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Course Description

LecturerPaolo Muratore-Ginanneschi 
Credits10, divided two units of 5 credits each. (grading info)
LanguageEnglish
Master's Programme MAST/ Theoretical and Computational Methods
Time span 03.09.2019 - 13.12.2019
VenueExactum Building
Lecture timetable
  • Tuesdays 10:15 - 12:00, Exactum, C129
  • Fridays 10:15-12:00, Exactum, C129

Questionnaire Please fill the form linked here
Exercise session

Wednesdays  12:15 - 14:00, Exactum, B120

Wednesday  11.9.2019 only: 12:15 - 14:00, Exactum, B119

Exams:

Exam Rules: you are entitled

to bring with you ONLY

1 A4 (both sides) of HANDWRITTEN NOTES

TCM320: E207, Physicum  Thu 24.10  at 14-16

TCM320: E207, Physicum, Mon 16.12 at 13-15

TCM321: E207, Physicum, Mon 16.12 at 13-15

TCM310: E207, Physicum, Mon 16.12 at 13-17


The course consists of two, as far as possible, parts I- II.

Part I (Stochastic Methods A, TCM320):

  • Overview of elementary probability
  • Finite Markov chains

Part II (Stochastic Methods B, TCM321)

  • Introduction to applied stochastic calculus.

It is possible to take credits separately for the two units of the course. Each unit is worth 5 credits. The course code for the first part is TCM320 (Stochastic Methods A), and TCM321 (Stochastic Methods B) for the second part.

Requirements

The course requires linear algebra and differential calculus as discussed in the basic courses. The course aims to be a self-contained introduction to stochastic methods. No previous expositions to probability and stochastic processes is assumed.

Literature

The following textbook contains all material (and much more) which will be discussed in the course:

  1. Grimmett, G. R. & Welsh, D. J. A. Probability. An introduction Oxford University Press, 2014, X, 270 (link to Helka-licensed e-book)

  2. Häggström, O. Finite Markov chains and algorithmic applications Cambridge University Press, 2002, 52, IX, 114.

  3. Gallager, R. Stochastic Processes: Theory for Applications Cambridge University Press, 2013.
  4. Jacobs, K. Stochastic Processes for Physicists. Understanding Noisy Systems Cambridge University Press, 2010, XIII, 204. (link to Helka-licensed e-book)

Gallager's book is based on the lecture notes of the course given for many years at MIT by the author. Lecture Notes and Video Lectures are available at this MITOPENCOURSEWARE web-page

Other useful references (more advanced) are

  1. van Handel, R. Stochastic Calculus and Stochastic Control, Lecture notes, CaliTech 2007 (recommended, download from van Handel's Princeton web-page).
  2. Norris, J. R. Markov Chains, Cambridge University Press, 1998, XVI, 237. (link to Helka-licensed e-book)

  3. Pavliotis, G. A. Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations Texts in Applied Mathematics, Springer New York, 2014, 339. (link to Helka-licensed e-book)

Special topics in part II of the course may-be found in

Detailed Schedule


DayTimePlaceCourse UnitTopic
Tue 3.9.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I1: Probability and random variables     (21.10)
Wed 4.9.201912:15 - 14:00Exactum, B120, Pietari Kalmin katu 5I2: Expectation values and basic inequalities   (21.10)
Fri 6.9.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I3: Conditional probability
Tue 10.9.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I4: Bernoulli process: basics     (18.09)
Wed 11.9.201912:15-14:00Exactum, B119, Pietari Kalmin katu 5I5: The gambler's ruin problem   (24.09)
Fri 13.9.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I

6: Asymptotic properties of the Bernoulli process  (18.09)

