Brownian motion by Antti Kemppainen

Stochastic methods, fall 2017

Lectures (Antti Kemppainen), Tuesdays 10:15-12:00, Thursdays 10:10-approx 11:45. Exactum CK111.

Exercises (Joonas Turunen) on Thursdays, 12-14 o'clock, Exactum D122 (1st and 2nd exercise session on Sep 14 and 21) and Physicum D210 (after that). [Notice that the timetable of exercises can be found by following the link "Ryhmä 1" on this page]

The midterm exam is Mon Oct 30 14:30-17:30.

The final exam is Thu Dec 21 9:00-12:00.


The course starts on September 5th and the exercises on the second week of the course.

In Fall 2017, the themes of the course touch the following topics (not necessary in this order)

1) Elements of probability
2) Stochastic processes
* Markov chains in discrete and continuous time, Poisson process, Brownian motion
3) Stochastic calculus
* Diffusion processes, integratration with respect to Brownian motion, stochastic differential equations
4) Simulation
5) Other topics

You can also check the Description panel on this page. Please remember to enrol! Notice that the information in Description panel is "permanent" and comes from Oodi. The information on the Fall 2017 course is described in the other panels. For instance, the conduct and the evaluation of the Fall 2017 course is under the "Conduct of the course" panel.

Contact information of the teachers can be found:
Antti Kemppainen,
Joonas Turunen

14.8.2017 at 09:00 - 14.12.2017 at 23:59


Antti Kemppainen's picture

Antti Kemppainen

Published, 28.11.2017 at 14:55

A small update on Chapter 6: a couple of typos were corrected, and the bibliography was added.

Antti Kemppainen's picture

Antti Kemppainen

Published, 30.10.2017 at 9:46

Reminder: the midterm exam is today Mon Oct 30 14:30-17:30 in Chemicum A129. Please preferably arrive some minutes early. I also wish you good luck for the exam!

Computer exercises will be posted today.

Antti Kemppainen's picture

Antti Kemppainen

Published, 24.10.2017 at 0:14

A file "Instructions to the midterm exam" was added to the material section. Please, read it before the exam.

The starting time of the midterm exam has been moved slightly to 14:30 (sharp, please arrive some minutes before preferably).

Antti Kemppainen's picture

Antti Kemppainen

Published, 15.10.2017 at 18:13

The latest lecture notes file got a small updated to the part on the gamler's ruin example (+ some typo corrected). This might help doing the exercises.

Antti Kemppainen's picture

Antti Kemppainen

Published, 13.10.2017 at 19:59

The midterm exam is Mon Oct 30 14:15-17:15 (provisional duration) in Chemicum A129.

If you cannot attend the exam, please contact the lecturer (Antti). Those who indicated that they cannot attend at the above given time have been sent an email.

Antti Kemppainen's picture

Antti Kemppainen

Published, 12.10.2017 at 10:10

A reminder: Please fill in the questionnaire on the midterm exam. Notice that the question is about which times are NOT suitable for you. (You can correct your answer, if needed, by filling in the questionnaire again.)

The date will be fixed today Thu Oct 12 around 15:30.

Antti Kemppainen's picture

Antti Kemppainen

Published, 10.10.2017 at 16:10

The midterm exam will be during the weeks Oct 23-27 or Oct 30-Nov 3. To choose a time slot that is suitable for most, a questionnaire has been posted to the Materials panel. If you want to influence the time, please fill in the questionnaire. There is also the option that all times fine. (Please tick a box of a time slot not suitable for you ONLY if you are really occupied for a very good reason. E.g. the lecture of another course is a bordeline case.)

The second purpose is that we use the number of "registered" participants when we estimate how many papers are printed.

For some information on the evaluation, check the "Conduct of the course" panel as well as the slides from the 1st exam. More information will be posted.

Antti Kemppainen's picture

Antti Kemppainen

Published, 1.10.2017 at 9:48

The problem sheet was posted (on Friday) as well as the computer exercises (on Saturday)---both for the Oct 5 exercise session. There was some mistakes in the Thu 28 version of lecture notes which were correted, so use an updated version when you are making the exercises.

