Stochastic integration 2009-2010
(code ST409018)

Contents

Stochastic calculus is an indispensable tool in modern financial mathematics. In this course we present this mathematical theory and apply it to the problem of pricing and hedging of financial derivatives. We treat the following topics from martingale theory and stochastic calculus: martingales in discrete and continuous time, construction and properties of the stochastic integral, Ito's formula, Girsanov's theorem, stochastic differential equations. As an application, we explain how stochastic differential equations are typically used to model financial markets and we discuss the problem of the pricing of derivatives such as stock options.

Prerequisites

Measure theory, stochastic processes at the level of the course Measure Theoretic Probability

Literature

Recommended background reading: I. Karatzas and S.E. Shreve, Brownian motions and stochastic calculus and D. Revuz and M. Yor, Continuous martingales and Brownian motion. The contents of the course are described in the lecture notes (which are based on these books). Updated version of the lecture notes by February 6. This mainly concerns a small reorganization, the BDG inequallty has been moved from Section 11 to Section 7.

Companion course

Students are recommended to take also the course on Stochastic Processes, see the Spring Courses of the Dutch Master Program in Mathematics.

Follow up courses

Courses that heavily rely on stochastic calculus are Financial Stochastics and Control of Stochastic Systems in Continuous Time.

Lecturer

P.J.C. Spreij

Homework

Strict deadlines: the lecture after you have been given the assignment, although serious excuses will always be accepted.

Schedule

Spring semester: usually Thursdays, 15:15-17:00 (note the changed hours), Room P.015B. First lecture on 4 February 2010, 15:00-18:00. From April 1 on, there will be three hours of class every week, 15:15 - 18:00; last class on April 22; exception: Friday 16 April 14:00 - 16:45 in room P.018, instead of Thursday 15. The schedule below will be adjusted during the remaining period.

Examination

The final grade is a combination of the results of the take home assignments and the oral exam. To take the oral exam, you make an appointment for a date that suits your own agenda (and mine as well). If it happens that you'd like to postpone the appointment, just inform me that you want so. This is never a problem! The only important matter is that you take the exam, when you feel ready for it. To plan your own agenda, please note that I am not always in Amsterdam. For the time being I am not available in the periods April 28 - May 7, May 20 - June 2, June 16 - June 19, June 28 - July 3. As soon as more is known, it will appear here.


Programme

(last modified: )

1 Lecture notes: Sections 1, 2.1
Homework: Exercises 1.1, 1.4, 1.5, 1.14
2 Lecture notes: Sections 2.2, 2.3 (Lemma 2.14, uniqueness of DM decomposition, Proposition 2.17 briefly)
Homework: make Exercises 2.1, 2.3, 2.5, 2.9, read the proof of Proposition 2.17
3 Lecture notes: Section 2.3 (existence part of the proof of Theorem 2.14, Theorem 2.15 mentioned)
Homework: make Exercises 2.6, 2.7, 2.12, 2.15; have a look at Appendix C
4 Lecture notes: most of Sections 3.2 and 4, but not in all details
Homework: Exercises 3.3 (total variation is the same as first order variation), 3.6 (this is a `stand alone' exercise), 3.9, 3.10 (last question only)
5 Lecture notes: Section 5 and the beginning of section 6
Homework: Exercises 4.2, 4.3, 5.1, 5.2
6 Lecture notes: Sections 6.1, 6.2 up to Theorem 6.11.
Homework: Read (optional, just needed in the proof of the K-W inequality) sections 3 and 6 of "The Radon-Nikodym theorem" (part of MTP lecture notes); make 6.1, 6.6, 6.9.
7 Lecture notes: Sections 6.2 (remaining parts), 6.3, 7.1
Homework: Make exercises 6.10, 6.11, 7.1 (note added on March 29: drop the last question!)
8 Lecture notes: Sections 7.2, 7.3, 7.4, 8.1 and a little bit of 8.2
Homework: Read the first example of a local martingale that is not a martingale and make Exercises 7.1 (only the last question, you may assume the other results to be known), 7.2, 7.3, 7.5, 8.1. (NB: This looks a lot, but I think that most things are completely straightforward. But if it is really too much work, you may drop one exercise.)
9 Lecture notes: Section 8.2, main results of Sections 9.1, 9.2, 9.3 and a little bit of Section 9.4 (survey)
Homework: Make exercises 8.3, 8.7, 9.4, 9.8
10 Lecture notes: Most of Sections 10.1 and 10.2
Homework: Make exercises 10.3, 10.4, 10.9, 10.16.
11 Lecture notes: sections 10.3 and 11
Homework: Make exercises 11.1, 11.2, 11.3, 11.4 and read the second example of a local martingale that is not a martingale.


To the Korteweg-de Vries Instituut voor Wiskunde or to the homepage of the master's programme in Stochastics and Financial Mathematics.