Numerical Methods for SDE's - M2 - 8EC

1) Aim of the Course:

Stochastic Differential Equations (SDEs) are pivotal tools for modeling system dynamics under uncertainty. Their applications span a wide range of fields, including finance, physics, renewable energy, power generation optimization and forecasting, weather and climate modeling, biology, and engineering.
This course offers a comprehensive introduction to the numerical methods essential for solving problems involving SDEs. Emphasizing a solid theoretical foundation, it equips students with the analytical insight and practical skills needed to construct accurate and efficient computational solutions, and to analyze their convergence and numerical complexity. Key topics covered include:

  1. Numerical Approximation of SDEs: We will focus on Itô SDEs, exploring weak and strong approximations and their convergence. The Euler–Maruyama and Milstein schemes will be studied in detail—covering formulation, implementation, and error analysis. We will derive error estimates and verify them numerically.

  2. Computing expectations of SDEs solutions: We will formulate and analyze numerical methods for computing expectations of functionals of SDE solutions. Beginning with the Monte Carlo (MC) method and its error and complexity analysis, we will then discuss variance reduction techniques to improve efficiency—such as control and antithetic variates, quasi-MC, importance sampling (for rare event estimation), and multilevel MC methods. Each method will be analyzed theoretically and compared numerically.

  3. Introduction to Optimal Control for SDEs: Students will gain an understanding of stochastic control problems and explore numerical approaches to solve them.

  4. Connections to Machine Learning and Stochastic Optimization: We will explore how stochastic optimization and modern machine learning methods build upon numerical methods for SDEs. In particular, we will discuss how algorithms such as stochastic gradient descent can be viewed as discretizations of Langevin-type SDEs, and how diffusion-based generative models rely on numerical SDE solvers for sampling.

 

2) Prerequisites:

  • Measure theory, stochastic processes at the level of the course measure theoretic probability, see Chapter 1--8 of 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.



3) Rules about Homework/Exam and Grading:

The grading consists of two components: Homework problems and a written exam.

Homework Assignments: Approximately every two weeks, there will be one homework assignment. These assignments are to be completed in groups of (2-3) students. 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. Note that exercises which are not seriously submitted for the first round will not have the opportunity for a better grade. 

The homework has two purposes: it poses exercises on new mathematical concepts or numerical method and gives the opportunity to practice written solutions. This means that a solution with just formulas is not acceptable. The solution should resemble the lecture notes' presentation of an example and not the teacher's shortened version of it when he presents on the blackboard.

 

You can submit your homework solutions by emailing the course teaching team (instructor and TAs).

 

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.

Grading and 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.



4) Lecture notes/Literature: 

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

The book “An introduction to the numerical simulation of stochastic differential equations” by  Higham, D., & Kloeden, P. (link: https://epubs.siam.org/doi/book/10.1137/1.9781611976434)