6 EC
Semester 1, period 1, 2
5334ATIS6Y
The two central themes of this course are backwards stochastic differential equations (BSDEs) and Malliavin calculus. BSDEs are stochastic differential equations for which not the initial, but the final value is given. Although this seems like a small difference, the implications are severe as the solution must satisfy adaptivity conditions. The martingale representation theorem (see `Stochastic integration') provides a solution to the simplest type of BSDE, and is a key tool for proving existence of a solution in general. The motivation for studying BSDEs comes from, e.g., stochastic control and hedging problems in finance. Another application lies in the link between BSDEs and non-linear PDEs, given by the extension of a Feynman-Kac formula.
Malliavin calculus is a tool to consider derivatives with respect to the Brownian motion--in a sense, it concerns an extension of the Ito calculus. It allows one to give an explicit expression of the solution to the martingale representation theorem. One can also determine the Malliavin derivative of the solution to a (B)SDE, which is useful for error analysis of stochastic simulations.
Topics that are covered in this course are:
The course is based on lecture notes.
The lecture notes are based on the following monographs (which you do not need to buy):
The second part of the course is based on, among others, the following article by Bouchard and Touzi
https://www.sciencedirect.com/science/article/pii/S0304414904000031
Activity |
Hours |
|
Hoorcollege |
28 |
|
Oral exam |
1 |
|
Self study |
139 |
|
Total |
168 |
(6 EC x 28 uur) |
The programme does not have requirements concerning attendance (OER-B).
Item and weight | Details |
Final grade |
There will be an oral exam for each part of the course. Sonja Cox will give an oral exam for the first part of the course. Asma Khedher will give an oral exam for the second part of the course. To have the exam, you make an appointment with the lecturers.
What do you have to know? The theory and homework, i.e. all important definitions and results (lemma's, theorems, etc.). The details about which theory will be discussed two weeks prior to the exam.
The final grade is a combination of the results of the take home assignments (which counts 40 %) and the first and second oral exam (together 60%). To pass the course the average grade of both oral exams should be higher than 5.0.
The same applies in case of a resit. That is the final grade will be a combination of the results of the take home assignments (which counts 40 %) and the resit grade (60%).
During the course, the students will have to hand in a total of six sets of homework. The assignments will contain theoretical and numerical exercises. The average homework grade will count for 40% in the final grade
The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl
Weeknummer | Onderwerpen | Studiestof |
1 | Stochastic integration recapitulation | Chapter 1 (first take home assignment) |
2 | Introduction and well posedness of BSDEs | Chapter 2, Sections 2.1-2.2 |
3 | Relation between BSDEs and non-linear PDEs | Chapter 2, Sections 2.3-2.5 (second take home assignment and deadline of the first one) |
4 | Wiener-chaos decomposition | Chapter 3, Section 3.1 |
5 | Malliavin derivative | Chapter 3, Section 3.2 (third take home assignment and deadline of the second one) |
6 | Divergence operator | Chapter 3, Section 3.3 |
7 | Malliavin calculus for SDEs | Chapter 3, Section 3.4 (deadline for the third take home assignment) |
8 | No lecture | |
9 | Euler scheme for SDEs (strong convergence study) | |
10 | Differentiability of SDEs | |
11 | Malliavin differentiability of BSDEs | |
12 | Representation theorems for solutions to BSDEs | |
13 | Numerical scheme for BSDEs | |
14 | Strong convergence study of the numerical scheme | |
15 | ||
16 |
The schedule for this course is published on DataNose.