Topic outline

  • General

    Brownian motion is a fundamentally important stochastic process, discovered in the contexts of financial markets and statistical physics. It relates to diverse mathematical topics from partial differential equations to constructive quantum field theory. This course introduces you to the key techniques for working with Brownian motion, including stochastic integration, martingales, and Ito's formula. 


    Contents
    • Brownian motion
    • Martingales
    • Stochastic integral
    • Itō's formula and applications


    Time
    2023 Period IV (6 weeks)

    • Lectures: Monday and Wednesday 10:15 - 12:00 in room M3 - M234
    • Exercise sessions Wednesday 14:15 - 16:00 in room Y229a
    • Oral exam: date to be determined (e.g. 17-18.4.2023)

    Grading
    Combination of oral exam (50% of the grade) and exercises (50% of the grade):
    • Exercises (with max 75 points): 5 returned Problem sets, each comprising 6 exercises. For each set, we randomly select 3 exercises that are graded with point scale 0-3 and from the other 3 exercises you will gain 0-2 point each if you returned a solution attempt. In total, each Problem set thus yields max: 3x3 + 3x2 = 15 points.
    • Oral exam (with max 45 points): There are 3 problems with increasing level of difficulty, that are worth 45 points in total (10 + 15 + 20 points).
    • Grading: Exercise points have weight 0.6 and exam points weight 1. To pass the course, you must return something for each exercise week 1-5, and participate to the oral exam.
    • Grade levels: Getting in total at least 72 weighted points (out of 90) gives grade 5; in total 64 weighted points gives grade 4; 56 weighted points gives grade 3; 48 weighted points gives grade 2; and 40 weighted points gives grade 1.


    Teachers

    Eveliina Peltola (lecturer)
    Osama Abuzaid (teaching assistant)


    Prerequisites
    Probability theory (MS-E1600), Stochastic processes (MS-C2111) or equivalent.


    Literature:


    Useful lecture notes:


    Useful textbooks:

    • Le Gall, J.-F. Brownian Motion, Martingales, and Stochastic Calculus. Graduate Texts in Mathematics, volume 274, 2016.

    • Williams, D. Probability with Martingales. Cambridge University Press, 1991.
    • Durrett, R. Stochastic calculus: a practical introduction. CRC Press, Probability and Stochastics Series, 1996.

    You may check out the recordings from the spring 2020 course, whose contents are rather similar.