| Date |
Reading |
Homework |
| Week
of Mon Dec 31 |
First day of class handout
here
§6.1
Markov property,
Time homogeneity,
Transition probability, Branching
process, Example 6.1.10, Chapman
Kolmogorov eqns
|
assignment
1,
see below for due date
|
| Week
of Mon Jan 07 |
Joint probability law,
past and future events,
review of conditional
probability,
pull through and other properties, answer
to pull-through question,
§6.2, first
passage times,
classification
of states Thm 6.2.3 |
assignment 1
due on Friday
assignment
2 |
Week
of Mon Jan 14
|
Abel thm
5.1.15, Corollary 6.2.4,
Expected number of
visits to a state,
§6.3, Communication,
classes, class properties, Dominated convergence,
monotone convergence, Fatou Lemma, interchange
of sums.
|
assignment 2 due on
Friday
assignment
3 |
Week
of Mon Jan 21
|
§6.4 stationary distributions,
Role of positive recurrence, irreducibility,
existence of and formula for stationary distribution
Thm 6.4.3 and proofs
|
assignment 3 due on
Friday
assignment
4 |
| Week
of Mon Jan 28 |
Convergence to stationary distribution, Thm
6.4.17
and proof by coupling in §6.4.
The time reversed Markov chain §6.5.
Equations of detailed balance,
application to
Ehrenfest model §6.5.
Finite state chains |
assignment 4 due
on Friday
no hw this week because of
midterm next week
|
Week
of Mon Feb 04
|
Random walk on a graph
MonteCarlo simulation of self-avoiding walk by the
pivot algorithm
MonteCarlo
Simulation of SAW
time averages and the stationary
distribution §6.4. The renewal theorem §10.2. Lectures 17=midterm
|
Midterm
Wed. Feb 09
[on assignments 1,2,3,4]
assignment
5 |
Week
of Mon Feb
11
Monday Family Day.
University closed
|
Defn of martingale §7.7
and §12.1. Examples based on simple random walk.
Failures of fairness in the limit. Discrete stochastic
integrals.
Defn of filtration, sub
and super martingales §12.1.
Doob upcrossing lemma, |
midterm
survey
please fill out and hand in
assignment 5 due
assignment
6
|
Week
of Mon Feb
18
Spring Break
|
|
|
| Week
of Mon Feb
25 |
supermartingale convergence theorem §12.3 and §7.8.
domination from above and below, Fatou
implies
Dominated convergence
thm,
Stopped martingales and optional sampling §12.4.
uniformly
square integrable martingales
are fair in the limit
§12.5 and §7.8.
|
assignment 6 due
assignment
7
|
| Week
of Mon Mar
04 |
Uses of martingale examples:
rw hits all points, expected time to exit a a set of
states.
Kolmogorov SLLN, difference between weak and strong
law.
Bellman optimality. |
assignment 7 due
assignment
8 |
Week
of Mon Mar
11
|
Financial
Math application.
Binary splitting models and representing rv by
stochastic integrals, self-financing portfolios.
Cox Rubinstein model, equivalent martingale measure,
present value of an option.
Review of Poisson rv, generating function, sum of
Poisson is Poisson.
|
assignment 8 due
assignment
9 |
Week
of Mon Mar
18
|
infinitely
divisible rv
and characterisation
of Poisson rv
§6.8 Poisson
process and relation to exp
times
strong Markov property of Poisson process
|
assignment 9 due
No assignment 10, instead
start learning everything for final
|
Week of Mon Mar 25
Friday Good Friday. University closed
|
Continuous time Markov processes
§6.9
time between jumps are i.i.d.
exp(lambda)
Claim 6.9.13
Jump time independent of
destination,
Claim 6.9.14
Exponential clock algorithm for X.
Explosions, birth process 6.8.11, Thm 6.8.19.
|
|
Week of Mon Apr 01
Easter Monday. University closed.
Friday Apr 5 Last day
Office
Hours for rest of term |
§6.10,
standard process Thm 6.10.1
Infinitesmal
generator G, computation
for birth process
6.9.7.
Kolmogorov
backward equations,
Thm 6.10.6, P
= exp(tG). |
|
Exam
schedule
|
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Final
Exams Information on final
|