# Math 635 - Introduction to Brownian Motion and Stochastic Calculus

Spring 2010

Meetings: TR 13:00-14:15, Van Vleck B329
Instructor: Benedek Valkó
Office: 409 Van Vleck
Phone: 263-2782
Email: valko at math dot wisc dot edu
Office hours: Tuesday 14:30-15:30 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, a midterm and the final exam. Late homework will not be accepted.

Announcements:
• There will be no class on March 11.
• The date of the take-home midterm exam is March 18.

The final exam will be available HERE on May 11. It is due back by 10AM, May 13.

Homework:

Schedule:

• Week 1. Basic principles of probability (a quick review)
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)
Brownian motion: definition, basic properties of multivariate Gaussians, a formal construction of BM (Chapter 3)
• Week 4. Brownian motion: 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)
• Week 7. Skorohod embedding (Sections 5.5)
Ito integral: definition for simple processes, extension via the Ito isometry, computing the integral of B(t) (Sections 6.1-6.4)
• 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. 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 11. 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 12. Functions of processes, the box calculus, general Ito, quadratic variation (Sections 8.4-8.6)
Stochastic Differential Equations, examples (Sections 9.1-9.3)
• Week 13. Stochastic Differential Equations, examples (Sections 9.1-9.3)
Existence and uniqueness (Section 9.4), weak and strong solutions, Levy's representation theorem
• Week 14. Black-Scholes formula (Sections 10.1-10.2)
Martingale representation theorem (Section 12.2)
• Week 15. Representation via time-change (Section 12.4)
Girsanov theory (Section 13.2-13.3)