Stochastic Differential Equations

 STAT 650, Spring 2004

 Rice University

The course will cover both theory and applications of stochastic differential equations. Topics include: the Langevin equation from physics, the Wiener process, white noise, the martingale theory, numerical methods and simulation, the Ito and Stratanovitch theories, applications in finance, signal processing, materials science, biology, and other fields.

Prerequisites: A course in stochastic processes and a graduate course in probability, or consent of instructor.

Last update: April 28, 2004

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


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: TBA, or by appointment
Time and Place
Tuesday Thursday   9:25 - 10:40 am,  Location DH 1042

Outline (to be refined as the course develops)

Indicate your intention to do a Project as early as possible and finalize arrangements
with one of the instructors by March 12.

Suggested reading

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

Reference on foundations of Probability (on reserve at Fondren)


2/3  Homework
1/3  Test 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.

Covered material Suggested reading
January 13 Orientation, history
January 15
review Measure theory Lecture notes Cox or Kurtz
January 20
continuity of Brownian motion, filtration, martingales
"Adventures..." (Resnick) for construction of BM
January 22
Langevin; Einstein: original approaches to SDEs
Oksendal Ch 1+2
January 27
White noise as a degenerate process, stoch integral with deterministic integrand

January 29
Ornstein-Uhlenbeck, interpretation of SDEs, informal definition of Ito and Stratanovitch integrals
Oksendal Ch 1+2
February 3
Explicit computation of \int W dW
Oksendal Ch 3 (p 28)
February 5 Adapted, mean square processes; approximation by simple functions
Oksendal Ch 3 (first half)
February 10 simple Ito Isometry; def of Ito integral; first extension Oksendal Ch 3 (second half)
February 12 Ito Isometry; first properties of Ito integral

February 17 Doob's martingale inequal; continuity of integral

February 19 cancelled

February 24
Ito formula; integration by parts
Oksendal Ch 4
March 9
Examples of Solutions to SDE-s
of solutions to SDE-s
Oksendal 5.1
March 16
Existence and Uniqueness
of solutions to SDE-s
Oksendal 5.2
March 18
Weak and Strong Solutions of SDE,
Time homogeneous Diffusions, Markov property
Oksendal 5.3
Oksendal 7.1
March 23, 25, 30
Generator (dual, acting on measures)

April 1
Generator (acting on functions),
Dynkin's formula, escape time from circle
Oksendal 7.2+3
April 6, 8
Poisson approximation of diffusion

April 13
Kolmogorov backwards equation,
Feynman-Kac equation
Oksendal 8.1
April 15
Black-Scholes equation,
Chemical system (Project)
Oksendal 12

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

(tex-source and solutions restricted to Rice University)

Homework sheet Due date (in class) Solutions
Problem Set 1 [ps] [pdf] [tex] Feb 5, 2004 included
Problem Set 2 [ps] [pdf] [tex] Feb 10, 2004 [ps] [pdf] [tex]
Problem Set 3 [ps] [pdf] [tex] Feb 12, 2004 See Dr. Cox
Problem Set 4 [ps] [pdf] [tex] Feb 22, 2004 [ps] [pdf] [tex]
Problem Set 5 [ps] [pdf] [tex] Mar 4, 2004 [ps] [pdf] [tex]
Problem Set 6 [ps] [pdf] [tex] Mar 30, 2004
[ps] [pdf] [tex]
Problem Set 7 [ps] [pdf] [tex] voluntary
office hours
TEST (Rice only) [ps] [pdf] [tex] last day of finals
office hours


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 January 11, 2004.  Drs. Dennis Cox and Rudolf Riedi