MS-E1604 - Brownian motion and stochastic analysis D, Lecture, 24.2.2025-7.4.2025

Topic outline

• Select topic General

  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

  2025 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 M2. 
  • NOTE: Wednesday  12.03.25 (week 11) Exercise session 14:15 - 16:00 in room  R001U264 K-RYHMÄ.
    
  • Oral exam: tentatively on Monday 14.04.25; to be confirmed (NB: MyCourses shows some exam dates but they are not correct; the course has no written exam.)

  
  

  Grading

  Combination of oral exam (50% of the grade) and exercises (50% of the grade):
  

  • 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 0-1 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): There are 3 problems with increasing level of difficulty, that are worth 45 points in total (5 + 15 + 25 points).
    
  • Grading: To pass the course, you must return something for each exercise week 1-5, and participate to the oral exam.
  • Grade levels: TBA
    

  
  

  Teachers

  Eveliina Peltola (lecturer)  (eveliina.peltola "at" aalto.fi)
  Johanna Immonen (teaching assistant)    (johanna.1.immonen "at" aalto.fi)
  

  
  

  
  

  Prerequisites

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

  
  

  Literature:

  
  

  • Lecture notes. 
    
  • For prerequisites, see Kytölä: Probability Theory (and also Leskelä: Stochastic Processes).
  • See also warm-up Exercise set 0.

  
  Useful lecture notes:
  
  

  • Berestycki: Stochastic Calculus
  • Norris: Advanced Probability

  
  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 somewhat (but not quite) similar. 
  

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