# Martingale Theory

## Academic Year 2018 - 2019: Semester I

Class Time: Tuesday and Friday 02:00 PM - 04:00 PM (4 hours per week).

Lecture Hall: Room # 520 on the Fifth Floor of the S. N. Bose Bhavan

Course Duration: July 16 - November 26, 2018
(Total of 18 weeks = 14 weeks of classes + 1 Midterm weeks + 1 week of Puja break + 2 Final weeks).

Midterm Examination:
Date: September 12, 2018
Time: 2:30 PM - 04:30 PM (2 hours)
Venue: Class Room 522 on the Fifth Floor of the S. N. Bose Bhavan

Final Examination:
Date: November 12 - 26, 2018
Time: TBA (4 hours)
Venue: TBA

Email: antar (at) isid (dot) ac (dot) in
Office: Room # 3.4 on the Third Floor of the A. N. Kolmogorov Bhavan.

Instructor's Office Hours: Friday 04:00 PM - 06:00 PM.

Teaching Assistant (TA): Partha Pratim Ghosh
Email: p (dot) pratim (dot) 10 (dot) 93 (at) gmail (dot) com
Office: Room # 3.18 on the Fifth Floor of the A. N. Kolmogorov Bhavan.

Course Outline:

• Absolute continuity and singularity of measures. Signed measures, Hahn-Jordon decomposition.
• Conditional expectation, definition, examples and special cases. Properties of conditional expectation, linearity, order-preserving, MCT, DCT, Jensen inequality. Conditional expectation as a projection.
• Regular conditional probability, criterion for existence. Regular conditional distribution, conditional expectation as integral with respect to the regular conditional distribution.
• Definition of a filtration and adapted sequence.
• Definitions and examples of discrete parameter sub-martingale, martingale, super-martingale. Basic properties. Doob's Maximal Inequality, Kolmogorov's Maximal Inequality.
• Definition of stopping time, stopped process, stopped σ-algebra, examples and properties.
• Predictable processes, Discrete martingale transform, Doob's Decomposition Theorem.
• Concept of up crossings. Doob's up crossing inequality, The (sub) Martingale Convergence Theorem, Convergence theorem for non-negative super-martingale.
• Lp-bounded martingales, for p > 1 and Lp-convergence.
• Uniform integrability, convergence in L1.
• Backward Martingales, Levy's Upward and Downward Theorems.
• Applications: SLLN for i.i.d. random variables, Hewitt-Savage 0-1 Law, de Finetti's Theorem, SLLN for U-Statistics.
• Optional Stopping Theorems.
• Randon-Nikodym Theorem through martingale.
• Introduction to continuous parameter martingales: definition, examples and basic properties.
• Martingale Central Limit Theorem and applications.
• Azuma's Inequality and some applications.

Prerequisites:

• Measure Theoretic Probability (at the level of Probability: Theory and Examples by R. Durrett and/or Probability and Measure by P. Billingsley).
• Real Analysis (at the level of Principles of Mathematical Analysis by W. Rudin).
• Liner Algebra (at the level of Finite Dimensional Vector Spaces, by P. R. Halmos).

References:

• Probability and Measure by P. Billingsley.
• Probability Theory and Examples by R. Durrett.
• Probability with Martingales by D. Williams
• Foundations of Calculus of Probability by J. Nevue.
• Discrete Parameter Martingales by J. Nevue.

• Assignments: 10% of the total credit.
• Quizzes: 15% of the total credit.
• Midterm Exam: 25% of the total credit.
• Final Exam: 50% of the total credit.

Assignment Policies:

• There will be a total of 14 sets of homework assignments each carrying a total of 10 points. The average of the 10 best assignment scores will be taken for the final grading.
• The assignments will be given in class on every Friday, starting from July 20, 2018. Each assignment will be due in class on the following Friday. For example the first assignment is due on Friday, July 27, 2018.
• Each assignment will be based on the course materials which will be covered in the class in the week of the assignment.
• Late submission of an assignment will NOT be accepted. If you can not submit an assignment on time, don't worry about it and try to do well in the others. It will not count in your final grade since you have four extra assignment anyway.
• Graded assignments will be returned in the class an week after their submission. For example, the assignment which is due on July 27, 2018 (Wednesday) will be returned after grading on August 03, 2018 (Wednesday).

Quiz Policies:

• There will be four or more quizzes as surprise tests given in the class. This means there shall be no pre-scheduling. A quiz will cover materials done in the lectures given in the weeks prior to it.
• Each quiz will be of 15 points and will be of 15 minutes duration.
• Final grade for the quizzes will be (Best Score Before Midterm + Best Score Between Midterm and Final)/2.
• There will be NO supplementary quiz given for any student who may miss a quiz for whatsoever reason. If you miss one then do not worry, try doing well in the others.
• All quizzes will be part of the final grading.
• All quizzes will be closed note and closed book examinations.

Exam Policies:

• The Midterm and the Final Examinations will be open notes examinations. That means, students will be allowed to bring his/her own hand written notes, study materials, list of theorems etc. But no printed or photocopied materials will be allowed.
• Any unfair means used by any students in the examinations will be dealt with the strictest possible measures, as per the Institute rules. In particular, if any student is found to be using any kind of unfair practice during any of the examinations (including the quizzes) then he/she will be awarded ZERO in that examination.
• Note that the homework assignments are part of the final examination and hence their grading and other policies will be same as that of the final examination.