MS-E199101 - Brownian motion and stochastic analysis, Lectures, 2.11.2021-16.12.2021

Topic outline

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  General
  Lectures (Tue 16-18 and Wed 16-18) will be online and exercises (Thu 16-18) in hybrid form. 

  
  

  NB: The events will not be recorded (in particular, questions during lectures are very welcome!). You may check out the recordings from the spring 2020 course, whose contents are rather similar. 
  

  In Zulip-chat we can discuss all around the course, ask hints to exercises, questions for lectures, etc. If you haven't received an invitation to Zulip but would like to, please contact Eveliina or Osama.

  
  

  Course description

  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

  2021 Period II (6 weeks): 2.11-9.12

  
  

  • Lectures: Tuesday and Wednesday 16:15 - 18:00 [zoom link]
    
  • Exercise sessions in room M1 - M232: Thursday 16:15 - 18:00 [zoom link]
    
  • Oral exam: date to be determined during 16.12-21.12.2021

  
  

  Grading

  Combination of oral exam and exercises (half-half):
  

  • Exercises (with max 45 points): 5 returned Problem sets, each comprising 6 exercises. For each set, we randomly select 3 exercises that are graded with point scale 0-2 and from the other 3 exercises you will gain 1 point each if you returned a solution attempt. In total, each Problem set thus yields max: 3x2 + 3x1 = 9 points.
  • Oral exam (with max 45 points)
    

  
  

  Teachers

  Eveliina Peltola (lecturer)
  Osama Abuzaid (teaching assistant)
  

  
  

  
  

  Prerequisites

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

  
  

  Literature:

  
  

  • The core contents of the course are similar to Kytölä: Brownian Motion and Stochastic Analysis.
    
  • For prerequisites, see Kytölä: Probability Theory (and also Leskelä: Stochastic Processes).
  • See also warm-up Exercise set 0.

  
  Useful lecture notes:
  
  

  • Berestycki: Stochastic Calculus and Applications
  • Norris: Advanced Probability

  
  Useful textbooks:
  

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

  • Durrett, R. Stochastic calculus: a practical introduction. CRC  Press,  Probability  and Stochastics Series, 1996.

  
  

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