Stochastic differential equations 2002-2003

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.

Literature

Brownian motions and stochastic calculus, I. Karatzas and S.E. Shreve Continuous martingales and Brownian motion, D. Revuz and M. Yor. Lecture notes (based on these books) will be written during the course and will be made available later on.

Lecturer

P.J.C. Spreij

Schedule

Spring semester, Thursdays 13-16, room P.015B, first class on February 6.
There will be no classes on May 1, 22, 29.

Participants




Programme

Week 1 Lecture notes: sections 1.1 - 2.1
Homework: 4 exercises from section 1.
Week 2 Lecture notes: section 2.2 up to theorem 2.15.
Homework: read the proof of theorem 2.15, exercises 2.3, 2.4, 2.6, 2.8
Week 3 Lecture notes: sections 3.1, 3.2.
Homework: exercises 3.2, 3.4, 3.5, 3.6
Week 4 Lecture notes: sections 4.1, 5.1, 5.3
Homework: exercises 4.2, 4.3, 4.4, 5.1
Week 5 Lecture notes: sections 6.1, 6.2, 6.3
Homework: exercises 6.1, 6.4, 6.6, 6.8
Week 6 Lecture notes: sections 7.1, 7.2, 7.3
Homework: exercises 6.9, 6.10, 7.1, 7.3
Week 7 Lecture notes: sections 7.4, 8.1, 8.2
Homework: exercises 7.2, 8.1, 8.2, 8.3
Week 8 Lecture notes: sections 9.1, 9.2, 9.3 up to corollary 9.12
Homework: exercises 9.1, 9.3, 9.4, 9.7
Week 9 Lecture notes: section 10.1 up to theorem 10.2
Homework: exercises 10.3, 10.4, 10.6, 10.7
Week 10 Lecture notes: proposition 10.3 and `weak solutions'
Homework: exercise 10.5
Week 11 Lecture notes: section 11
Homework: exercises 10.2, 10.8, 10.9, 10.10
Week 12 Lecture notes: section 12
Homework: exercises 12.1, 12.4, 12.5
Week 13


To Korteweg-de Vries Instituut voor Wiskunde or to the homepage of the master's programme.

Email: spreij@science.uva.nl