Stochastic integration 2005-2006

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.

Until 31 July 2006 Springer Verlag holds its Yellow Sale with many mathematics books at low prices. The above two reference books are both available, see Brownian motions and stochastic calculus (€ 24,56) and Continuous martingales and Brownian motion (€ 53,45).

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 (odd weeks) and Peter Spreij (even weeks)

Schedule

Spring semester, Thursdays 10.15-12.00 in P.015B, except the first class on February 9, 2006, from 09.15 to 12.00. No classes on March 9 and on April 27. Special lecture by Professor A.N. Shiryaev on May 4, at 10.30 in room P.227. On May 11 and 18 there will be three hours of class, from 10.15 until 13.00.


Programme

1 Lecture notes: section 1, section 2.1
Homework: 4 exercises from section 1.3
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
3 Lecture notes: section 2.3 (UI part of the proof of theorem 2.14, theorem 2.15 (only statement), proposition 2.16), section 3.2: definition of quadratic variation, statement of proposition 3.8
Homework: select four exercises from 2.6, 2.7, 2.12, 2.14, 3.9 and 3.10 (new in the lecture notes!)
4 Lecture notes: most of sections 3.1 and 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 3.1, 3.2, 3.3, and two from 3.4, 3.5, 3.6, 3.9
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
6 Lecture notes: section 6.1 (except proposition 6.7), lemma 6.9.
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
7 Lecture notes: sections 6.2 (remaining parts), 6.3, section 7.1 (items 7.1-3, 7.5-7)
Homework: three exercises from the set 6.7, 6.8, 6.10 - 6.14
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
9 Lecture notes: section 8 until theorem 8.6 (with only a sketch of the proof)
Homework: four exercises from section 8
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
11 Special lecture by Professor A.N. Shiryaev: Stochastic integral representations of functionals of Brownian motion (with applications to proofs of Poincare-Chernof and Log-Sobolev inequalities and to financial mathematics), room P.227.
11 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
12 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.
13 No class



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