Lecture 1 Notes: Metric spaces and Topology
1
Metric spaces. Open, closed and compact sets
When we discuss probability theory of random processes, the underlying sample
spaces and -eld structures become quite complex. It helps to have a unifying
framework
Lecture 2 Notes: Large Deviations for i.i.d. Random Variables
1
Preliminary notes
The Weak Law of Large Numbers tells us that if X1 , X2 , . . . , is an i.i.d. sequence of random variables with mean E[X1 ] < then for every E > 0
P(|
X1 + . . . + Xn
| > E
Lecture 6 Notes: Brownian motion
1
Historical notes
1765 Jan Ingenhousz observations of carbon dust in alcohol.
1828 Robert Brown observed that pollen grains suspended in water perform a continual swarming motion.
1900 Bacheliers work The theory of spe
Lecture 9 Notes: Conditional Expectations
1
Conditional Expectations
1.1
Denition
Recall how we dene conditional expectations. Given a random variable X
and an event A we dene E[X|A] = E[X1cfw_A] .
Also we can consider conditional expectations with respec
Lecture 10 Notes: Martingales
1
Martingales
We continue with studying examples of martingales.
Brownian motion. A standard Brownian motion B(t) is a martingale
on C[0, ), equipped with the Wiener measure, with respect to the
ltration Bt , t R+ , dened as
Lecture 8 Notes: Quadratic Variation Property
1
Unbounded variation of a Brownian motion
Any sequence of values 0 < t0 < t1 < < tn < T is called a partition =
(t0 , . . . , tn ) of an interval [0, T ]. Given a continuous function f : [0, T ]
R its total
Lecture 3 Notes: Large deviations Theory. Cramers Theorem
1
Cramers Theorem
We have established in the previous lecture that under some assumptions on
the Moment Generating Function (MGF) M (), an i.i.d. sequence of random
variables Xi , 1 i n with mean s
Lecture 5 Notes: Extension of LD to Rd and dependent Process
1
Large Deviations in Rd
Most of the developments in this lecture follows Dembo and Zeitouni book [1].
Let Xn Rd be i.i.d. random variables and A Rd . Let Sn =
Xn
1in
The large deviations questi
Lecture 4 Notes: Applications of the large deviation technique
1
Safety capital for an insurance company
Consider some insurance company which needs to decide on the amount of capital S0 it needs to hold to avoid the cash ow issue. Suppose the insurance p
Lecture 7 Notes: The Reection Principle
1
Technical preliminary: stopping times
Stopping times are loosely speaking rules by which we interrupt the process
without looking at the process after it was interrupted. For example sell your
stock the rst time i