Stochastic Processes 2009-2010

Stochastic Processes and Itô Calculus
2009-2010
Duisenberg School of Finance

Aim

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

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.

Prerequisites

Basic concepts of probability and measure theory, as given in the course on Measure and Probability by Laurens de Haan.

Literature

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

Examination

Oral exams, made by individual appointments. What you have to know: You will be partly asked to explain me the theory (I had 21 hours, you have less than 1 hour, so I will not ask you all kinds of details). In particular you have to know the main definitions and theorems or other main results. As an example, you have to know how to construct stochastic integrals w.r.t. Brownian motion, or other martingales (which also implies that you what a martingale is). You may also be asked what partial differential equations have to do with stochastic differential equations. Why is a solution to the latter equation a Markov process? What is the content of Girsanov's theorem in the context of Brownian motion? What is the related result for jump processes (with finitely many different jump types)? Why are W and N independent? What is Ito's formula? What is the core idea behind the proof?
There are plenty more questions of this type. I might also ask you something about the homework assignments. It all depends on my mood.

Planning: You are welcome any time, taking the exceptions below into account. You can make an appointment for a date that suits you and on which you think that you will be well prepared. Made appointments can always be shifted to a later date in the future, should you wish so (it doesn't make sense to have the exam, while you think that you need more time for preparation). I will not be available in the periods 22 November - 2 December and 13 - 20 December. Appointments between Christmas and New Year are in principle not excluded. In 2010 the dates 18-20 January are blocked.

People

Lectures by Peter Spreij.

Schedule

Fall semester, 2nd half, Fridays 09:00 - 12:00. The course will start on October 30. Extra class on November 5, 19:00 - 22:00, no class on November 27.

Location

Duisenberg school of finance, Roetersstraat 33, Amsterdam (travel directions).

Programme
(last modified: )

1
Class: Random walk, Central limit theorem, Brownian motion, martingale, quadratic variation; from Shreve, most (but not all!) of sections 3.1-3.4.
Homework: Exercises 3.1, 3.2, 3.4 (read also the additional text in Section 3.4.2), 3.5.

2
Class: Integral of simple functions, construction of stochastic integral w.r.t. Brownian motion, properties (isometry), general integrands; from Shreve, most of sections 4.1 - 4.3
Homework: Exercises 4.1, 4.2

3
Class: Itô-formula, bivariate extension, product rule, Lévy's characterization; from Shreve, theory of Section 4.4, parts of Section 4.6
Homework: Understand the examples in Section 4.4, pay attention to the parts of Section 4.6 that I skipped and make exercises 4.7, 4.14

4
Class: Change of measure, Girsanov's theorem, Martingale representation theorem, relevance for risk neutral pricing; from Shreve, Sections 5.2.1-5.2.4, 5.3
Homework: Read Section 5.4.1 and (if you like) more on derivative pricing, have a look at the Summary (Section 5.7) and the Notes (Section 5.8) as well. Make Exercises 5.2, 5.5, 5.8 and Exercise 8.3 from the lecture notes of another course in which you have to find the process Γ of Theorem 5.3.1 for the martingales in the exercise. Don't get confused by the notation X • W. It means stochastic integral, not a product (this notation is not uncommon, by the way).

5
Class: Stochastic differential equations, connections to Partial differential equations, Feynman-Kac formula, stopping times; from Shreve, Sections 6.2 - 6.4, 8.2 and Definition 8.3.1
Homework: Read Sections 6.7, 6.8 (as far as applicable), 6.5 ; make Exercises 6.1, 6.3, 6.8, 6.9; extra question: suppose that X is an adapted continuous process. Define τ as the first moment t that X(t) is at least equal to some level m, τ = min{t≥ 0: X(t)≥ m}. Give a simple argument that shows that τ is a stopping time. (Distinguish between X(0)≥ m and X(0) < m.)

6
Class: (Compound) Poisson process, integrals w.r.t. jump processes; from Shreve, Chapter 11 up to Theorem 11.4.5
Homework: Exercises 11.1(ii), 11.2, 11.3

7
Class: quadratic variation, Ito-formula for processes with jumps, change of measure for (compound) Poisson process; from Shreve, Chapter 11, Theorem 11.4.5 - Theorem 11.6.5 with an informal discussion of Theorems 11.6.7 and 11.6.9
Homework: Exercises 11.4, 11.5, 11.6 (optional, it is more of the same), 11.7 and Watanabe's characterization of a Poisson process.

Links

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

1	Class: Random walk, Central limit theorem, Brownian motion, martingale, quadratic variation; from Shreve, most (but not all!) of sections 3.1-3.4. Homework: Exercises 3.1, 3.2, 3.4 (read also the additional text in Section 3.4.2), 3.5.
2	Class: Integral of simple functions, construction of stochastic integral w.r.t. Brownian motion, properties (isometry), general integrands; from Shreve, most of sections 4.1 - 4.3 Homework: Exercises 4.1, 4.2
3	Class: Itô-formula, bivariate extension, product rule, Lévy's characterization; from Shreve, theory of Section 4.4, parts of Section 4.6 Homework: Understand the examples in Section 4.4, pay attention to the parts of Section 4.6 that I skipped and make exercises 4.7, 4.14
4	Class: Change of measure, Girsanov's theorem, Martingale representation theorem, relevance for risk neutral pricing; from Shreve, Sections 5.2.1-5.2.4, 5.3 Homework: Read Section 5.4.1 and (if you like) more on derivative pricing, have a look at the Summary (Section 5.7) and the Notes (Section 5.8) as well. Make Exercises 5.2, 5.5, 5.8 and Exercise 8.3 from the lecture notes of another course in which you have to find the process Γ of Theorem 5.3.1 for the martingales in the exercise. Don't get confused by the notation X • W. It means stochastic integral, not a product (this notation is not uncommon, by the way).
5	Class: Stochastic differential equations, connections to Partial differential equations, Feynman-Kac formula, stopping times; from Shreve, Sections 6.2 - 6.4, 8.2 and Definition 8.3.1 Homework: Read Sections 6.7, 6.8 (as far as applicable), 6.5 ; make Exercises 6.1, 6.3, 6.8, 6.9; extra question: suppose that X is an adapted continuous process. Define τ as the first moment t that X(t) is at least equal to some level m, τ = min{t≥ 0: X(t)≥ m}. Give a simple argument that shows that τ is a stopping time. (Distinguish between X(0)≥ m and X(0) < m.)
6	Class: (Compound) Poisson process, integrals w.r.t. jump processes; from Shreve, Chapter 11 up to Theorem 11.4.5 Homework: Exercises 11.1(ii), 11.2, 11.3
7	Class: quadratic variation, Ito-formula for processes with jumps, change of measure for (compound) Poisson process; from Shreve, Chapter 11, Theorem 11.4.5 - Theorem 11.6.5 with an informal discussion of Theorems 11.6.7 and 11.6.9 Homework: Exercises 11.4, 11.5, 11.6 (optional, it is more of the same), 11.7 and Watanabe's characterization of a Poisson process.

Aim

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