Tue 17.9.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I7: Empirical measure process  (21.10)
Fri  20.9.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I8: Central limit theorem and large deviations for the empirical measure process
Tue 24.9.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I9: Markov property
Fri 27.9.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I10: An overview of limit theorems for sequences of i.i.d. random variables with infinite variance
Tue 1.10.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I11: Time-homogeneous Markov chains, definition of hitting time  (09.10)
Fri 4.10.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I12: Hitting time statistics   (21.10)
Tue 8.10.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I13: Strong Markov property and applications  (18.10)
Fri 11.10.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I14: Classification of states   (21.10)
Tue 15.10.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5I15: Invariant measure  (29.10)
Fri 18.10.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5IInvariant measure (continued)
Tue 29.10.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II16: Recap
Wed 30.10.201912:15-14:00Exactum, B120, Pietari Kalmin katu 5II17: Perrron-Frobenius theorem (10.11)
Fri 1.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II18: Time reversal
Tue 5.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II19: Borel-Cantelli and convergence with probability one   (10.11)
Wed 6.11.201912:15-14:00Exactum, B120, Pietari Kalmin katu 5II20: Ergodicity of Markov chains
Fri 8.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II21: Poisson process     (12.12)
Tue 12.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II22: Approximation of a Poisson by a Bernoulli process
Wed 13.11.201912:15-14:00Exactum, B120, Pietari Kalmin katu 5II23: Continuous time Markov chains
Fri 15.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II24: Embedded Markov chain  (21.11)
Tue 19.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II25: Differential calculus with the Poisson process   (12.12)
Fri 22.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II

26: Calculus with independent Poisson processes

Tue 26.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II27: Stochastic calculus with the Wiener process     (12.12)
Fri 29.11.2019
10:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II28: Stochastic equations with Poisson noise  
Tue 3.12.201910:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II29: Stochastic equations with Wiener noise 
Tue 10.12.201910:15 - 12:00Exactum, C129, Pietari Kalmin katu 5II30: Forward and Backward Kolmogorov equations for Wiener SDEs
Fri 13.12.201910:15 - 12:00Exactum, C129, Pietari Kalmin katu 5IIRecap

Exercises

Exercise Session I-01:Wed 4.9.2019Exactum, B120, Pietari Kalmin katu 5

Review of basic probability, see lectures' calendar.

Exercise Session I-02:Wed 11.9.2019Exactum, B119, Pietari Kalmin katu 5

see lectures' calendar.

Exercise Session I-03Wed 18.9.2019Exactum, B120, Pietari Kalmin katu 5

Exercise set I (delivery deadline 18.09 12:15)

Exercise Session I-04Wed 25.9.2019Exactum, B120, Pietari Kalmin katu 5

Exercise set II (delivery deadline 25.09 12:15), (25.09:  typos corrected)

Exercise Session I-05Wed 2.10.2019

Exactum, B120, Pietari Kalmin katu 5

Exercise set III (delivery deadline 02.10 12:15)  (01.10: typos corrected)

Exercise Session I-06Wed 9.10.2019Exactum, B120, Pietari Kalmin katu 5

Exercise set IV (delivery deadline 09.10 12:15)  (09.10: typos corrected)

Exercise Session I-07

Wed 16.10.2019

Exactum, B120, Pietari Kalmin katu 5

Exercise set V (delivery deadline 16.10 12:15) 

Further model exam questions

Exercise set VI (no delivery deadline) (18.10: typos corrected)
Exercise Session II-01Wed 30.10.2019Exactum, B120, Pietari Kalmin katu 5

see lectures' calendar.

Exercise Session II-02Wed 6.11.2019Exactum, B120, Pietari Kalmin katu 5

see lectures' calendar.

Exercise Session II-03Wed 13.11.2019Exactum, B120, Pietari Kalmin katu 5

see lectures' calendar.

Exercise Session II-04Wed 20.11.2019Exactum, B120, Pietari Kalmin katu 5Exercise  set I   (22.11 typos corrected)
Exercise Session II-05

Wed 27.11.2019

Exactum, B120, Pietari Kalmin katu 5Exercise set II   (updated version)
Exercise Session II-06Wed 4.12.2019Exactum, B120, Pietari Kalmin katu 5Exercise set III
Exercise Session II-07Wed 11.12.2019Exactum, B120, Pietari Kalmin katu 5Exercise set IV  (22.11 typos corrected)



Exercise set V (no delivery)
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