Mathematical Modeling in Finance with Stochastic Processes

5.7. QUADRATIC VARIATION OF THE WIENER PROCESS 193

Problems to Work for Understanding

1. Show that a monotone increasing function has bounded variation.


2. Show that a function with continuous derivative has bounded variation.


3. Show that the function

f(t) =

{

t^2 sin(1/t) 0< t≤ 1

0 t= 0

is of bounded variation, while the function

f(t) =

{

tsin(1/t) 0< t≤ 1

0 t= 0

is not of bounded variation.

4. Show that a continuous function of bounded variation is also of quadratic
   variation.


5. Show that the fourth momentE[Z^4 ] = 3 whereZ ∼N(0,1). Then
   show that
   E

[

Wnk^2

]

= 2t^2 / 4 n

Outside Readings and Links:

1. Yuval Peres, University of California Berkeley, Department of Statistics
   Notes on sample paths of Brownian Motion. Contributed by S. Dunbar,
   October 30, 2005.


2. Wikipedia, Quadratic variation Contributed by S. Dunbar, November
   10, 2009.


3. Michael Kozdron, University of Regina, Contributed by S. Dunbar,
   November 10, 2009.