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.
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
1. L`vy Processes and Poisson processes.
Denition: A function f : [0, ) R is c`dl`g if it is right-continuous at all t [0, ) and if the
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