FinM 34500/ Stat 39000
STOCHASTIC CALCULUS
Lecture 2
Greg Lawler
January 10-11, 2011
1 / 43
Discrete Stochastic Integral
Let X1 , X2 , . . . be independent, identically distributed random
variables with mean zero and variance 2 . Two main
examples are:
Bi
FinM 34500/ Stat 39000
STOCHASTIC CALCULUS
Lecture 1
Greg Lawler
January 3-4, 2011
1 / 37
STOCHASTIC CALCULUS (FINM 34500 and STAT 39000
Instructor: Greg Lawler, 415 Eckhart
Information for the course will be posted on my web page:
There will be no text f
FinM 34500/ Stat 39000
STOCHASTIC CALCULUS
Lecture 3
Greg Lawler
January 17-18, 2011
1 / 30
Review: denition of (one-dimensional) Brownian motion
Two parameters (drift) and 2 ( variance parameter or volatility)
For s < t , the random variable Bt Bs has a
FinM 34500/ Stat 39000
STOCHASTIC CALCULUS
Lecture 6
Greg Lawler
February 14February 15, 2011
1 / 40
Axioms of (measure theoretic) probability theory
A probability space is a set of outcomes , a collection F of
events (subsets of ), and a probability (mea
FinM 34500/ Stat 39000
STOCHASTIC CALCULUS
Lecture 5
Greg Lawler
January 31February 1, 2011
1 / 37
Reviewing the denition of the stochastic integral
Suppose At is adapted to cfw_Ft such that for each n,
t
E [ Z t] = E
0
A2 ds =
s
Then the integral
t
0
E[
FinM 34500/ Stat 39000
STOCHASTIC CALCULUS
Lecture 4
Greg Lawler
January 24-25, 2011
1 / 43
It stochastic integral
o
Bt will be a standard (one dimensional) Brownian motion with
B0 = 0. Ft denotes the information contained in cfw_Bs : s t .
At is a proces
FinM 34500/ Stat 39000
STOCHASTIC CALCULUS
Lecture 7
Greg Lawler
February 21February 22, 2011
1 / 47
The main goal of todays lecture is to discuss the
martingale approach to option pricing.
Before doing this we will discuss some results about stochastic
i