| Date |
Reading |
Homework |
| Week
of
Mon Dec 31 |
First day of class handout here
Thm 6.25, uniformly
integrable families of rv |
assignment
1,
see below for due date
|
| Week
of
Mon Jan 07 |
Doob upcrossing inequality
Martingale convergence Thms 11.4, 11.7
upward theorem for uniformly
integrable submartingales,
optional sampling, Thms 10.11 and 10.21
(unbounded stopping times and UI)
Doobs maximal bound,
|
assignment 1
due on Friday
assignment
2 |
Week
of
Mon Jan 14
|
Product spaces, Polish spaces and the role
of topology,
Kolmogorov-Daniell extension theorem.
biography
of Daniell
Corrected notes on
proof of Daniell-Kolmogorov Extension Theorem
|
assignment 2 due
on Friday
assignment
3 |
Week
of
Mon Jan 21
|
Defn of Brownian motion as Gaussian process
Holder continuity of Brownian motion restricted
to dyadic numbers
|
assignment 3 due
on Friday
assignment
4 |
| Week
of
Mon Jan 28
|
Stopping times and right continuous
filtration
Strong Markov Property of Brownian motion
Blumenthal 0-1 law and zero sets of BM
|
assignment 4 due
on Friday
assignment
5
|
Week
of
Mon Feb 04
|
The reflection principle
path properties of Brownian motion
|
assignment 5
due
assignment
6 |
Week
of
Mon Feb
11
Monday Family Day.
University closed
|
One dimensional potential theory
Balayage and other properties
|
assignment 6 due
assignment
7
midterm
survey
please fill out and hand
in
|
Week
of
Mon Feb
18
Spring Break
|
|
|
| Week
of
Mon Feb
25 |
Skorohod embedding and the Donsker Theorem
Markov Processes
sections 17.1, 17.2
|
assignment 7 due
assignment
8 |
| Week
of
Mon Mar
04 |
semigroups
and the Q matrix
Markov chains, section 17.4
recurrence,transience and other properties of states
|
assignment 8
due
assignment
9 |
Week
of
Mon Mar
11
|
invariant
measures
uniqueness and irreducibility
existence of invariant distributions
|
assignment 9 due
assignment
10 |
Week
of
Mon Mar
18
|
time averages converge to equilibrium expectations,
renewal argument
Couplings |
assignment 10
due |
Week of Mon Mar
25
Friday Good Friday. University closed
|
Convergence
of probabilities,
proof by coupling
jump times for continuous
time markov
chains. |
|
Week of Mon Apr 01
Easter Monday. University closed.
Friday Apr 5 Last day
Office
Hours
for
rest of
term |
Poisson
clock picture for continuous time Markov
chains. Local
times. |
|
| Exam
schedule |
final exam
|