Stochastic integration 2019-2020
(code 5374STIN8Y)

Contents

Stochastic calculus is an indispensable tool in modern financial mathematics. In this course we present this mathematical theory. We treat the following topics from martingale theory and stochastic calculus: martingales in discrete and continuous time, the Doob-Meyer decomposition, construction and properties of the stochastic integral, Itô's formula, (Brownian) martingale representation theorem, Girsanov's theorem, stochastic differential equations and we will briefly explain their relevance for mathematical finance.

Aims

At the end of the course, students
  • can explain the theory and construction of stochastic integrals,
  • are able to apply the Itô formula,
  • can explain different solution concepts of SDEs,
  • know how to apply measure changes for continuous semimartingales (Girsanov's theorem) and are able to calculate the new drift,
  • are able to compare SDEs and PDEs, and can calculate the probabilistic representation of solutions to PDEs,
  • are able to solve problems, where knowledge of the above topics is essential.

Prerequisites

Measure theory, stochastic processes at the level of the course Measure Theoretic Probability (2010 version)

Literature

Recommended background reading: I. Karatzas and S.E. Shreve, Brownian motions and stochastic calculus and D. Revuz and M. Yor, Continuous martingales and Brownian motion. These books form the main basis for the lecture notes that we use for the course.

Companion course

In the past students were recommended to take also the course on Stochastic Processes, up to 2018 by Floske Spieksma (UL). The current course of the Dutch Master Program in Mathematics is taught by Daniel Valesin (RuG) and Christian Hirsch (RuG).

Follow up courses

A course that heavily relies on stochastic calculus is Interest rate models (the webpage is a bit outdated, but still fine for a first impression). Another interesting course, more theoretical, is Advanced Topics in Stochastic Analysis.

Lecturers

Asma Khedher (first half) and Peter Spreij (second half), assistance by Sven Karbach.

Homework

Compulsory! Strict deadlines: hand in during the lecture after you have been given the assignment, although serious excuses will always be accepted. You are allowed to work in pairs (a pair means 2 persons, not 3 or more), in which case one set of solutions should be handed in.

Schedule

Spring semester: Thursdays, 09:00-10:45 in room G3.10 (exception: G5.29 on March 5), first lecture on Thursday 6 February 2020. Tutorials biweekly after the lectures. For up to date information on the lecture rooms, see datanose.nl. See also the map of Science Park and the travel directions. See below for the "corona" adjustments.

Examination

The final grade is a combination of the results of the take home assignments and the written or oral exam (first part, to be decided) and oral exam (second part). The homework results count for 40% of the final grade. The partial oral exams have equal weight. For the oral exams appointments with the lecturers will be scheduled.

What do you have to know? The theory, i.e. all important definitions and results (lemma's, theorems, etc.), but not those in sections that have not been treated. Optional: for each part you may prepare three theorems together with their proofs. You select your favourite ones! Criteria to consider: they should be interesting, non-trivial and explainable in a reasonably short time span. You will be asked to present one of them, after which you will be questioned on different topics.

There is a schedule, that will regularly be updated. Warning: sometimes there are problems with synchronizing files on surfdrive, so the file may not always be up to date, or a previous web address (url) has to be replaced with a newer one.

Registration

The UvA now wants all participants to be registered four weeks before the start of the course. If you missed this deadline you can use the late registration form. Note that a UvAnetID is required, so at least you have to be registered as a UvA student.

Important!

Due to the immensely fast spreading corona virus there will no regular lectures from week 8 on. Instead, there will be screen recordings of the lecture notes, together with detailed comments. The recordings will appear on Canvas and should be available every week at the originally scheduled class hours (or earlier). More information will follow. The study programme is as below, in particular valid for the second half.
Note that there is no lecture scheduled for April 2, the lecture of week 9 is scheduled for April 9.
The programme will be regularly updated!

