Imperial College London

Professor Xue-Mei Li

Faculty of Natural SciencesDepartment of Mathematics

Chair in Probability and Stochastic Analysis
 
 
 
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Contact

 

+44 (0)20 7594 3709xue-mei.li Website

 
 
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Location

 

6M51Huxley BuildingSouth Kensington Campus

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Summary

 

Courses


Measure and Integration(MAG_P19)(96031/97040/97149), offered to year 3/4/MSc

Measure and Integration is a foundational course for advanced studies using Analysis or Probaiblity.

The aim of the course is to understand measures, measurable maps, integration with respect to a measure, and various useful techniques. Here one can consolidate knowledge on first year integration theory and on taking expecatation from elementary probaiblity class.


Assessments:  There will be two courseworks  and one written exam in May.  

CW1 (5%).  Thursday 31/10/2019 (4pm), to be submitted through the U/G office.

CW2(5%)   Thursday  28/11/2018 (4pm),  to be submitted through the U/G office.    

May Exam:

2-hour Written exam (90%) for thirsd years; 2.5-hour exam (90%) for 4th years and MSc students.


Refrences:  Refer to library and blackboard page.

Course Structure: 

There are nine weekly assignments (not assessed).

Lecture Schedules:  

                  Wed: 9-10, HXLY 145,

                   Fri. 12-2 HLX 213 Clore


Markov Processes (M345P70) is offered to 3rd, 4th and P/G students.

Markov processes are widely used to model random evolutions with the Markov property `given the present, the future is independent of the past’.  Even a non-Markov process is a component of a Markov process. The theory connects with many other subjects in mathematics and has vast applications, in this course we aim to give a comprehensive introduction to this fascinating subject, building intuitions and foundations for further studies in stochastic analysis and in stochastic modelling. Markov processes are memoryless. The value of a Markov chain depend on its past only through its most recent value. 

The aim of the course is to understand the notion of Markov processes. We will  learn important properties of Markov chains (on general state spaces) and their long time behaviour.  For finite state Markov chains, we will familiar ourselves with computations, e.g.  the n-step transition probabilities and hitting probabilities. We will learn the notion of  invariant measures, their existence, uniqueness, and structures. We will learn techniques (Lyapunov functions, minorization), ergodic theorems (Perron-Frobenius theorem, Prohorov's theorem, Krylov-Bogolubov theorem,Birkhoff ergodic theorems). 


Main Reference:   The course will be based the lecture notes    Ergodic Properties of Markov Processes  by Martin Hairer.  The handout with some late additions, updated notes,   and additional material can be found on  Balckboard Learn.

Further references can be found on the Library course page. A good reference to Integration is Chapter 11 (especially section 3) of the third edition Real Analysis by Royden. A good introduction to Conditional Expectation is Chapter 4 of Probability by Leo Breiman.


Assessments:  There will be two courseworks  and one written exam in May.  

CW1 (5%).  31/10/2018 (4pm), to be submitted through the U/G office (room 649)

CW2(5%)   30/11/2018 (4pm),  to be submitted through the U/G office.    

May Exam:

2-hour Written exam (90%) for thirsd years; 2.5-hour exam (90%) for 4th years and MSc students.



Course Structure.

There are nine weekly assignments (not assessed).

Andris Gerasimovics will  offer a weakly support class. Time and location to be confirmed. 

Lcture Schedules.    Mon 10-10:50 (wks 2-11), Huxley 311; Tue 12:00-12:50 (wks 2-11), Huxley 144; Fri  12:00-12:50 (wks 1-10),  Huxley Clore.