**Course Announcement**

**Time:**Monday, Wednesday, Friday at 2:00 p.m. - 2:50 p.m.

**Location:**Taft Hall 300**Instructor:**Jie Yang

**Office:**SEO 513

**Phone:**(312) 413-3748

**E-Mail:**jyang06 AT math DOT uic DOT edu

**Office Hours:**Monday, Friday at 1:00 p.m. - 2:00 p.m.; Wednesday at 11:00 a.m. - 12:00 p.m.**Textbook:**Sidney I. Resnick,*A Probability Path*, Birkäuser, 1999.

**Content:**Convergence concepts, laws of large numbers, convergence in distribution, characteristic functions, central limit theorem, conditional expectation, Martingale, fundamental theorems of mathematical finance

**Prerequisite:**STAT 501 or consent of instructor**Homework:**Turn in every Wednesday before class; half of the grade counts for completeness; half of the grade counts for correctness of one selected problem.

**Short Presentations:**Each student is required to do two short presentations during the course period. One presentation should be done in front of the whole class. Another presentation should be done in office hours. Class presentation may last up to 15 minutes. Office hour presentation may last up to 20 minutes. The topics of presentations may come from the optional part of homework assignments.

**Grading:**Homework 50%, presentations 25% each

**Grading Scale:**90% A , 75% B , 60% C , 30% D

**Course Syllabus****WEEK****SECTIONS****BRIEF DESCRIPTION**01/09 - 01/13 6.1, 6.2; 6.3; 6.4, 6.5 Almost Sure Convergence, Convergence in Probability; Connections between a.s. and i.p. Convergence; Quantile Estimation, Lp Convergence 01/16 - 01/20 Holiday; 6.5; 6.6 Lp Convergence; More on Lp Convergence 01/23 - 01/27 7.1; 7.2; 7.3 Truncation and Equivalence; A General Weak Law of Large Numbers; Almost Sure Convergence of Sums 01/30 - 02/03 7.3; 7.4; 7.5 Almost Sure Convergence of Sums; Strong Laws of Large Numbers; Strong Law of Large Numbers for IID Sequences 02/06 - 02/10 7.6; 8.1; 8.2 Kolmogorov Three Series Theorem; Definition of Convergence in Distribution 02/13 - 02/17 8.3; 8.4; 8.5 Baby Skorohod Theorem; Weak Convergence Equivalences, Portmanteau Theorem; More Relations among Modes of Convergence 02/20 - 02/24 8.6; 8.7; 8.7 New Convergences from Old; Convergence to Types Theorem 02/27 - 03/02 9.1, 9.2; 9.3; 9.4 Moment Generating Functions and Central Limit Theorem, Characteristic Functions; Expansions; Moments and Derivatives 03/05 - 03/09 9.5; 9.6; 9.7 Uniqueness and Continuity; Selection Theorem, Tightness, and Prohorov's Theorem; Classical CLT for IID Random Variables 03/12 - 03/16 9.8; 10.1; 10.2 Lindeberg-Feller CLT; Radon-Nikodym Theorem; Definition of Conditional Expectation 03/26 - 03/30 10.3; 10.4; 10.5 Properties of Conditional Expectation; Martingales; Examples of Martingales 04/02 - 04/06 10.6; 10.7; 10.8 Connections between Martingales and Submartingales; Stopping Times; Positive Super Martingales 04/09 - 04/13 10.8; 10.9; 10.10, 10.11 Positive Super Martingales; Examples; Martingale and Submartingale Convergence, Regularity and Closure 04/16 - 04/20 10.12; 10.13; 10.14 Regularity and Stopping; Stopping Theorems; Wald's Identity and Random Walks 04/23 - 04/27 10.15; 10.16; 10.16 Reversed Martingales; Fundamental Theorems of Mathematical Finance

**Homework**- Homework #1, due 01/18/2012

- Homework #2, due 01/25/2012

- Homework #3, due 02/01/2012

- Homework #4, due 02/08/2012

- Homework #5, due 02/15/2012

- Homework #6, due 02/22/2012

- Homework #7, due 02/29/2012

- Homework #8, due 03/07/2012

- Homework #9, due 03/14/2012

- Homework #10, due 03/28/2012

- Homework #11, due 04/04/2012

- Homework #12, due 04/11/2012

- Homework #13, due 04/18/2012

- Homework #14, due 04/25/2012

- Homework #1, due 01/18/2012
**Relevant Course Materials**- Patrick Billingsley,
*Probability and Measure*, 3rd edition, John Wiley & Sons, 1995 -- a good reference book

- Jeffrey S. Rosenthal,
*A First Look at Rigorous Probability Theory*, 2nd edition, World Scientific Publishing Company, 2006 -- a good reference book for readers without sufficient mathematical background

- William Chen,
*Introduction to Complex Analysis*-- good online notes for beginners in complex analysis

- Yuan Shih Chow and Henry Teicher,
*Probability Theory: Independence, Interchangeability, Martingales*, Springer, 3rd edition, 1997 -- a good reference book

- Alison Etheridge,
*A Course in Financial Calculus*, Cambridge University Press, 2002 -- a short introductory book on mathematical finance

- Steven E. Shreve,
*Stochastic Calculus for Finance II: Continuous-Time Models*, Springer-Verlag, 2004 -- a good textbook for stochastic Calculus and finance

- Steven Lalley,
*Skorohod's Theorem: Statement and Discussion*-- reading material for §8.3

- Patrick Billingsley,

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