# Math 635 Introduction to Brownian Motion and Stochastic Calculus

### Spring 2014

 Meetings: MWF 1:20-2:10, B139 Van Vleck Instructor: Timo Seppäläinen Office: 419 Van Vleck, Office Hours: after class MW or by appointment Phone: 263-2812 E-mail: seppalai@math.wisc.edu

This is the course homepage that also serves as the syllabus for the course. Here you will find updates on scheduling matters.

Math 635 is an introduction to Brownian motion and stochastic calculus without a measure theory prerequisite. Topics touched upon include sample path properties of Brownian motion, Itô stochastic integrals, Itô's formula, stochastic differential equations and their solutions. As an application we will discuss the Black-Scholes formula of mathematical finance.

### Prerequisites

Math 521 and Math 632 (that is, a good level of mathematical maturity and an introductory course on stochastic processes). If you need to review basics of probability theory, here is a brief handout.

### Textbook

Stochastic Calculus and Financial Applications by J. Michael Steele. Springer.

The heart of the material is in Chapters 6 and 8 of the book. Before we can define the Itô integral, we need to cover preliminaries such as conditional expectations, martingales and Brownian motion from Chapters 2-5.

### Evaluation

Course grades will be based on a combination of homework and two exams.

Exam 1. In-class portion on Wednesday March 5, take-home portion due two days later.

Exam 2. Saturday May 3, 1-3 PM, VV B139, the usual classroom.

### Spring 2014 Schedule

We will record our actual schedule here.
• Week 1. 1/21-24. Probability spaces, measure spaces, Lebesgue integral and expectation.
• Week 2. 1/27-31. Conditional expectation. Martingales.
• Week 3. 2/3-7. Martingales. Borel-Cantelli lemma.
• Week 4. 2/10-14. Gaussian distributions and processes. Brownian motion: definition, canonical probability space, standard filtration, roughness of paths.
• Week 5. 2/17-21. Quadratic variation of Brownian motion. The first stochastic integral. Completeness of Lp spaces. Begin development of Itô integral.
• Week 6. 2/24-28. Development of the Itô integral. For a proof of the approximation lemma, see Lemma 4.2 and Proposition 4.3 in these lecture notes.
• Week 7. 3/3-3/7. Localization. Midterm exam.
• Week 8. 3/10-14. Itô's formula.
• Spring Break March 17-21.
• Week 9. 3/24-28. Itô's formula. Examples of stochastic differential equations: Ornstein-Uhlenbeck process, geometric Brownia motion, Brownian bridge.
• Week 10. 3/31-4/4. SDEs: (strong) existence and uniqueness.
• Week 11. 4/7-11. Weak and strong solutions to SDEs. Arbitrage. Begin Black-Scholes model. Derivation of the heat equation.
• Week 12. 4/14-18. The heat kernel and existence and uniqueness for the heat equation. Solving the Black-Scholes p.d.e.
• Week 13. 4/21-25. Girsanov's theorem.
• Week 14. 4/28-5/2. Finish proof of Girsanov's theorem, applications.
• Week 15. 5/5-9. Going over exam 2. Feynman-Kac formula for Brownian motion and diffusions, application to the Black-Scholes model of a European call option. Representation theorems.

### Instructions for Homework

• Homework must be handed in by the due date, either in class or by 3 PM in the instructor's office or mailbox. Late submissions cannot be accepted.
• 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.