Advanced Topics in Stochastic Analysis

Course manual 2024/2025

Course content

The aim of this course is to give an introduction to some currently active areas of research in stochastic analysis, namely backward stochastic differential equations (BSDEs), numerical approximation of (B)SDEs, and Malliavin calculus. As we will see in the course, these topics are closely related, and have several applications in financial mathematics.

Study materials

Literature

• The course is based on lecture notes that are made available over canvas. These 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)

Objectives

• The student can reproduce the polynomial chaos decomposition both in the abstract setting and in the setting of an L^2(0,T)-cylindrical Gaussian process.
• The student can provide the definition of a Malliavin derivative, knows its properties (chain rule etc).
• The student can provide the definition of the divergence operator and explain its relation to the Ito integral.
• The student can explain the relation between the Malliavin derivative, the divergence operator, and the polynomial chaos decomposition.
• The student can calculate the Malliavin derivative and the divergence of appropriate `easy' random variables.
• The student can explain what a BSDE is, give conditions under which it is well-posed, and sketch the proof of well-posedness.
• The student is able to apply the theory of BSDEs in hedging problems and control theory in finance.
• The student is able to explain the relation between BSDEs and non-linear PDEs.
• The student is able to apply the theory of Malliavin calculus on forward stochastic differential equations and BSDEs.
• The student is able to introduce discretization techniques for BSDEs and use the Malliavin calculus to compute convergence rates for the approximation.
• The student is able to solve BSDEs numerically and use this to compute hedging positions in financial contracts.

Teaching methods

• Lecture
• Self-study
• Computer lab session/practical training

The theory is explained during the lectures. Exercises (both computer and theoretical) help the students master the material during self-study.

Learning activities

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

You need a passing grade (6 or higher) for the oral exams in order to pass the course.

Inspection of assessed work

Homework is graded and returned to students. If desired, the students can request a meeting to reflect on the oral exam(s).

Assignments

Homework may be handed in in pairs.

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

Honours information

Not applicable.

Contact information

Coordinator

• dr. A. Khedher

Staff

• dr. A. Khedher
• dr. S. Cox