MATH-487 / 5 crédits

Enseignant: Hairer Xue-Mei

Langue: Anglais

## 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

## Dans les plans d'études

• Semestre: Printemps
• Forme de l'examen: Oral (session d'été)
• Matière examinée: Topics in stochastic analysis
• Cours: 3 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Semestre: Printemps
• Forme de l'examen: Oral (session d'été)
• Matière examinée: Topics in stochastic analysis
• Cours: 3 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Semestre: Printemps
• Forme de l'examen: Oral (session d'été)
• Matière examinée: Topics in stochastic analysis
• Cours: 3 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines

## Semaine de référence

 Lu Ma Me Je Ve 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

Mardi, 10h - 12h: Cours MAA330

Mercredi, 13h - 14h: Cours MAA330

Vendredi, 13h - 15h: Exercice, TP MAA112