This is the second half of the basic graduate course in probability theory. The goal of this course is to understand the basic theory of stochastic calculus. We will cover the following topics:

1. Brownian motion;
2. Continuous Time Matingales;
3. Markov processes;
4. Stochastic Integrals;
5. Ito's formula;
6. Representation of Martingales;
7. Girsanov theorem and
8. Stochastic Differential Equations.

If time allows, we will give a brief introduction to mathematical finance.

Instructor	Partha Dey
Office	35 CAB
Contact	By email with subject line: "Math 562:"
Class	TR 9:30am -10:50am in G30 Literatures, Cultures and Linguistics Building.
Website	go.illinois.edu/math562
Office Hrs	Wednesday 4-5:50pm or By appointment. I will be happy to answer your questions in my office anytime as long as I'm not otherwise engaged.
References	1. J. F. Le Gall: Brownian motion, martingales, and stochastic calculus. (2016), Springer; 2. I. Karatzas and S. E. Shreve: Brownian motion and stochastic calculus (2nd ed), Springer; 3. P. Mörters and Y. Peres: Brownian Motion, Cambridge University Press, Cambridge.
Prerequisite	Math 540 Real Analysis I - we will review measure theory topics as needed. Math 541 is also nice to have, but not necessary. Math 561 Probability Theory I - you should be willing to spend time and effort on this background material if necessary.
Grading Policy	Report: 25% of your grade will be based on detailed LaTex scribe on a relevant topic. Weekly Homework (Due Thursday 9:30am via Canvas): 50% of the course grade. You are encouraged to work together on the homework, but I ask that you write up your own solutions and turn them in separately. There will be few problems assigned; emphasis will be placed on clear, concise, and coherent writing. Final Exam: 25%. Take home final exam due TBA on Canvas.

Week		Date	Due		Content
1	T	Aug 22			Introduction to Brownian Motion.
	R	Aug 24			Construction of Pre-Brownian Motion.
2	T	Aug 29			Kolmogorov's Continuity Theorem.
	R	Aug 31			Properties of Brownian Motion.
3	T	Sep 05			Properties of Brownian Motion.
	R	Sep 7			Filtrations and Stopping Times.
4	T	Sep 12			Martingales and Upcrossings.
	R	Sep 14			Martingale Convergence Theorems.
5	T	Sep 19			Strong Markov Property.
	R	Sep 21			Finite Variation processes.
6	T	Sep 26			Continuous Local Martingales.
	R	Sep 28			Quadratic Variation Process.
7	T	Oct 03			Continuous Semi-martingales.
	R	Oct 05			Stochastic Int for L2-Bounded Martingales.
8	T	Oct 10			Stochastic Int for Semimartingales.
	R	Oct 12			Itô's formula.
9	T	Oct 17			Consequences of Itô's formula.
	R	Oct 19			BDG inequality.
10	T	Oct 24			Martingales as Stochastic Integrals .
	R	Oct 26			Girsanov's Theorem.