Mathematical Modeling in Finance with Stochastic Processes

188 CHAPTER 5. BROWNIAN MOTION

Mathematical Ideas

Variation

Definition.A functionf(x) is said to havebounded variationif, over the
closed interval [a,b], there exists anM such 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.

The idea is that we measure the total (hence the absolute value) up-and-
down movement of a function. This definition is similar to other partition
based definitions such as the Riemann integral and the arclength of the graph
of the function. A monotone increasing or decreasing function has bounded
variation. A function with a continuous derivative has bounded variation.
Some functions, for instance Brownian Motion, do not have bounded varia-
tion.

Definition.A functionf(t) is said to havequadratic variationif, over the
closed interval [a,b], there exists anM such 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.

Again, the idea is that we measure the total (hence the positive terms
created by squaring) up-and-down movement of a function. However, the
squaring will make small ups-and-downs smaller, so that perhaps a function
without bounded variation may have quadratic variation. In fact, this is the
case for the Wiener Process.

Definition.The total quadratic variation of Q of a function f on an
interval [a,b] is

Q= sup

P

∑n

i=0

(f(ti+1)−f(ti))^2

where the supremum is taken over all partitionsPwitha=t 0 < t 1 < ... <
tn< tn+1=b, with mesh size going to zero as the number of partition points
ngoes to infinity.