8 EC
Semester 2, period 4, 5
5374STIN8Y
| Owner | Master Mathematics |
| Coordinator | Peter Spreij |
| Part of | Master Mathematics, year 1Master Stochastics and Financial Mathematics, year 1 |
Stochastic calculus is an indispensable tool in modern financial mathematics. In this course we present this mathematical theory. We treat the following topics from martingale theory and stochastic calculus: martingales in discrete and continuous time, the Doob-Meyer decomposition, construction and properties of the stochastic integral, Itô's formula, (Brownian) martingale representation theorem, Girsanov's theorem, stochastic differential equations and we will briefly explain their relevance for mathematical finance.
Recommended background reading: I. Karatzas and S.E. Shreve, Brownian motions and stochastic calculusand D. Revuz and M. Yor, Continuous martingales and Brownian motion.
https://staff.fnwi.uva.nl/p.j.c.spreij/onderwijs/master/si.pdf
At end of the course, students
|
Activity |
Number of hours |
|
Lectures |
26 |
|
Self-study |
224 |
The programme does not have requirements concerning attendance (OER-B).
| Item and weight | Details |
|
Final grade | |
|
0.5 (50%) Homework | |
|
0.5 (50%) Oral and/or written exam | Allows retake |
The manner of inspection will be communicated via the lecturer's website.
Graded homework will be weekly returned.
The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl
| Weeknummer | Lectures | Exercises |
| 1 | Sections 1 and 2.1 (very briefly). | Read the lecture notes, including the superficially treated Section 2.1 and make Exercises 1.4, 1.5, 1.8, 1.14. |
| 2 | Most of Section 2.2, Section 2.3: definitions, Lemma 2.15, proof of uniqueness of DM decomposition, Theorem 2.17 mentioned and Proposition 2.18. | Make yourself familiar with the contents of (Appendix) Sections B and D; just the big picture, no details. Make Exercises 2.1, 2.3 (optional), 2.5, 2.10. |
| 3 | Remainder of Chapter 2 (proof of existence of DM decomposition, Proposition 2.18), perhaps introductory remarks on Chapter 3. | Make Exercises 2.7, 2.8, 2.13, and (optional) 2.16. |
| 4 | Chapters 3 and 4 (rather briefly). | Make Exercises 3.3 (a,b,c), 3.9, 4.3. |
| 5 | Chapter 5 and perhaps a quick introduction to Chapter 6. | Make Exercises 4.10, 5.1, 5.2. |
| 6 | Most of Sections 6.1, 6.2. | Read (optional, just needed in the proof of the Kunita-Watanabe inequality) Sections 6.3 and 6.9 of the MTP lecture notes; also look at the proof of this inequality. Make Exercises 6.1, 6.6, 6.8. |
| 7 | Sections 6.2 (from Theorem 6.11), 6.3, 7.1. | Make Exercises 6.9, 6.10, 6.13. |
| 8 | Sections 7.2, 7.3, 7.4. | Read the first example of a local martingale that is not a martingale and make Exercises 7.1, 7.4, 7.5, 7.6 (restrict yourself to ftwice continuously differentiable in both variables and depart from the formula in Remark 7.12). |
| 9 | Section 8. | Read also the parts of section 8 that have been skipped, look at exercise 8.1 (it tells you why the ordinary Brownian filtration is not right-continuous) and make exercises 8.2, 8.3 (only for t < T), 8.6 (optional). To be prepared for next week, read the lecture notes on the Radon Nikodym theorem (mainly the introduction and the statement of the theorem). |
| 10 | Sections 9.1 - 9.3. | Make Exercises 9.4, 9.6, 9.8, 9.9. |
| 11 | Section 9.4, Sections 10 and 10.1 up to Theorem 10.2. | Make Exercises 9.5, 9.12, 10.3, 10.9. |
| 12 | Most of Sections 10.1 (remainder) and 10.2 | Make Exercises 10.4, 10.7, 10.18; read Proposition 10.3 and the second example of a local martingale that is not a martingale. |
| 13 | Sections 10.3, 11.1 (without the proof of Theorem 11.2) . | Make Exercises 11.1, 11.3, 11.4. |
| 14 | ||
| 15 | ||
| 16 |
The schedule for this course is published on DataNose.