General information

Course Description

• Wednesday
  08:15-10:00
  room 734 SB2.

• Friday
  14:15-16:00
  room 734 SB2.

• First lecture: Friday 23.08.

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

• Espen Robstad Jakobsen
  
• Espen [dot] Jakobsen [at] math [dot] ntnu [dot] no
  
• Office 1148, SBII
  
• Office hours: Wednesday 14:15-15:00, room 1148 SB2 (my office)

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.

• 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).

• Øksendal: Stochastic Differential Equations, Springer Verlag.
  
  Available on Springer Link: https://link.springer.com/book/10.1007/978-3-642-14394-6

• Measure And Probability
• Lebesgueintegralet
• Brownian Motion
• Martingales And Ito
• Mean Square Continuous Processes and Proof of Ito's formula
• Linear SDEs and Physical Brownian Motion
• Informal Comments on Ito Diffusions
• Kolmogorov and Fokker-Planck equations

• Strong solution of SDEs. Uniqueness and Existence

• 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.

• 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

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:

• Measure And Probability
• Brownian Motion
• Martingales And Ito
• Mean Square Continuous Processes and Proof of Ito's formula
• Strong solution of SDEs. Uniqueness and Existence
• Linear SDEs and Physical Brownian Motion
• Informal Comments on Ito Diffusions
• Kolmogorov and Fokker-Planck equations

All homework problem sets.

• More Brownian motion: See e.g. the book Revuz-Yor Continuous martingales and Brownian motions.
• Levy processe: The easiest generalisation of Brownian motion.
  (Classical book: Satt Levy processes and infinitely divisible distributions)
• The martingale problem: Weak solutions of SDEs, fine analysis of diffusion processes using PDE/potential theory/harmonic analaysis.
  (Classical book: Stroock-Varadhan Multidimensional diffusion processes)
• Stochastic control theory: Controling SDEs, Bellman principle, optimal controls, Bellman PDE.
  (Books: Young-Zhou Stochastic Controls, Fleming-Soner Controlled markov processes and viscosity solutions)
• General Markov processes: Many more processes than Brownian motion and Ito processes. Potential theory, semigroup theory, Dirichlet forms.
  (Classical book: Blumenthal-Getoor Markov processes and pontial theory, Fukushima et al. Dirichlet forms and symmetric markov processes)
• Pseudo differential operators and Markov processes: See book series by Niels Jacob.
• Numerical methods: See e.g. the book Kloden-Platen Numerical solution of stochastic differential equations.
• Other: spatial noise/random fields, stochastic PDEs (many and very different theories), applications in economy, physics, engineering, sciences, …