Stochastic Differential Equations (Spring 2004)
Description
Stochastic differential equations arise when some randomness
is allowed in the coefficients of a differential equation.
They have many applications, including mathematical biology, theory of
partial
differential equations, differential geometry and mathematical finance.
This is an introduction at the advanced undergraduate/beginning graduate
level to the theory and applications of stochastic differential equations.
A knowledge of measure theory is strongly recommended but not required.
First we will review probability spaces, random variables and stochastic processes.
Brownian motion will be discussed in some detail before we
introduce the Ito integral and relevant aspects of
martingale theory as a method to formulate and solve stochastic differential
equations. Applications will include mathematical finance (arbitrage and
option pricing).
Recommended Texts
Stochastic Differential Equations, An Introduction with Applications.
Bernt Oksendal, Springer, Sixth Edition, 2003. This is the main text for
the course.
Arbitrage Theory in Continuous Time. Thomas Bjork, Oxford University Press,
1998. For applications.
Measure and Integral, R. Wheeden and A. Zygmund, Dekker, 1997.
Useful as a supplementary text on measure theory.
and Complex Analysis, W. Rudin, 3rd Edition, McGraw Hill, 1987.
Useful as a supplementary text on measure theory.
Probability and Statistics I (Fall 2004)
Probability and Statistics I
This course is intended to help students build a solid foundation in
probability. Emphasis will be placed on a thorough understanding of
the basic concepts as well as developing problem solving
skills. Topics covered include: axioms of probability; conditional
probability and independence; discrete and continuous random
variables; main discrete and continuous probability distributions
(Bernoulli, Binomial, Poisson, Exponential etc); jointly
distributed random variables; conditional expectation; moment
generating function; classical limit theorems (strong and weak law of large numbers,
central limit theorem etc); techniques of simulation, including Monte Carlo simulation.
After a grounding in basic probability we will consider, depending
upon time constraints and class interest, some more advanced topics
such as random walks, Markov chains and
Brownian motion.
Recommended Texts
A First Course in Probability, Sixth Edition by Sheldon
Ross, 2002, Prentice Hall
An Introduction to Probability Theory and Its Applications, Vol
1, 3rd edition, 1968 by William Feller (any edition would be fine).
Probability by Leo Breiman, 1968, Addison- Wesley.
A First Look at Rigorous Probability Theory by Jeffrey
Rosenthal, 2000. Useful as a supplementary text but unfortunately
not yet in library.
Problems Sheets
Problem sheet 1
Problem sheet 2
Probability and Statistics II (Spring 2005)
Description
This course is an introduction to mathematical
statistics. It assumes a knowledge of probability at the
level of Sheldon Ross (A First Course in Probability, Prentice Hall)
which is the set text for Math 6382. Topics covered include random samples,
principles of data reduction, point and interval estimation, hypothesis
testing, regression models and asymptotic evaluations.
Recommended Texts
Statistical Inference, 2nd Edition by George Casella and
Roger Berger, 2002, Duxbury Press.
Handout 3 Solutions
Problem sheet 3 Solutions