~~NOTOC~~

====== General information ======

[[http://www.ntnu.edu/studies/courses/MA8109/|Course Description]]

=====Lectures:=====  

  * Wednesday 14:15-16:00, room 734 SB2
  * Thursday 15:15-17:00, room 734 SB2
  
  * First lecture: Thursday August 24

=====Exercises:=====
  * Every 2-3 weeks there will be homework exercises.
  * Solutions will be posted on the webpage.
  * See "Homework" in left menu.

===== Lecturer =====

  * [[http://www.math.ntnu.no/ansatte/v/?erj|Espen Robstad Jakobsen]]\\ 
  * <Espen.Jakobsen@math.ntnu.no>\\
  * Office 1148, SBII\\
  * Office hours: Mondays 14:15-15:00

===== What is this course about?=====
Differential equations with noisy/uncertain coefficients (stochastic differential equations), and their solutions, continuous time stochastic processes: We give a mathematical background, the main results, and some applications. Of the multitude of applications in science, engineering and other disciplines, the most famous one is perhaps the Black-Scholes model for option pricing in finance.

=====Who can take this course?=====
    * Interested students at Master or PhD level.
    * The level should be suitable for good 4th year students in the industrial mathematics program.
    * It can be taken as a regular course or a 'fordypningsemne' (TMA4505).

===== Books and reading material =====
=== Main textbook ===
  * B. Øksendal: //Stochastic Differential Equations,// Springer Verlag.\\ \\ Available on Springer Link: [[https://link.springer.com/book/10.1007/978-3-642-14394-6|https://link.springer.com/book/10.1007/978-3-642-14394-6]]

=== Notes by Krogstad ===
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/MeasureAndProbability2007.pdf|Measure And Probability]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/Lebesgueintegralet.pdf|Lebesgueintegralet]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/BrownianMotion2009.pdf|Brownian Motion]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/MartingalesAndIto2011.pdf|Martingales And Ito]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/MeanSquareContinuousProcesses2011.pdf|Mean Square Continuous Processes and Proof of Ito's formula]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/LinearSDEandPhysicalBrownianMotion.pdf|Linear SDEs and Physical Brownian Motion]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/DiffusionComments2011.pdf|Informal Comments on Ito Diffusions]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/KolmogorovFokkerPlanck2011.pdf|Kolmogorov and Fokker-Planck equations]]

=== Supplementary reading ===
  * Easy introduction:
    * T. Mikosch: //Elementary Stochastic Calculus with Finance in View//, World Scientific, 1998. 
  * Intermediate level:
    * L. C. Evans: An Introduction to Stochastic Differential Equations, 2013, AMS (a very readable lecture note)
    * J. Jacod and P. Protter: //Probability Essentials//, 2004, Spinger.
    * R. Schilling and L. Partzsch: //Brownian Motion//, 2012, De Gruyter.
    * H.-H. Kuo: //Introduction to Stochastic Integration//,2006, Springer.
  * Advanced level:
    * D. Revuz and M. Yor: //Continuous Martingales and Brownian Motion//, 2005, Springer.
    * I. Karatzas and S. E. Shreve: //Brownian Motion and Stochastic Calculus//, 1991, Springer.

===== Contents:=====

  *   Probability and measure theory (background)
  *   Independence and conditional expectation (main theorems)
  *   Differential equations with stochastic loading
  *   Brownian motion
  *   Martingale theory
  *   The Itô integral
  *   Itô calculus
  *   Stochastic differential equations 
  *   Optimal stopping
  *   Diffusions
  *   Limit theorems
  *   Stochastic modelling applications  

===== Final Curriculum =====

From Øksendal:
  * Chp. 2
  * Chp. 3 
  * Sec. 4.1-4.2
  * Sec. 4.3 ideas and results, no proofs (self-study)
  * Sec. 5.1-5.2
  * Sec. 5.3 ideas and results, no proofs (self-study) 
  * Sec. 7.1 
  * Sec. 7.2 (up till but not including the proof of Thm. 7.2.4)
  * Sec. 7.3
  * Sec. 7.4
  * Sec. 8.1  
  * Sec. 8.2 (not the proof)
  * Sec. 8.3 (only Thm. 8.3.1)
  * Appendix A 
  * Appendix B 

Notes:
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/MeasureAndProbability2007.pdf|Measure And Probability]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/BrownianMotion2009.pdf|Brownian Motion]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/MartingalesAndIto2011.pdf|Martingales And Ito]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/MeanSquareContinuousProcesses2011.pdf|Mean Square Continuous Processes and Proof of Ito's formula]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/LinearSDEandPhysicalBrownianMotion.pdf|Linear SDEs and Physical Brownian Motion]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/DiffusionComments2011.pdf|Informal Comments on Ito Diffusions]]
  * [[http://www.math.ntnu.no/emner/MA8109/2017h/notes/HEK2011/KolmogorovFokkerPlanck2011.pdf|Kolmogorov and Fokker-Planck equations]]

All homework problem sets.