Rice University
Instructors
Dennis CoxTime and Place
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
Tuesday 12:40  1:40 pm, Thursday 12:10  1:40 pm, Location HZ 122
Outline
Grading
1/4 Homework
1/4 Exam 1 on Basics (Martingale theory, Ito calculus, KalmanBucy, 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. L2norm 
Lecture
notes by Dennis Cox or "A Probability Path" by Resnick, [Introd. + properties of integrals + p 345348] 
January 19 (R)  Brownian motion, Construction, Martingale and Markov property 
Oksendal ch 1+2 "Adventures..." (Resnick) ch 6, p482499 
January 24 (R)  Brownian motion: reflection, extreme
values Simple Processes: Ito and Stratonovich 
Resnick "Adventures Stoch.
Processes" Oksendal, ch 3 first 4 pages (pp 2124) 
January 26 (R)  Ito integral of simple processes are
continuous martingales adapted L2 processes can be approximated by simple processes 
Oksendal ch 3 pp 2528 
January 31 (R)  General Ito integral, Ito isometry, int W dW; multidimensional Ito Integral  Oksendal pp 2930 ; pp 3435 
February 2 (R)  Ito integrals are continuous
martingales; compare with Stratonovich 
Oksendal pp 3033; Oksendal pp 3637 
February 7 (R)  Ito processes, Ito formula; Integration by parts  Oksendal Ch 4: pp 4345;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 FokkerPlanck or forward equations Motivation and Intuition 

April 4 (R) 
forward & backward equations,
formal proof Selffinancing Portfolio, Options 
Oksendal Ch 8.1 Ch 12.1 
April 6 
Spring Recess 

April 11 (R) 
BlackScholes PDE for option
pricing 
Oksendal Ch 12.3 
April 13 (R) 
Change of Measure 
Mikosch 
April 18 (R) 
BlackScholes 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
(texsource 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] 
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]