The course is on Canvas
Homework
Homework #1 – Review of Expectation and Conditional Expectation
Homework #2 – Review of Discrete-time Martingales
Homework #3 – Review of Brownian Motion
Homework #4 – Stochastic Integration and Ito’s formula
Lecture Notes
First week’s Lecture Notes: Notes-1, Notes-2, Notes-3, Notes-4, Notes-5, Notes-6, Notes-7
Martingales and Uniform Integrability: Notes
Brownian Motion: Notes
Syllabus
Time and Location
M-W-F, Period 5 (11:45 AM – 12:35 PM), LIT 219
Office Hours
Monday 1:55pm – 2:45pm, Wednesday 1:55pm – 2:45pm, or by appointment
Textbook
There is no required text, but the following textbooks are suggested:
- G. F. Lawler, Stochastic Calculus: An Introduction with Applications (PDF available on Prof. Lawler’s website).
- P. E. Protter, Stochastic Integration and Differential Equations, Springer.
- R. Durrett, Probability: Theory and Examples, 5th edition (PDF available on Prof. Durrett’s website)
- P. Morters and Y. Peres, Brownian Motion, Cambridge University Press (PDF available on Prof. Morters’ website)
Scope of the course
The aim of the course is to provide students with strong foundations in the area of probability theory. At the end of the course, students will be acquainted with the language of probability and will gain sufficient experience to successfully apply probabilistic tools to most areas of pure and applied sciences.
The course is intended for graduate students as part of their PhD requirement, and for students considering studying probability theory at a research level.
Prerequisite
MAP 6472-6473 — Probability and Potential Theory
Topics Covered
Topics include continuous time Martingales, Brownian motion, Stochastic integral, Itô calculus, Stochastic differential equations. Below is the weekly schedule:
W1: Review of Lebesgue integral.
W2: Review of conditional expectation.
W3: Continuous time martingales.
W4: Backward martingale and Lévy’s downward theorem.
W5: Review of Brownian motion.
W6: Review of Riemann integral.
W7: Construction of stochastic integral.
W8: Properties of the stochastic integral.
W9: Itô calculus.
W10: Stochastic differential equations.
W11: Ornstein-Ulhenbeck process, Black-Scholes model in finance.
W12: Simulation.
W13-W14: Change of measure, Girsanov theorem.