Advanced Topics in Stochastic Analysis

Course manual 2019/2020

Course content

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:

• well-posedness of BSDEs with Lipschitz coefficients
• the relation between BSDEs and (deterministic) PDEs- examples of BSDEs arising from applications
• the Wiener Chaos decomposition (useful for Malliavin calculus)
• Malliavin calculus (derivative and Skorohod integral): this involves the question of taking derivatives with respect to a Brownian motion and allows us to analyse the regularity of a (B)SDE
• approximation of BSDEs by numerical methods
• well-posedness of BSDEs with quadratic coefficients.

Study materials

Literature

• The course is based on lecture notes. 

  The lecture notes are based on the following monographs (which you do not need to buy):

  • David Nualart, 'The Malliavin Calculus and Related Topics', Springer, ISBN 978-3-540-28329-4 (not necessary to buy this).
  • Huyên Pham, 'Continuous-time stochastic control and optimization with financial applications', Springer, ISBN 978-3-540-89500-8 (not necessary to buy this)

  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

   

   

Other

• https://pdfs.semanticscholar.org/6a8e/8ddb18c4cd34f2dea5e3fd0c444665fddf65.pdf

  https://projecteuclid.org/euclid.aop/1019160253

Objectives

• understand the theory of BSDEs
• apply the theory of BSDEs in hedging problems and control theory in finance
• explain the relation between BSEDs and non-linear PDEs
• understand the theory of Malliavin calculus
• apply the theory of Malliavin calculus on forward stochastic differential equations and BSDEs
• introduce discretization techniques for BSDEs and use the Malliavin calculus to compute convergence rates for the approximation
• solve BSDEs numerically and use this to compute hedging positions in financial contracts.

Teaching methods

• Lecture

Learning activities

Attendance

The programme does not have requirements concerning attendance (OER-B).

Assessment

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%).

Assignments

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

Fraud and plagiarism

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

Course structure

Timetable

The schedule for this course is published on DataNose.

Contact information

Coordinator

• dr. A. Khedher