BROWNIAN MOTION, MATH 642:592, Spring 2008
1. Stochastic Processes in Continuous Time.
A stochastic process in continuous time is a family X = cfw_X (t); t 0 of random variables on a
xed probability space, indexed by all real t 0. Given , the function t X

Notes on the Dunford-Pettis compactness criterion The Dunford-Pettis compactness criterion implies that uniform integrability is a necessary and sufficient condition for weak sequential compactness of a family of integrable random variables. The theorem a

NOTES AND EXERCISES, LECTURE 1, MATH 642:592, Spring 2008
1. Probability space, random variables and expectation.
We summarize the formal mathematical setting of the course:
(1) All analysis takes place in a probability space. This is a triple (, F , P),

NOTES AND EXERCISES, MARTINGALES, MATH 642:592, Spring 2008
The material covered in lectures 2, 3, and 4 on conditional expectation and discrete time
martingales is standard and may be found in RW, Volume I. Therefore, we mainly summarize
here and give in

BROWNIAN MOTION-continued, MATH 642:592, Spring 2008
1. Lvys construction of Brownian motion. This is nicely covered in RW, Chapter 1, section 6.
e
The approach we took in lecture was dierent only in the way we expressed it. (The particular
way we did the

Lvy Processes, Math 642:592, Spring 2008
e
1. L`vy Processes and Poisson processes.
e
Denition: A function f : [0, ) R is c`dl`g if it is right-continuous at all t [0, ) and if the
aa
limit from the left
f (t) = lim s t, s < tf (s)
exists and is nite for

Introduction to stochastic integration: Math 642:592, Spring 2008
I. Bounded variation functions and Lebesgue-Stieltjes integrals.
As a preliminary to the theory of stochastic integration, we recall the theory of Lebesgue-Stieltes
integrals and its relati

Filtrations, Stopping times, and some applications: Math 642:592, Spring 2008
1. Filtrations for continuous time processes.
A ltration F = cfw_Ft t0 in a measure space (, F ) is an increasing family of sub- -algebras of
F . Here t ranges through all non-n