Numerical Methods for Stochastic Differential Equations - M2 - 8EC

Stochastic Differential Equations (SDEs) are pivotal tools for modeling systems dynamics affected by uncertainty. Their applications span a wide spectrum of fields, including finance, physics, renewable energy, power generation optimization and forecasting, weather and climate modeling, biology, and engineering. This course aims to offer a comprehensive introduction to the numerical methods essential for solving problems involving SDEs. Emphasizing the foundational understanding of SDEs, the course equips students with the theoretical insights and practical skills required to construct accurate and efficient computational solutions, and analyze their convergence and numerical complexity.

Key topics covered in this course include:

  1. Numerical Approximation of SDEs: Particularly, we will focus on Itô SDEs, exploring weak and strong approximations and their convergence. We will consider the Euler-Maruyama and Milstein approximations: Formulation, implementation, and error analysis. We will derive the different error estimates and show how to check them numerically.

  2. Computing expectations of SDEs solutions: We will formulate and analyze numerical methods for computing expectations of a functional of a solution to an SDE. We will first focus on the Monte Carlo (MC) method with error and complexity analysis. Then, we will cover variance reduction techniques to improve the MC method’s efficiency, such as control and antithetic variates, quasi-MC, importance sampling (which is relevant for rare events estimation), and multilevel MC methods. We will explain the construction of these methods, derive their error convergence and complexity results, and then compare them via numerical examples.

  3. Introduction to Optimal Control for SDEs: Students will gain a basic understanding of control problems in the context of SDEs and explore their numerical solutions.

Prerequisites:

  • Measure theory, stochastic processes at the level of the course measure theoretic probability, see lecture notes written by Peter Spreij.  It is recommended to take the course `Measure Theoretic Probability' before this course.
  • Introductory courses on numerical analysis and differential equations.
  • Experience with computer programming in Matlab or Python (or a similar language, for example, C++, ...): this will be needed for the assignments/homeworks.

Homework Assignments: Approximately every two weeks, there will be one homework assignment. These assignments are to be completed in pairs. Each group is responsible for submitting a written report for each assignment. Additionally, for each assignment, one group will be chosen to present their solution to the class. It is important that all group members understand the entire solution, as the presenting member may be asked questions by the instructors, who may also question the non-presenting group member. Groups are allowed one opportunity to revise and resubmit their solution within one week following the oral presentation for a possible improved grade.
Concerning the Exam: The final exam will be a closed book, in-class test held in the classroom. A tentative list of questions will be provided beforehand.
Retake Policy: Students must achieve a minimum score of 5.0 on the final exam to pass. The retake will be a closed book exam conducted in the classroom. Scores from homework assignments will still contribute to the final grade after the retake. 

The final numerical course grade will be determined using the following formula: Total Score = 0.6 ×
Final Exam Score + 0.4 × Average Homework Score.

Literature notes & lectures

The course will be based on lecture notes authored, among others, by the lecturer. The following references have been useful for preparing these notes and are recommended for further studies:
1- The notes “AN INTRODUCTION TO STOCHASTIC DIFFERENTIAL EQUATIONS” by Lawrence C. Evans. You can find the pdf file here.
2- The book “Numerical solution of stochastic differential equations” by Peter E. Kloeden and Eckhard Platen.
3- The book “Stochastic differential equations” by Bernt Øksendal. You can find the pdf file here.

(http://www.stat.ucla.edu/~ywu/research/documents/StochasticDifferentialEquations.pdf) 
Lecturers:
Chiheb Ben Hammouda (Utrecht University) (and possibly guest lecturers)