Antti Kemppainen's picture

Antti Kemppainen

Published, 28.9.2017 at 12:35

Exercises are in Physicum D210. There was an unfortunate typo in Problem sheet 3.

Antti Kemppainen's picture

Antti Kemppainen

Published, 8.9.2017 at 11:03

The first problem sheet was posted yesterday. It is in the Tasks pane (of this course page) and you should login to see it.

Notice the exceptional location Exactum D122 which is valid at least for the first exercise session on Sep 14.

Problems are usually posted on Thursdays so that there is approximately one week for solving the problems. For more instrcution see Problem sheet 1.



You can use the Presemo room for interaction.


Here is the course’s teaching schedule. Check the description for possible other schedules.

Tue 5.9.2017
10:15 - 12:00
Thu 7.9.2017
10:15 - 12:00
Tue 12.9.2017
10:15 - 12:00
Thu 14.9.2017
10:15 - 12:00
Tue 19.9.2017
10:15 - 12:00
Thu 21.9.2017
10:15 - 12:00
Tue 26.9.2017
10:15 - 12:00
Thu 28.9.2017
10:15 - 12:00
Tue 3.10.2017
10:15 - 12:00
Thu 5.10.2017
10:15 - 12:00
Tue 10.10.2017
10:15 - 12:00
Thu 12.10.2017
10:15 - 12:00
Tue 17.10.2017
10:15 - 12:00
Thu 19.10.2017
10:15 - 12:00
Mon 30.10.2017
14:15 - 18:00
Tue 31.10.2017
10:15 - 12:00
Thu 2.11.2017
10:15 - 12:00
Tue 7.11.2017
10:15 - 12:00
Thu 9.11.2017
10:15 - 12:00
Tue 14.11.2017
10:15 - 12:00
Thu 16.11.2017
10:15 - 12:00
Tue 21.11.2017
10:15 - 12:00
Thu 23.11.2017
10:15 - 12:00
Tue 28.11.2017
10:15 - 12:00
Thu 30.11.2017
10:15 - 12:00
Tue 5.12.2017
10:15 - 12:00
Thu 7.12.2017
10:15 - 12:00
Tue 12.12.2017
10:15 - 12:00
Thu 14.12.2017
10:15 - 12:00
Thu 21.12.2017
09:15 - 13:00

Other teaching

14.09. - 21.09.2017 Thu 12.15-14.00
28.09. - 19.10.2017 Thu 12.15-14.00
02.11. - 14.12.2017 Thu 12.15-14.00
Joonas Turunen
Teaching language: English


Lecture notes (NOTE changes on Oct 4: theorem (etc.) numbering has been changed to better match the structure of the lecture notes. Sorry for inconvenience.):
* Foils of the first lecture - includes some information on the practical matters
* Chapter 1 - the entire chapter will be posted in parts during the first weeks of the lectures - please, check this section for updates
* Chapter 2 - the entire chapter will be posted in parts - please, check this section for updates
* Chapter 3 - some typed text will be posted in parts - please, check this section for updates

Selected bibliography: You can find some reference texts in this file. The main sources are under the title Stochastic processes, stochastic analysis and simulation. Many of these texts you can find online, for instance, as their author's own preprint copy or as ebook in the collections of the library of the University of Helsinki ( ). Measure theoretic probability -titles are for additional information (see lectures).


Notice the material mentioned above is listed below. You need to LOGIN TO SEE THE FILES.


Problem sheets for the exercise sessions

Below you can find problems for the exercise sessions. The date in the link below as well as in the problem sheet indicates the date that the exercise session takes place.

The exercise sessions on Sep 14 and 21 take place in Exactum D122 and the following sessions take place in Physicum D210.


To see the files login.

Matlab files for computer exercises

For technical purposes the files are .txt files. Please rename them as .m files.

Submitting solutions

In the case you can't attend the exercise session, you may contact Joonas to inform him and use the provided link for returning your solutions in a pdf file.

