Math 635 - Introduction to Brownian Motion and Stochastic Calculus

Spring 2012

I will use the class email list to send out corrections, announcements, please check your wisc.edu email regularly.

Announcements:
  1. Exam 1 information: The exam will be a take-home exam.  It will cover chapters 1 - 4.  You will receive the exam on Monday, March 5th at the end of class.  The exam is due by class time (beginning of class) on Wedensday, March 7th.  If you want to turn it in earlier, you can either email me your solutions (if typed up) or by getting it to me some other way.

Course description: Math 635 serves a few purposes.  First and foremost, it serves as the second course (with 632 or 605 being the first) in a sequence aimed at introducing students to stochastic processes.  Whereas both Math 632 and 605 focus on processes with discrete state spaces, 635 focuses on processes with a continuous state space and, in particular, on Brownian motion. Sample path properties of Brownian motion, Ito stochastic integrals, Ito's formula, stochastic differential equations, and properties of their solutions will be discussed. Other important concepts we will discuss are general notions of stopping times and martingales. 

Required textbook: Stochastic Calculus and Financial Applications, by J. Michael Steele.  I plan to cover chapters 1 - 9 of the text, plus selected topics from later sections.

Prerequisites: Math 521 *and* Math 632 or 605 (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 (25%), two midterms (25% each) and the final exam (25%).

Final Exam information:
Date: 5/13/2012
Time: 2:45 - 4:45 PM.
Place: TBD.


Instructions for homework: 

Homework Assignments:

  1. Homework 1. Due February 6th.  The .tex file is here. Solutions are available at the Learn @ UW website.
  2. Homework 2.  Due February 24th.  The .tex file is here. Solutions are available at the Learn @ UW website.
  3. Homework 3.  Due March 12th.  The .tex file is here.
  4. Homework 4.  Due March 19th.  The .tex file is here.
  5. Homework 5.  Due March 26th.  The .tex file is here.
  6. Homework 6.  Due April 18th.  The .tex file is here.
  7. Homework 7.  Due May 11th. 

Lectures.
  1. Chapter 4 lecture notes (slides) are here. 
  2. The computation showing f(B_t) - \int_0^t (1/2)f '' (B_s) ds is a martingale is here.
  3. Chapter 5 slides are here.
  4. Chapter 6 slides are here.
  5. Chapter 8 slides are here.
  6. Written notes on numerical methods are here.  The .tex file is here.

Spring 2011 Schedule: This schedule is tentative and is subject to change.  Section numbers refer to Steele's book.

Week
 Monday
 Wednesday
 Friday
1
 Jan. 23rd
Random walk and first step analysis.
Readings: Chapter 1.
 Jan. 25th
Finish Chapter 1.  Begin martingales.
Readings: Chapter 1, Sections 2.1 - 2.2.
 Jan 27th
Martingales.
Readings: Sections 2.2.
2
 Jan. 30th
Readings: Sections 2.3.
 Feb. 1st
Readings: Sections 2.4, 2.5.
 Feb. 3rd
Readings: Section 2.5 (finish Doob's Maximal inequality).  Start 2.6.
3
 Feb. 6th  HW 1 Due.
Readings: Section 2.6
 Feb. 8th
Readings: Section 3.1.
 Feb. 10th Class Cancelled.
Readings: N.A.
4
 Feb. 13th
Readings: Section 3.1 - 3.5.
 Feb. 15th
Readings: Section 3.1 - 3.5.		 Feb. 17th
Readings: Section 3.1 - 3.5.
5
 Feb. 20th
Readings: Sections 3.5 and 4.1-4.2.
 Feb. 22nd
Readings: Sections 4.2 - 4.3.
 Feb. 24th HW 2 Due.
Readings: Section 4.4.
6
 Feb. 27th
Readings: Section 4.5.
 Feb. 29th
Readings: out with bad back.
 March 2nd
Readings: Sections 5.1 - 5.2.
7
 March 5th 1st Take Home Exam Given
Readings: Sections 5.1 - 5.2.
 March 7th 1st Take Home exam Due
Readings: Sections 5.3 - 5.4.
 March 9th
Readings: Section 6.1 and 6.6.
8
 March 12th HW 3 Due.
Readings: Sections 6.1 - 6.2, 6.6.
 March 14th
Readings: Sections 6.2 - 6.4.		 March 16th
Readings: Sections 6.4 - 6.5.
9
 March 19th  HW 4 Due.
Readings: Sections 7.1.
 March 21st
Readings: Section 7.2.		 March 23rd
Readings: Sections 7.2, 7.4.
10
 March 26th HW 5 Due.
Readings: Section 7.4.
 March 28th
Readings: Sections 7.4, 8.1-8.2.		 March 30th
Readings: Sections 8.1 - 8.2.
N.A.
 April 2nd
No class due to spring break.
 April 4th
No class due to spring break.
 April 6th
No class due to spring break.
11
 April 9th
Readings: Sections 8.2 - 8.3.
 April 11th
Readings: Sections 8.3-8.4.		 April 13th
Readings: Sections 8.4-8.6, 9.1.
12
 April 16th
Readings: Sections 9.1 - 9.3.
 April 18th HW 6 Due.
Readings: Sections 9.3 - 9.4.		 April 20th
Readings: Sections 9.4 - 9.5.
13
 April 23rd 2nd Exam Given
Readings: Sections 9.4 - 9.5. Start numerical methods.
 April 25th 2nd Exam Due.
Topic:  Numerical methods.		 April 27th
Topic:  Numerical methods.
14
 April 30th
Topics: Financial math.  Replicating portfolios.
Readings: Sections 10.1 - 10.3.
 May 2nd
Topics: Replicating portfolios and Black-Scholes.
Readings: Sections 10.1 - 10.3.		 May 4th
Topics: Black-Scholes, Martingale Representation Theorem.
Readings: Sections 10.1 - 10.3, and 12.2, 13.1.
15
 May 7th
Readings: Sections 13.1 - 13.3.
 May 9th
Readings: Sections 13.1 - 13.3.		 May 11th  HW 7 Due.
Topic: Inference for SDEs.
Guest Lecturer: Bret Hanlon, UW Stats Dept.
Readings: None for now.


Miscellaneous material:

  1. Here is a writeup on solving diference equations.  Here is the .tex file.
  2. Here is a writeup on conditional expections from a perspective of Math 431.
  3. Here is a writeup concerning the Wright-Fischer model where some hitting probabilities are computed using martingale methods.
  4. Here are the matlab codes I used to simulate the relevant stochastic processes on Wednesday, February 8th:  Polya Urn, Polya Urn Histogram, Product example, Random Walk.
  5. Robert Brown's original paper can be found here.
  6. Here are some codes running Brownian motions.  The first two were used to do the example where I estimated the probability of a path hitting the level 5 by time 1000.  The third was used to demonstrate Donsker's theorem:  BM.m, BM_Average.m, ScaledRandomWalk.m.
  7. Here are some Matlab codes implementing Euler's method and Milstein's method on the SDE dX_t = -X_t dt + 2X_t^2 dB_t.  The codes simulate a single trajectory and plots it.  Here is a code implementing both Euler's method and Milstein's method on the same SDE.  The methods are using *the same noise*.  Thus, we should expect both to approximate the exact solution, with Milstein's method approximating it closer.  Play with these codes.  Feel free to mess around with the SDE and initial conditions.  For example: test what happens when you change -X_t dt to -10 X_t^3 dt and choose an initial condition of X_0 = 1 and h = 1/5.  You'll see these methods are not so good then.  Can you explain why?
  8. Here is the code that approximates rare events for normal random variables.