1st half, old programme

1 Lecture: Sections 1 and 2.1 (very briefly).
Homework: Read the lecture notes, including the superficially treated Section 2.1 and make Exercises 1.4, 1.5, 1.8, 1.14.
2 Lecture: most of Section 2.2, Section 2.3: definitions, Lemma 2.15, proof of uniqueness of DM decomposition, Theorem 2.17 mentioned and Proposition 2.18.
Homework: Make yourself familiar with the contents of (Appendix) Sections B and D; just the big picture, no details. Make Exercises 2.1, 2.3, 2.5, 2.10.
3 Lecture: Remainder of Chapter 2 (proof of existence of DM decomposition, Proposition 2.18), perhaps introductory remarks on Chapter 3.
Homework: make Exercises 2.7, 2.8, 2.13, and 2.16 (skip the optional part).
4 Lecture: Chapters 3 and 4 (rather briefly).
Homework: Exercises 3.3 (a,b,c), 3.9, 4.3.
5 Lecture: Chapter 5 and perhaps a quick introduction to Chapter 6.
Homework: Exercises 4.10, 5.1, 5.2.
6 Lecture: Most of Sections 6.1, 6.2.
Homework: Read (optional, just needed in the proof of the Kunita-Watanabe inequality) Sections 6.3 and 6.9 of the MTP lecture notes; also look at the proof of this inequality. Make 6.1, 6.6, 6.8.
7 Lecture: Sections 6.2 up to Proposition 6.10.
Homework (updated on 5 April 2018): Make Exercises 6.5, 6.10, 6.13. (It is not necessary to make the distinction between bounded processes and the general case (as stated in the exercise) to give the proof. Just rely on the relevant results in Chapter 5.)

2nd half, Intended programme

8 Lecture: Sections 6.3, 7.1, 7.2.
Homework: Make Exercises 6.9, 6.14 (due April 9, you work in pairs). Optional, not to be handed in: (1) give the details of the proof of Corollary 7.9, (2) verify the expression for $X_t^2-X_0^2$ in the proof of Proposition 7.10 and finish that proof.
9 Lecture: Sections 7.3, 7.4, 8.1, 8.2 up to Lemma 8.4.
Homework: Read (just read, you can also do this a week later) the first example of a local martingale that is not a martingale and make Exercises 7.1 (for part (a) you are not obliged to follow the hint), 7.4, 7.7 (restrict yourself to $f$ twice continuously differentiable in both variables and depart from the formula in Remark 7.12, then there will not be much left to do) and 8.7. Look (just look) also at exercise 8.1 (it tells you why the ordinary Brownian filtration is not right-continuous). To be prepared for next week, read the lecture notes on the Radon Nikodym theorem (mainly the introduction and the statement of the theorem) if you are not familiar with it.
10 Lecture: Sections 8.2 (from Lemma 8.6), 9.1, 9.2 (most of it), 9.3 (some parts briefly) up to Corollary 9.13.
Homework: Read also the parts that I skipped in the recordings. Make Exercises 8.2, 8.3(a,b,c,f) with b,c only for $t < T$, 9.4 (only prove the statement "If $XZ$ is a local martingale under $\mathbb{P}$ then $X$ is a local martingale under $\mathbb{Q}$"), and 9.6.
11 Lecture: Most of Sections 9.3, 9.4, Section 10.1 at top speed up to Theorem 10.2 with proof.
Homework: Pay attention to parts in the lecture notes that I skipped and make Exercises 9.7 (you are not obliged to follow the hint), 9.9, 9.12, 10.3.
12 Lecture: Remainder of Section 10.1, most of Section 10.2.
Homework: Make Exercises 10.4, 10.7, 10.9(a) (you may assume that $g$ is differentiable, but note part (b)!), 10.18 (in part (a) you have to show that the iteration results in the formula for $X^n_t$); read the second example of a local martingale that is not a martingale.
13 Lecture: Sections 10.3, 11.1 (most of it).
Homework: Make Exercises 10.14, 11.1, 11.3, 11.4.



To the Korteweg-de Vries Instituut voor Wiskunde or to the homepage of the master's programme in Stochastics and Financial Mathematics.
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