Solutions for the exercises

Topics of next lecture and past lectures

Here we summarize the topics covered on the lectures.


Next lecture:

* Dec 12: Stochastic calculus
* Dec 14: The final lecture. Summary of the course and we go through some applications.


Past lectures:

* Sep 5 and Sep 7: Elements of probability
* Sep 12 and Sep 14: Elements of probability (Lectured by Joonas Turunen)
* Sep 19 and Sep 21: Elements of probability
* Sep 26 and Sep 28: Elements of probability, Simulation: Part I
* Oct 3 and Oct 5: A couple of words on Simulation: Part I, then Markov chains
* Oct 10 and Oct 12: Markov chains, irreducibility and aperiodicity, stationaty measure, long-time behaviour, first step analysis
* Oct 17 and Oct 19: Markov chains, the definition of Poisson process
* Oct 31 and Nov 2: Poisson process,
* Nov 7: continuous-time Markov chains
* Nov 9:
- First half: some comments to the solutions of the midterm exam problems
- Second half: continuous-time Markov chains
* Nov 14: continuous-time Markov chains
* Nov 16: last comments on Markov chains. Simulation, part II: Markov chain Monte Carlo
* Nov 21 and Nov 23: Simulation, part II: Markov chain Monte Carlo. Conditional probability and expectation revisited. Martingales.
* Nov 28 and Nov 30: Martingales (quick overview, questions are welcome). Brownian motion. Stochastic integration.
* Dec 5 (Lectured by Joonas Turunen) and Dec 7: Stochastic calculus

Conduct of the course

The course consists of lectures and exercises which are supported by lecture notes and possibly other named literature. The points from the exercises are bonus points towards the final grade. The required reading for the exams include the all topics covered in lectures, exercises as well as the specified literature (which is mostly the lecture notes and announced clearly before the exam).


The midterm exam is Mon Oct 30 14:30-17:30 (provisional duration) in Chemicum A129. If you cannot attend the exam, please contact the lecturer (Antti).


The final exam is held on Thu Dec 21 9:00-12:00 in the room Exactum CK112.


Any feedback is welcome! You can send it using the form below.

Use the form below also for reporting typos in the lecture notes and other material. If you have an urgent issue related to problem sheets, it might be better to contact directly Joonas or Antti.


Master’s Programme in Theoretical and Computational Methods is responsible for the course.

Module where the course belongs to:

  • TCM300 Advanced Studies in Theoretical and Computational Methods

The course is available to students from other degree programmes.

  • Differential and integral calculus, linear algebra, complex numbers, (ordinary and partial) differential equations.
  • Elementary probability (either Todennäköisyyslaskenta I and/or II or similar knowledge, possibly from Fysiikan Matemaattiset Menetelmät IIa/b).
  • Courses in stochastics (Probability theory, Stochastic analysis etc.)
  • Courses in mathematical physics
  • You will learn to know the essential theories and methods of stochastics as well as stochastic models frequently encountered in applications.
  • You will understand how to treat mathematically random phenomena and related models.
  • If the implementation of the course emphasises more computational methods, you’ll learn to them also in practise in small tasks and/or a project work.

Can be taken in the early or later stages of studies. This can even be the only stochastics course you study.

Lectured every second year.

  • The course covers essential (theoretical and/or computational) methods in stochastics and emphasises their applications to natural sciences (and possibly also to other fields, e.g. finance and insurance mathematics). Although the topics may vary between years, we aim to cover the essential parts of background in the probability theory and topics in stochastic processes (e.g. Markov chains, random walks, Brownian motion, Poisson process) and stochastic calculus (integration with respect to random processes, Ito’s formula etc.).
  • The course is aimed to be self-contained for those students that are more interested in the applications. The course can be complementary to the other courses in stochastics (for those students that have studied or plan to study more stochastics courses) in that it emphasises more the applications.

Weekly lectures and exercises. Other teaching activities will be announced at the beginning of the course.

The course is graded based on the exam, exercises and the other compulsory course work announced at the beginning of the course.

Exam and exercises.