Stochastic Processes and Itô Calculus
2009-2010
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Aim

To make students familiar with the mathematical fundamentals of stochastic processes and stochastic integrals.

Contents

Construction of Brownian motion from symmetric walk by the central limit theorem, martingale property and quadratic variation; construction of the Itô integral, fundamental properties (Itô isometry); Itô rule (in one and more dimensions), stochastic product rule, Lévy's characterization of Brownian motion; absolutely continuous change of measure, Girsanov's theorem, martingale representation theorem; stochastic differential equations, diffusions and partial differential equations, Feynman-Kaç formula, stopping times; Poisson and compound Poisson process, decomposition of compound Poisson processes, quadratic variation, Itô formula for processes with jumps, change of measure for jump-diffusions, extension of Girsanov's theorem.

Literature

The course is mainly based on Steve Shreve, Stochastic Calculus for Finance II, Continuous-Time Models, Springer (2004).

Prerequisites

Basic concepts of probability and measure theory, as given in the first two chapters of the book by Shreve,

People

Lectures by Peter Spreij.

Schedule

Wednesday afternoons. First meeting on January 13, 2010 (question session on Shreve, Chapters 1 and 2) at 16:00. Then from January 27 on according to the schedule below.

Programme
(last modified: )

1
 Random walk, Central limit theorem, Brownian motion, martingale; from Shreve, most (but not all!) of Sections 3.1-3.3.
2
 Quadratic variation, stochastic integral of simple functions; from Shreve, Sections 3.4, 4.1 - 4.2
3
 Construction of stochastic integral w.r.t. Brownian motion, properties (isometry), general integrands; from Shreve, Section 4.3
4
 Itô-formula, bivariate extension, product rule; from Shreve Section 4.4
5
 Lévy's characterization, Change of measure; from Shreve, parts of Section 4.6, Section 5.1
6
 Girsanov's theorem, Martingale representation theorem, relevance for risk neutral pricing; from Shreve, Sections 5.2.1-5.2.4, 5.3
7
 Stochastic differential equations; from Shreve, Sections 6.2, 6.3
8
 Connections to Partial differential equations, Feynman-Kac formula, stopping times; from Shreve, Sections 6.3, 6.4, 8.2 and Definition 8.3.1
9
 (Compound) Poisson process, integrals w.r.t. jump processes; from Shreve, Section 11.1 - 11.3
10
 Quadratic variation, Itô-formula for processes with jumps, change of measure for (compound) Poisson process; from Shreve, Sections 11.4, 11.5
11
 Change of measure for (compound) Poisson process; from Shreve, Section 11.6



Links

Korteweg-de Vries Institute for Mathematics
Master Stochastics and Financial Mathematics