General information

Lectures:

  • Thursday 15:15 - 17:00 in room 734 SB2
  • Friday 08:15 - 10:00 in room 734 SB2
  • First lecture: August 29, 2013

Room 734 is on the same floor as the Math. Department office.

Exercises:

  • Solutions will be posted on the webpage.

Lecturer

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

  • Øksendal: Stochastic Differential Equations, Springer Verlag (4-6th editions).

Notes by Krogstad

Evans lecture note

Supplementary reading

  • Easy introduction:
    • T. Mikosch: Elementary Stochastic Calculus with Finance in View, World Scientific, 1998.
  • Intermediate level:
  • Advanced level:
    • D. Revuz and M. Yor: Continuous Martingales and Brownian Motion, 2005, 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

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 proof of Thm. 7.2.4 p. 111)
  • Sec. 7.3
  • Sec. 7.4
  • Sec. 8.1 (not the proofs of Lemma 8.1.3, 8.1.4 and Theorem 8.1.5)
  • Sec. 8.2 (not the proof)
  • Sec. 8.3 (only Thm. 8.3.1)
  • Appendix A
  • Appendix B

From Evans:

  • Sec. 2.G
  • Sec. 3.A
  • Sec. 4.D-4.E

Notes:

All homework problem sets.

2013-11-11, Espen Robstad Jakobsen