Math 2940: Applied Stochastic Methods Spring 2013

Instructor: David Swigon

Office: Thackeray 511, 412-624-4689, swigon@pitt.edu

Lectures:  MWF 9:00-9:50pm, Thackeray 525

Office Hours: MW 1:30-3:00pm, Thackeray 511, or by appointment.

Course Web Page: (check frequently for changes and updates) http://www.math.pitt.edu/~swigon/math2940.html

Course Description

In this course we will discuss the fundamental theory and applications of stochastic processes. The topics will include continuous/discrete state and continuous/discrete time Markov chains, with specializations for birth and death processes, and basic stochastic differential equations. We will use both analytical and computational tools. Most of the applications will be in biology, chemistry or physics.

Prerequisites

Advanced calculus (Math 1530, or equivalent) including Riemann and Lebesgue integral, Linear algebra (Math 1180, 1185 or equivalent) including eigenvalue and eigenvector analysis.

Textbook

Linda J.S. Allen: An Introduction to Stochastic Processes with Applications to Biology, Second Edition, CRC Press, 2011. ISBN:978-1-4398-1882-4

Additional Suggested Reading

E. Parzen, Stochastic Processes, SIAM 1999.

C. W. Gardiner, Handbook of Stochastic Methods, Springer 2004.

N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier, 1992.

E. Nummelin, General Irreducible Markov Chains and Non-negative Operators, Cambridge University Press, 1984.

S.P. Meyn & R.L. Tweedie, Markov Chains and Stochastic Stability, Springer, London, 1993.

Grading Scheme

Homework Assignments: 30%

Midterm Exam: 30%

Final Exam: 40%

Disability Resource Services

If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and Disability Resources and Services, 140 William Pitt Union, 412-648-7890 or 412-383-7355 (TTY) as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course.

Academic Integrity

Cheating/plagiarism will not be tolerated. Students suspected of violating the University of Pittsburgh Policy on Academic Integrity will incur a minimum sanction of a zero score for the quiz, exam or paper in question. Additional sanctions may be imposed, depending on the severity of the infraction. On homework, you may work with other students or use library resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly.

 


 

Syllabus

Week

Reading

Topics

Homework

Notes

Jan 7 Jan 11

1.1-1.6

Review of Probability Theory, distributions, generating functions, Central limit theorem, stochastic processes

HW #1

 

Jan 14 Jan 18

2.1-2.5

Discrete-time Markov chains, first passage time

 

Jan 21

NO CLASSES

 

Jan 23 Jan 25

2.6-2.10

Stationary probability distribution, MC simulation, random walk in 2D and 3D

HW #2

 

Jan 28 Feb 1

3.1-3.8

Restricted random walk, self-avoiding walk, absorbing boundary, reflecting boundary

LatticeRW_1D.m

LatticeRW_2D.m

Feb 4 Feb 8

[Saloff-Coste]

Convergence rate, Random walks on groups, card shuffling

HW #3

CardShuffling_single.m

Feb 11 Feb 15

[Diaconis]

[Tierney]

[Roberts & Rosenthal]

[Brooks & Roberts]

Metropolis algorithm,
Discrete-time continuous-space Markov Chains, Markov kernels, Convergence diagnostics

MHMCvisual2D.tar

MHMCvisual2D.zip

Feb 18 22

5.1-5.6

Continuous time Markov-Chains, Poisson process

 

Feb 25 Mar 1

5.7-5.11

Kolmogorov differential equation, Generating function technique

Midterm Exam

Exam review sheet

Solutions

 

 

Mar 4 Mar 8

6.1-6.11

Birth and death processes, First passage time

HW #4

 

Mar 11 Mar 15

SPRING BREAK

 

Mar 18 Mar 22

7.1-7.8

[Gillespie]

Applications of continuous-time Markov chains

Gillespie algorithm

 

Mar 25 Mar 29

8.1-8.5

[Doering]

Diffusion processes, boundary conditions, eigenfunction method

 

Apr 1 Apr 5

8.6-8.10

Wiener process, Ito integral, Stochastic differential equations

HW #5

notes

 

Apr 8 Apr 12

9.1-9.5

Numerical methods, Applications of SDEs

 

Apr 15 Apr 19

9.6-9.9

Applications of SDEs, Review

 

 

 

Final Exam