Stochastic Processes 2011-2012

Stochastic Processes and Itô Calculus
2011-2012
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

The plan is to have weekly a set of homework assignments and a final written exam at the end. Starting with the assignments of week 2, you are required to work in pairs (and a pair means two people only!). The date of the written exam will be made known later.

People

Lectures by Peter Spreij.

Schedule

Fall semester, 2nd half; Tuesdays 17:00-20:00, except starting lecture on November 8 (19:00-22:00), and (last lecture) on Friday December 9, 13:00-16:00.

Location

Duisenberg school of finance, Gustav Mahlerlaan 117, 1082 MS 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: make Exercises 3.1, 3.2, 3.4. Read Section 3.2.7, a motivation for the use of Brownian motion in finance (but you have to know the Binomial model for stock prices...).

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, 4.4

3
Class: Itô-formula, 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 in Section 5.4 to get an idea why the change of measure is useful and necessary in finance, have a look at the Summary (Section 5.7) and the Notes (Section 5.8) as well. Make Exercises 5.2, 5.5 and Exercise 8.3(a,b,c) 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 (do this for t < T only). Don't get confused by the notation X • W. It means stochastic integral, not a product (this notation is not uncommon, by the way, see also Williams for a discrete time version).

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 6.5, 6.7, 6.8 (as far as applicable); make Exercises 6.1, 6.3, 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 (no complicated computations) 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 (first (ii), than (i)), 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 - Corollary 11.5.3, Lemma 11.6.1 - Theorem 11.6.5 with a very informal discussion of Theorems 11.6.9, 11.6.10.
Homework: Exercises 11.4, 11.5, 11.7, Watanabe's characterization of a Poisson process and only read 11.6 in order to know the result.

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: make Exercises 3.1, 3.2, 3.4. Read Section 3.2.7, a motivation for the use of Brownian motion in finance (but you have to know the Binomial model for stock prices...).
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, 4.4
3	Class: Itô-formula, 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 in Section 5.4 to get an idea why the change of measure is useful and necessary in finance, have a look at the Summary (Section 5.7) and the Notes (Section 5.8) as well. Make Exercises 5.2, 5.5 and Exercise 8.3(a,b,c) 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 (do this for t < T only). Don't get confused by the notation X • W. It means stochastic integral, not a product (this notation is not uncommon, by the way, see also Williams for a discrete time version).
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 6.5, 6.7, 6.8 (as far as applicable); make Exercises 6.1, 6.3, 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 (no complicated computations) 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 (first (ii), than (i)), 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 - Corollary 11.5.3, Lemma 11.6.1 - Theorem 11.6.5 with a very informal discussion of Theorems 11.6.9, 11.6.10. Homework: Exercises 11.4, 11.5, 11.7, Watanabe's characterization of a Poisson process and only read 11.6 in order to know the result.

Aim

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