )−f(t 1 ))^2 +···+ (f(b)−f(tn))^2 ≤M

for all partitionsa=t 0 < t 1 < t 2 < ... < tn< tn+1=bof the interval.

3. Themesh sizeof a partitionPwitha=t 0 < t 1 < ... < tn< tn+1=b
   is maxj=0,...,n{tj+1−tj|j= 1,...,n}.


4. Thetotal quadratic variationof a functionfon an interval [a,b] is

sup

P

∑n

j=0

(f(tj+1)−f(tj))^2

where the supremum is taken over all partitionsP witha=t 0 < t 1 <

... < tn < tn+1 =b, with mesh size going to zero as the number of

partition pointsngoes to infinity.