Math 635 - Introduction to Brownian Motion and Stochastic Calculus

Meetings: MWF 11:00-11:50, Van Vleck B123

Instructor: Wai-Tong (Louis) Fan

Office: Van Vleck 725

Email: louisfan at math dot wisc dot edu

Office hours: Fridays 5-6pm or by appointment

I will use the class email list to send out corrections, announcements, please check your wisc.edu email from time to time.

Textbook Stochastic Calculus and Financial Applications, by M. Steele

Course description Math 635 is an introduction to Brownian motion and stochastic calculus. Sample path properties of Brownian motion, Ito stochastic integrals, Ito's formula, stochastic differential equations, and properties of their solutions will be discussed. If we have time we will also discuss some financial applications.

Prerequisites Math 521 and Math 632 (that is, a good level of mathematical maturity and an introductory course on stochastic processes)

This is a write-up by Prof. Seppäläinen on some of the basic concepts of probability theory.

Evaluation

Course grades will be based on homework assignments (20%) and exams (Midterm 30% + Final 50%). Late homework will not be accepted.

A Bonus project is available, which will be considered in your final grade.

Homework:

All due Monday at the beginning of lecture.

• • Homework 1 due February 5

  • Homework 2 due February 19

  • Homework 3 due March 5

  • Homework 4 due March 19

  • Homework 5 due April 16

  • Homework 6 due April 30 (postponed to May 4, Friday)

Learning Outcomes: At the end of this course students should be able to:

- Define, construct and derive the properties of basic stochastic processes, including discrete time martingales, continuous time martingales and Brownian motions.

- Construct stochastic integration with respect to Brownian motions, more specifically the Ito's integral.

- State, prove and apply Ito's formula to derive properties of Ito's integrals.

- Prove existence and uniqueness of stochastic differential equations (SDEs), solve and study basic properties of various linear SDEs.

- Model random processes in finance and economics using SDEs, derive and solve the Black-Scholes partial differential equations.

Tentative Schedule (What we actually covered are summarized in "lecture_overview.pdf" on CANVAS)

Part 0: Discrete time martingales

• • Week 1. (Jan 24, 26) Motivation (gambler's ruin, hitting time),

  • Martingales: definition, examples, martingale transform (Sections 2.1-2.2)

  • Week 2. Martingales: stopping times, submartingales, Doob's maximal inequality, L2 convergence theorem (Sections 2.2-2.6)

  • Week 3. Martingales: L1 convergence theorem, upcrossing inequality (Sections 2.6)

Part I: Continuous time martingales and Brownian motions

• • Week 4. Brownian motion: basic properties of multivariate Gaussians, a formal construction of BM, rigorous construction, scaling properties, Brownian bridge (Chapter 3)

  • Rigorous definition of the conditional expectation, uniform integrability (Sections 4.2-3)

  • Week 5. Uniform integrability: various results (Section 4.3)

  • Martingales in continuous time: the continuous analogues of the discrete results (Section 4.4)

  • Classic BM martingales (Section 4.5)

  • Week 6. Hitting times for BM (Section 4.5)

  • Properties of the Brownian path: Hölder continuity, non-differentiability, modulus of continuity (Sections 5.1-2)

  • Reflection principle, Invariance principle (Sections 5.3-5.4), Skorohod embedding (Sections 5.5)

Part II: Stochastic integration with respect to Brownian motions

• Week 7. (March 5,7,9) Ito integral: definition for simple processes, extension via the Ito isometry, computing the integral of B(t) (Sections 6.1-6.4). Midterm on March 7 (Wed)

  • Week 8. Ito integral as a process, the approximation operator (Sections 6.2, 6.6)

  • Week 9. Persistence of identity, localizing sequence, extension to L^2_{LOC} (Sections 6.5, 7.1)

• Week 10. Spring Break March 24-April 1.

