MATH-487 / 5 credits

Teacher: Hairer Xue-Mei

Language: English

Summary

This course offers an introduction to Markov processes, a widely used model for random evolutions with no memory.

Keywords

Probability, Conditional Expectation, Markov Property, Conditional expectations, Chapman-Kolmogorov equation,  Feller Property, Strong Feller property, Kolmogorov's theorem, stopping times, strong Markov property,  stationary processes, weak convergence and Prohorov's theorem, invariant measures,  Krylov- Bogolubov method, Lyapunov method. Ergodicity by contraction method and Doeblin's criterion.  Structures of invariant measures, ergodic theorems.

Optional: Diffusion Processes, Markov semigroups and Markov generators, Browniam motions, relation with second order parabolic differential equations, and Brownian motions.

Required courses

The folowing courses or knowledge on the content of the course will be very helpful: Analysis, Metric and topological spaces, probability, Linear Algebra, Measure and Integration (see comment below concernign the last). Also useful are: ODEs, PDES, and Functional analysis.

The following courses wil be helpful:

Measure ans Integration (Math 303) -- this year I will give a short introduction on this to motivated students who hasnt taken on a course on the Measure and Integration.

Probability Theory (Math 432)

Stochastic Processes (Math 332)

Recommended courses

The courses below are on the pathway of Stochastic Abalysis.

Introduction to stochastic PDEs (Math 485)

Martingales et mouvement brownien (MATH-330)

Stochastic Calculus (Math 431)

Numerical Solutiosn fo Stochastic Differential Equations (Math 450)

Stochastic Simulation (Math 414)

Stochastic epidemic model (Math 560)

Martingales in Mathematical finance (Math 470)

Learning Outcomes

By the end of the course, the student must be able to:

• Demonstrate understanding of the concepts and results from the syllabus includign the proofs of a variey of results
• Apply these concepts and results to tackle a range of problems, including previously unseen ones
• Apply their understanding to develop proofs of unfamiliar results
• Explain thier knowledge of the area in a concise, accurate and coherent manner
• Demonstrate additional competence i nthe subject through the study of more advanced material

Teaching methods

Lectures and Exercise classes

Expected student activities

Attend lectures, problem classes, do exercises and extra reading

Oral

Supervision

 Office hours No Assistants Yes

Bibliography

-- Stewart N. Ethier and Thomas G. Kurtz. Markov processes.
-- Markov Chains and Mixing Times, by David A. Levin Yuval Peres Elizabeth L. Wilmer
-- Markov Chains, James Norris
-- Markov Chains and stochastic stability, Meyn and Tweedie
-- Bremaud: Markov chains

In the programs

• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Topics in stochastic analysis
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Topics in stochastic analysis
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Topics in stochastic analysis
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks

Reference week

 Mo Tu We Th Fr 8-9 9-10 10-11 MAA330 11-12 12-13 13-14 MAA330 MAA112 14-15 15-16 16-17 17-18 18-19 19-20 20-21 21-22

Tuesday, 10h - 12h: Lecture MAA330

Wednesday, 13h - 14h: Lecture MAA330

Friday, 13h - 15h: Exercise, TP MAA112