Stochastic integration 2004-2005
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) are available
both in pdf and in ps. Readers are kindly requested to report
errors of any kind.
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
Grading by Shota Gugushvili (odd weeks) and Said el Marzguioui (even weeks)
Schedule
Spring semester, Wednesdays 09.15-12.00 in P.015A, first class on February 9, 2005.
Three hours of class on February 16. There will be no classes on March 2, March 30 and not on May 11.
Programme
| Week 1 |
Lecture notes: section 1
Homework: 4 exercises from section 1.3
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| Week 2 |
Lecture notes: sections 2.1, 2.2, 2.3 (uniqueness of DM
decomposition and part of the proof of existence, up to page 11, line 8)
Homework: choose 4 exercises from 2.2, 2.3, 2.4, 2.7, 2.8, 2.13
|
| Week 3 |
Lecture notes: section 2.3 (UI part of the proof of theorem 2.14, proposition 2.16, statement of theorem 2.15), sections 3.1 and 3.2 (proof of theorem 3.8 partly)
Homework: select four exercises
from 2.5, 2.6, 2.9, 2.10, 2.11, 2.12
|
| Week 4 |
Lecture notes: examples of martingales and their quadratic variation process, proof of theorem 2.15, some remarks on lemma 3.7 and the proof of proposition 3.8
Homework: Two exercises from 3.1, 3.2, 3.3, and two from 3.4, 3.5, 3.9
|
| Week 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
|
| Week 6 |
Lecture notes: sections 6.1
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
|
| Week 7 |
Lecture notes: sections 6.2,6.3
Homework: 4 exercises from the set 6.7, 6.8, 6.10 - 6.14
|
| Week 8 |
Lecture notes: sections 7.1, 7.2, 7.3 (not all proofs)
Homework: three exercises from section 7, read the first example of a local martingale that is not a martingale
|
| Week 9 |
Lecture notes: sections 7.4, 8 until theorem 8.6 (with only a sketch of the proof)
Homework: four exercises from section 8
|
| Week 10 |
Lecture notes: (most of) sections 9.1-9.3
Homework: read section 9.4 (superficially, but at least the Novikov condition!) and make three exercises from 9.2-9.8
|
| Week 11 |
Lecture notes: (most of) section 10.1
Homework: four exercises from 10.3, 10.4, 10.5 (BTW, existence of a solution follows from proposition 10.3), 10.6, 10.7, 10.8
|
| Week 12 |
Lecture notes: sections 10.2 and 10.3
Homework: three exercises from 10.2, 10.9, 10.10 (10.9 is probably useful here), 10.11, 10.13, 10.14 and read the second example of a local martingale that is not a martingale.
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| Week 13 |
Lecture notes: section 11
Homework: four exercises from section 11.2 or three from section 11.1 and one (not made before) from section 10.4
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| Week 14 |
No class
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To Korteweg-de
Vries Instituut voor Wiskunde or to the homepage
of the master's programme.
Email: spreij@science.uva.nl
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