  • Week 11. Integrating f(B_t) or f(t) with respect to B_t, time changed BM, local martingales, Ito formula (Sections 7.2-7.4, 8.1)

  • Week 12. Various versions of the Ito formula, martingale condition, application to the ruin problem, vector extension, rucurrence and transience of the BM in R^2 and R^d (d>2) (Sections 8.1-8.3)

  • Week 13. Functions of processes, the box calculus, general Ito, quadratic variation (Sections 8.4-8.6)

  • Stochastic Differential Equations, examples (Sections 9.1-9.3)

Part III: Stochastic Differential Equations & other applications

• • Week 14. Stochastic Differential Equations, examples (Sections 9.1-9.3)

  • Existence and uniqueness (Section 9.4), weak and strong solutions, Levy's representation theorem

  • Week 15. Black-Scholes formula (Sections 10.1-10.2)

  • Martingale representation theorem (Section 12.2)

  • Week 16. Representation via time-change (Section 12.4)

  • Girsanov theory (Section 13.2-13.3)

Instructions for Homework

• • Homework must be handed in by the due date (Monday) in class. Late submissions cannot be accepted.

  • If you cannot attend that Monday lecture, you should have it sent to the grader on or before 11:00am on the same day.

  • Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions. You are encouraged to discuss problems with your fellow students, but in the end you must write your own personal work.

  • Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Staple you sheets together.

  • It is not trivial to learn to write solutions. You have to write enough to show that you understand the flow of ideas and that you are not jumping to unjustified conclusions, but not too much to get lost in details. If you are unsure of the appropriate level of detail to include, you can separate some of the technical details as "Lemmas" and put them at the end of the solution. A good rule of thumb is if the grader needs to pick up a pencil to check your assertion, you should have proved it. The grader can deduct points in such cases.

  • You can use basic facts from analysis and the theorems we cover in class without reproving them. If you find a helpful theorem or passage in another book, do not copy the passage but use the idea to write up your own solution. If you do use other literature for help, cite your sources properly. However, it is better to attack the problems with your own mental resources instead of searching the literature or the internet. The purpose of the homework is to strengthen your problem solving skills, not literature search skills.

Instructions for in-class midterm and for final

• • Exams will be closed book and closed note. Some questions will be taken from homework.

ACADEMIC INTEGRITY

By enrolling in this course, each student assumes the responsibilities of an active participant in UW-Madison's community of scholars in which everyone's academic work and behavior are held to the highest academic integrity standards. Academic misconduct compromises the integrity of the university. Cheating, fabrication, plagiarism, unauthorized collaboration, and helping others commit these acts are examples of academic misconduct, which can result in disciplinary action. This includes but is not limited to failure on the assignment/course, disciplinary probation, or suspension. Substantial or repeated cases of misconduct will be forwarded to the Office of Student Conduct & Community Standards for additional review. For more information, refer to https://conduct.students.wisc.edu/academic-integrity/.

ACCOMMODATIONS FOR STUDENTS WITH DISABILITIES

McBurney Disability Resource Center syllabus statement: “The University of Wisconsin-Madison supports the right of all enrolled students to a full and equal educational opportunity. The Americans with Disabilities Act (ADA), Wisconsin State Statute (36.12), and UW-Madison policy (Faculty Document 1071) require that students with disabilities be reasonably accommodated in instruction and campus life. Reasonable accommodations for students with disabilities is a shared faculty and student responsibility. Students are expected to inform faculty [me] of their need for instructional accommodations by the end of the third week of the semester, or as soon as possible after a disability has been incurred or recognized. Faculty [I], will work either directly with the student [you] or in coordination with the McBurney Center to identify and provide reasonable instructional accommodations. Disability information, including instructional accommodations as part of a student's educational record, is confidential and protected under FERPA.” http://mcburney.wisc.edu/facstaffother/faculty/syllabus.php

DIVERSITY & INCLUSION

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The University of Wisconsin-Madison fulfills its public mission by creating a welcoming and inclusive community for people from every background – people who as students, faculty, and staff serve Wisconsin and the world.” https://diversity.wisc.edu/