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
- 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
- Øksendal: Stochastic Differential Equations, Springer Verlag (4-6th editions).
Notes by Krogstad
Evans lecture note
- L. C. Evans: An Introduction to Stochastic Differential Equations (only a few sections)
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 (a very readable lecture note)
- J. Jacod and P. Protter: Probability Essentials, 2004, Spinger.
- 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.