Stochastic integration 2006-2007
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 reading: I. Karatzas and S.E. Shreve,
Brownian motions and stochastic calculus and D. Revuz and M. Yor,
Continuous martingales and Brownian motion. Lecture notes (based on these books) can be downloaded.
Companion course
Students are recommended to take also the course on Stochastic Processes, see also 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
Grading by Shota Gugushvili
Schedule
Spring semester, Wednesdays 14.15-16.00 in room I.402 (first two weeks 14.15-17.00). No class on March 7 and on May 2. From March 28 on, classes will start at 15.00 (sharp) in P.016. Final class on May 16.
Programme
Please check the programme below. It is not definitive and adjustments will be made throughout the course
| 1 |
Lecture notes: section 1, section 2.1
Homework: 4 exercises from section 1.3
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| 2 |
Lecture notes: sections 2.1 (what remained), 2.2, 2.3 (uniqueness of DM
decomposition and part of the proof of existence, up to claim (2.11))
Homework: choose 3 exercises from 2.1, 2.3, 2.5, 2.8, 2.9
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| 3 |
Lecture notes: section 2.3 (UI part of the proof of theorem 2.14, theorem 2.15 (weaker statement), proposition 2.16), section 3.1, definition of quadratic variation
Homework: select four exercises
from 2.6, 2.7, 2.12, 2.14, 3.1 and 3.10
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| 4 |
Lecture notes: most of section 3.2; section 4.1 (definition of local martingales under the restriction that they start in zero) and section 4.2 up to proposition 4.6.
Homework: One exercise from 3.2, 3.3, and two from 3.4, 3.5, 3.6, 3.9
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| 5 |
Lecture notes: sections 4 and 5
Homework: Two exercises from 4.1 - 4.4 and two from 4.5 - 4.8, 5.1-5.2
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| 6 |
Lecture notes: section 6.1 (except proposition 6.7), lemma 6.8 (partly).
Homework: read sections 3 and 6 of "The Radon-Nikodym theorem", make three exercises from 6.1, 6.3, 6.4,
6.6, 6.9
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| 7 |
Lecture notes: sections 6.2 (remaining parts), 6.3, section 7.1 up to Proposition 7.7
Homework: three exercises from the set 6.7, 6.8, 6.10 - 6.14
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| 8 |
Lecture notes: remainder of section 7.1, sections 7.2, 7.3, 7.4
Homework: three exercises from section 7, read the first example of a local martingale that is not a martingale
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| 9 |
Lecture notes: section 8 until theorem 8.6 (with only a sketch of the proof)
Homework: four exercises from section 8
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| 10 |
Lecture notes: (most of) sections 9.1-9.3
Homework: make three exercises from 9.2-9.8
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| 11 |
Lecture notes: remainder of section 9.3, section 9.4 and definition 10.1 plus examples of ODEs.
Homework: none
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| 12 |
Lecture notes: (most of) section 10.1, parts of section 10.2
Homework: three exercises from 10.3, 10.4, 10.6, 10.8, 10.16
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| 13 |
Lecture notes: sections 10.3 and 11
Homework: one exercises from 10.2, 10.14, 10.15, two from 11.1, 11.2, 11.3, 11.4, 11.5 and read the second example of a local martingale that is not a martingale.
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To Korteweg-de
Vries Instituut voor Wiskunde or to the homepage
of the master's programme in Stochastics and Financial Mathematics.
Email: spreij@science.uva.nl
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