Math 735 Stochastic Analysis

Fall 2016

Course description

Stochastic analysis is a term that refers to stochastic integration and stochastic differential equations and related themes. Here is a list of topics we expect to cover. The amount of time devoted to the fundamentals in the beginning will depend on the level of background that the audience possesses.

• 1. Foundations of probability theory, especially conditional expectation

  2. Generalities about stochastic processes, Brownian motion, Poisson process

  3. Martingales

  4. Stochastic integral with respect to Brownian motion (quick overview of the Math 635 stochastic integral)

  5. Predictable processes and stochastic integral with respect to cadlag martingales and semimartingales

  6. Itô's formula

  7. Stochastic differential equations

  8. Martingale problem method

  9. Local time for Brownian motion, Girsanov's theorem

  10. White noise integrals and a stochastic partial differential equation

Prerequisites

Section 1.1 (Measure theory and integration) of the textbook. You should be familiar with all concepts in this section, including sigma-algebras, measure spaces and BV functions.

The ideal background would be one or two semesters of graduate measure-theoretic probability theory, such as our 733 or 733-734. An essential prerequisite is a certain degree of mathematical maturity, so familiarity with advanced probability is not absolutely necessary. The course will rely on modern integration theory (measure theory covered in Math 629 and 721) and advanced probability, and we can cover some of these points quickly in the beginning.

Evaluation

Course grades will be based on in-class quizzes, a take-home exam and a possible in-class exam. Homework will be posted on Learn@UW. You can also see your score record on Learn@UW.

• • Problems of quizzes are from homework problems (with possible slight modifications).

  • Dates of quizzes will NOT be announced, but it will be on Tuesdays.

  • It is your responsibility to attend lectures and take the quizzes. No make up will be provided and the lowest score will be dropped.

Lecture notes

The course is based on lecture notes written by Professor Timo Seppalainen, available on Learn@UW. No textbook purchase is required.

Please read Section 1.1 (Measure theory and integration).

Tentative schedule

• • Week 1. Probability spaces, σ-algebras as information, conditional expectations.

  • Week 2. Stochastic processes, filtrations, stopping times, quadratic variation.

  • Week 3. Quadratic variation, path spaces, Markov processes.

  • Week 4. Strong Markov property. Brownian motion.

  • Week 5. Brownian motion, Poisson process, martingales. Friday September 10: no class on account of Midwest Probability Colloquium.

  • Week 6. Martingales. No class on Oct 13 on account of Midwest Probability Colloquium.

  • Week 7. Stochastic integral with respect to Brownian motion.

  • Week 8. Stochastic integral with respect to cadlag L² martingales and local L² martingales.

  • Week 9. Stochastic integral with respect to cadlag local L² martingales and semimartingales.

  • Week 10. Itô's formula for cadlag semimartingales: proof of the single variable case. Applications of Itô's formula. Lévy's characterization of Brownian motion.

  • Week 11. Bessel process. Part of Burkholder-Davis-Gundy inequalities. SDEs, first example Ornstein-Uhlenbeck process.

  • Week 12. Geometric Brownian motion. Strong existence and uniqueness for Itô equations.

  • Week 13. Weak uniqueness and strong Markov property for Itô equations.

  • Week 14. Local time for Brownian motion. Tanaka's formula. Skorohod reflection problem.

  • Week 15. Girsanov's theorem. Martingale problem method (if time permits)

Other material

• • A modern, rather deep treatment of the subject can be found in P. Protter: Stochastic Integration and Differential Equations, Springer.

  • An easier read is K. Chung and R. Williams: Introduction to Stochastic Integration, Birkhäuser.

  • A carefully written book is Y. Karatzas and S. Shreve: Brownian Motion and Stochastic Calculus, Springer. This book covers integrals with respect to continuous martingales.

  • Concise lecture notes are available on T. Kurtz's homepage: http://www.math.wisc.edu/~kurtz/m735.htm

Instructions for homework

• • Homework will not be collected but you are strongly encouraged to do all of them since problems of quizzes are from homework problems (with possible slight modifications).

  • Neatness and clarity are essential. Write one problem per page except in cases of very short problems. 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.

  • You can use basic facts from analysis and measure theory in your homework, and the theorems we cover in class without reproving them. If you do use other literature for help, cite your sources properly. However, it is better to attack the problems with your own resources instead of searching the literature.

  • It is extremely valuable, maybe essential, to discuss ideas for homework problems with other students.

Check out the probability seminar for talks on topics that might interest you.