Stochastic integration 2012-2013
(code ST409018)

Contents

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, construction and properties of the stochastic integral, Itô's formula, Girsanov's theorem, stochastic differential equations and we will briefly explain their relevance for mathematical finance.

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 (based on these books) lecture notes.

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

A course that heavily relies on stochastic calculus is Interest rate models.

Lecturers

Peter Spreij and Neil Walton

Homework

Strict deadlines: the lecture after you have been given the assignment, although serious excuses will always be accepted. You are allowed to work in pairs (a pair means 2 persons, not 3 or more), in which case one set of solutions should be handed in.

Schedule

Spring semester: First lecture on Thursday 6 February 2013, then next Thursdays, 13:00-15:00 up to March 21, Room G2.02 (Science Park); from April 4 on, Thursdays 11:00-13:00 in A1.14. see the map of Science Park and the travel directions. On May 16: 10:15 - 12:00.

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. What do you have to know? The theory, i.e. all important definitions and results (lemma's, theorems, etc.). Optional: you may prepare three theorems together with their proofs. You select your favorite ones! Criteria to consider: they should be interesting, non-trivial and explainable in a reasonably short time span. You will be asked to present two of them. Unavailable periods are June 1-5, 10-15, 20-28, July 20 - August 10.

Registration

The UvA now wants all participants to be registered four weeks before the start of the course. If you missed this deadline you can use the late registration form. Note that a UvAnetID is required, so at least you have to be registered as a UvA student.


Programme

(last modified: )

1 Lecture: Sections 1 and 2.1.
Homework: Exercises 1.4, 1.5, 1.8, 1.14
2 Lecture: 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.
Homework: Make yourself familiar with the contents of (Appendix) Sections B (the inequality in Lemma B.1 should be reversed) and D; just the big picture, no details. Make Exercises 2.1, 2.3 (optional!), 2.5, 2.9.
3 Lecture: Remainder of Chapter 2 (proof of existence of DM decomposition, Proposition 2.18), introductory remarks on Chapter 3.
Homework: make Exercises 2.6, 2.7, 2.12, and (optional) 2.15
4 Lecture: Chapter 3 and a bit of Chapter 4
Homework: Exercises 3.3 (total variation is the same as first order variation), 3.6 (optional,this is a `stand alone' exercise), 3.9, 3.10
5 Lecture: Chapter 4 and Chapter 5
Homework: Exercises 4.1, 4.3, 5.1, 5.2
6 Lecture: Most of Sections 6.1, 6.2
Homework: 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 6.1, 6.6, 6.9.
7 Lecture: Sections 6.2 (remaining parts), 6.3, 7.1
Homework: Make exercises 6.8, 6.10, 6.11
8 Lecture: Sections 7.2, 7.3, 7.4
Homework: (plan) 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 f twice continuously differentiable in both variables and depart from the formula in Remark 7.12).
9 Lecture: Section 8
Homework: Read also the parts of section 8 that I 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, 8.6.
10 Lecture: Sections 9.1 - 9.3, quick mentioning of section 9.4
Homework: Make exercises 9.4, 9.6, 9.8, 9.9 (assume that M is continuous!!)
11 Lecture: Sections 9.4, 10.1 (up to Theorem 10.2)
Homework: Read the end of the proof of Theorem 10.2 and make exercises 9.10, 9.11, 10.3, 10.7. Corrected on May 10. Earlier there was a mistake in the selection of the exercises: 9.11 and 9.12 instead of 9.10, 9.11, but 9.12 doesn't exist. Keep an eye on the time; if it is too much work, drop one exercise and inform me on May 16.
12 Lecture: Sections 10.2, beginning of 10.3
Homework: Make exercises 10.4, 10.12, 10.16 and read the second example of a local martingale that is not a martingale.
13 Lecture: Sections 10.3, 11.1 (without the proof of Theorem 11.2)
Homework: 11.1, 11.3, 11.4


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