Not offered in 2017-2018.

Wai Tong Fan (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.

- 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

#### Prerequisites

This course has flexible prerequisites. 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 homework, quizes and a possible mini project. You can see your score record on Learn@UW.

#### Lecture notes

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

#### Tentative schedule for Fall 2016.

**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.

#### 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