Stochastic Differential Equations

 STAT 650, Spring 2006

Rice University

The course will cover both theory and applications of stochastic differential equations.
Topics include:
Wiener process, Brownian motion, Ito and Stratonovitch integral, Ito Calculus,
Markov properties, Kolmogorov and Fokker-Planck equations, Girsanov transforms
Applications in Finance, Signal Processing, Materials science, other fields.

Prerequisites: A course in stochastic processes and a graduate coursein probability, or consent of instructor.
Last update: June 24, 2006
[Outline] [Textbooks] [Grading] [Reading] [Homework problems and solutions]

Instructors

Dennis Cox
Duncan Hall 2080, 713 / 348 6007
Office Hours: T Th 10:50 - 12:00, or by appointment

Dr. Rudolf Riedi
Duncan Hall 2082, 713 / 348 3020,
Office Hours: T Th 10:50 - 12:00, or by appointment
Time and Place
Tuesday 12:40 - 1:40 pm, Thursday   12:10 - 1:40 pm,  Location  HZ 122

Outline

Indicate your intention to do a Project as early as possible and finalize arrangements
with one of the instructors by mid March.


Course book and notes
Suggested reading
Books on reserve at Fondren Library Other excellent reading on Stochastic Differential equation which
is in close agreement with the course (try Amazon.com to
buy or one of the instructors to borrow a copy)

Reference on foundations of Probability

Grading

1/4  Homework
1/4  Exam 1 on Basics (Martingale theory, Ito calculus, Kalman-Bucy, Details TBA)
1/2  Exam 2 on Advanced topics (Dynkin, Kolmogorov, Girsanov, Details TBA)
        or Course Project [agree with instructor on project topic by March 12]

[Outline] [Textbooks] [Grading] [Reading] [Homework problems and solutions]

Material covered during class

This doubles as a calendar for the course.
Check the schedule of 2004 for an idea on the course contents and progression.

 Covered material Suggested reading
January 12 (C)
 Orientation, history
January 17 (R)
 I. Stochastic Integral. Basics
!.1. Probability background
Conditional Expectation. L2-norm		 Lecture notes by Dennis Cox
or "A Probability Path" by Resnick,
[Introd. + properties of integrals + p 345-348]
January 19 (R) Brownian motion,
Construction, Martingale and Markov property		 Oksendal ch 1+2
"Adventures..." (Resnick) ch 6, p482-499
January 24 (R) Brownian motion: reflection, extreme values
Simple Processes: Ito and Stratonovich		 Resnick "Adventures Stoch. Processes"
Oksendal, ch 3 first 4 pages (pp 21-24)
January 26 (R) Ito integral of simple processes are continuous martingales
adapted L2 processes can be approximated by simple processes		 Oksendal ch 3 pp 25-28
January 31 (R) General Ito integral, Ito isometry,  int W dW; multidimensional Ito Integral Oksendal pp 29-30 ; pp 34-35
February 2 (R) Ito integrals are continuous martingales;
compare with Stratonovich		 Oksendal pp 30-33;
Oksendal pp 36-37
February 7 (R) Ito processes, Ito formula; Integration by parts Oksendal Ch 4: pp 43-45;46
February 9 (R) Solutions of Stochastic Differential Equations;
Review		 Oksendal Ch 5
February 14 (C) Filtering: Problem formulation;
Best Linear Estimation in the Gaussian case		 Oksendal Ch 6
February 16 (C) Filtering: Best Linear Prediction for Gaussian processes Oksendal Ch 6
February 21 (C) Filtering: Innovation Process Oksendal Ch 6
February 23 (C) Filtering: Riccatti equation for mean square error Oksendal Ch 6
February 28 (C)
 Diffusion processes, Markov property
 Oksendal Ch 7
March 2 (C)
 Stopped Filtration
March 7 (C)
 Strong Markov Property
March 9 (C)
 Generator
March 14+ 16
 Spring Break
March 21 (C)
 Dynkin's formula
March 23 (C)
 Exit Time of BM from Disc and Ring
March 28 (C)
 Characteristic Operator

March 30 (R)
 Kolmogorov backward equations
Fokker-Planck or forward equations
Motivation and Intuition
April 4 (R)
 forward & backward equations, formal proof
Self-financing Portfolio, Options
 Oksendal Ch 8.1
Ch 12.1
April 6
 Spring Recess
April 11 (R)
 Black-Scholes PDE for option pricing
 Oksendal Ch 12.3
April 13 (R)
 Change of Measure
 Mikosch
April 18 (R)
 Black-Scholes Price via Pricing Measure Mikosch
April 20 (R)
 Girsanov Transform, Novikov, weak solutions
 Oksendal Ch 8.6
April 25 + 27 (joint)
 Projects (topics from Physics, Bayesian analysis, Computational Finance)

[Outline] [Textbooks] [Grading] [Reading] [Homework problems and solutions]

Homework
(tex-source and solutions restricted to Rice University)

Homework sheet Due date (in class) Direct questions to
 Solutions
Problem Set 1 [ps] [pdf] [tex] Feb 9, 2006 Dr. Riedi
 Solution 1 [ps] [pdf] [tex]
Problem Set 2 [ps] [pdf] [tex] Feb 21, 2006 Dr. Riedi
 Solution 2 [ps] [pdf] [tex]
Problem Set 3: Oksendal 6.1 6.13 Feb 23, 2006 Dr. Cox
 In class
Problem Set 4: [pdf]
 March 21, 2006
 Dr. Cox
 In class Mar 23
Problem Set 5 [ps] [pdf] [tex] April 27, 2006 Dr. Riedi
 [ps] [pdf] [tex]

 

Late Homework Policy
 

Homework is due at the beginning of class on the due date. After the due date, but before solutions are handed out, homework can be turned in for 50% credit. In this case, please slip your homework under the instructors's office door, or bring it to class. After solutions are handed out, 0% credit will be issued. You are encouraged to work in groups for homeworks but you will hand in your own solution which you are expected to understand.

[Outline] [Textbooks] [Grading] [Reading] [Homework problems and solutions]

Any student with a documented disability needing academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain confidential. Students with disabilities should also contact Disabled Student Services in the Ley Student Center.
  Created December 24, 2005. Dr. Rudolf Riedi