Meetings: MWF 9:55-10:45 Van Vleck B119 |

Instructor: Timo Seppäläinen |

Office: Van Vleck 425. Office Hours: MW after class, other times by appointment. |

Phone: 263-3624 |

E-mail: seppalai math wisc edu |

- Foundations of probability theory, especially conditional expectation
- Generalities about stochastic processes, Brownian motion, Poisson process
- Martingales
- Stochastic integral with respect to Brownian motion (quick overview of the Math 635 stochastic integral)
- Predictable processes and stochastic integral with respect to cadlag martingales and semimartingales
- Itô's formula
- Stochastic differential equations
- Local time for Brownian motion, Girsanov's theorem
- White noise integrals and a stochastic partial differential equation

**Week 1.**Measures and integration, BV functions.**Week 2.**Probability spaces, σ-algebras as information, conditional expectations.**Week 3.**Stochastic processes, filtrations, stopping times, quadratic variation.**Week 4.**Quadratic variation, path spaces, Markov processes.**Week 5.**Strong Markov property. Brownian motion.**Week 6.**Brownian motion, Poisson process, martingales. Friday September 10: no class on account of Midwest Probability Colloquium.**Week 7.**Martingales.**Week 8.**Stochastic integral with respect to Brownian motion.**Week 9.**Stochastic integral with respect to cadlag*L*martingales and local^{2}*L*martingales.^{2}**Week 10.**Stochastic integral with respect to cadlag local*L*martingales and semimartingales.^{2}**Week 11.**Itô's formula for cadlag semimartingales: proof of the single variable case. Applications of Itô's formula. Lévy's characterization of Brownian motion.**Week 12.**Bessel process. Part of Burkholder-Davis-Gundy inequalities. SDEs, first example Ornstein-Uhlenbeck process.**Week 13.**Geometric Brownian motion. Strong existence and uniqueness for Itô equations. (Thanksgiving week.)**Week 14.**Weak uniqueness and strong Markov property for Itô equations. Local time for Brownian motion.**Week 15.**Local time for Brownian motion. Tanaka's formula. Skorohod reflection problem. In-class exam on Wednesday.

- 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

- 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.
- 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.**The grader can deduct points in such cases. - 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. But it is
**not acceptable**to write solutions together or to copy another person's solution. In the end you have to hand in your own**personal**work. Similarly, finding solutions on the internet is tantamount to cheating. It is the same as copying someone else's solution.

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