Stochastic integration 2010-2011
(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 composed during this spring).

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.

Lecturers

Michel Mandjes and Peter Spreij, assisted by Florian Simatos and Sjors van der Stelt

Homework

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

Schedule

Spring semester: Tuesdays, 10:00-13:00, Room G3.13 (Science Park); see the map of Science Park and the travel directions. First lecture on 1 February 2011, no class on March 29. Starting April 19: lectures in D1.115

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. If it happens that you'd like to postpone the appointment, just inform us 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. Unavailable periods: May 19-25, June 2-8, 15-20, 24-29, July 25 - August 20.


Programme

(last modified: )

1 Lecture: Section 1
Homework: Exercises 1.1, 1.4, 1.5, 1.14
2 Lecture: Most of Sections 2.1, 2.2; Of Section 2.3 Lemma 2.15, proof of uniqueness of DM decomposition
Homework: Make yourself familiar with the contents of (Appendix) Sections A and C (just the big picture, no details). Make Exercises 2.1, 2.5, 2.9.
3 Lecture: Remainder of Chapter 2
Homework: make Exercises 2.6, 2.7, 2.12, 2.15
4 Lecture: Chapters 3 and 4
Homework: Exercises 3.3 (X,Y are square integrable martingales, total variation is the same as first order variation; in (c) you should read M=X, Y=N (typo)), 3.6 (this is a `stand alone' exercise), 3.9, 3.10 (last question only)
5 Lecture: Section 5 and the beginning of section 6
Homework: Exercises 4.2, 4.3, 5.1, 5.2
6 Lecture: Sections 6.1, 6.2
Homework: Read (optional, just needed in the proof of the K-W inequality) sections 6.3 and 6.6 of the MTP lecture notes; make 6.1, 6.6, 6.9.
7 Lecture: Sections 6.2 (remaining parts), 6.3, 7.1, 7.2, 7.3
Homework: Make exercises 6.10, 6.11, 7.1
8 Lecture: Sections 7.4, 8.1, 8.2 until Lemma 8.4
Homework: (plan) 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), 7.4, 7.5, 8.1 (there is not so much to do here).
9 Lecture: Remainder of Section 8, Sections 9.1, 9.2 (except Propositions 9.7, 9.8), Theorem 9.3.
Homework: Make exercises 8.2, 8.3, 9.4
10 Lecture: Sections 9.3 (from Proposition 9.10), 9.4
Homework: Make exercises 9.6, 9.9, 9.11
11 Lecture: Section 10.1
Homework: Make exercises 10.8, 10.9, 10.16.
12 Lecture: Sections 10.2, 10.3
Homework: Make exercises 10.2, 10.3, 10.4, 10.17 and read the second example of a local martingale that is not a martingale.
13 Lecture: Section 11
Homework: 11.1, 11.2, 11.3, 11.4


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