Stochastic Differential Equations (SDEs) have become a standard tool to model differential equation systems subject to noise. Applications range from Neuroscience or Polymeric Chemistry to Finance or Mechanical Engineering. Treating practical problems requires analytic techniques to understand and investigate properties of SDEs and stochastic numerical methods to compute quantities of interest, where the latter and the former often go hand in hand. During this course we will discuss efficiency of Monte Carlo methods for SDEs and how to improve it by variance reduction techniques and Multi-level Monte Carlo, and we will explore structural properties of SDEs and numerical methods that preserve these properties. Further, we will discuss jump-diffusion and Stratonovich SDEs and how to adapt SDE numerics to treat these.

Contents

• Monte Carlo methods for SDEs: Weak approximations for SDEs and their efficiency
• Variance reduction techniques and Multi-level Monte Carlo
• Structural properties for SDEs
• Numerical methods that preserve structural properties, in particular splitting methods
• Jump-diffusion and Stratonovich SDEs and their numerics

Prerequisistes

Standard analysis and linear algebra, Numerical analysis of ordinary differential equations (including the corresponding programming skills), Basic probability theory, fundamentals of the concepts of SDEs and how to develop and analyse numerical methods for their simulation.

Schedule

The course schedule should involve 9 lectures and 5 exercises:

• Week 45:  2 Lectures
• Week 46:  1 Lecture, 1 Exercise
• Week 47:  1 Lecture, 1 Exercise
• Week 48:  1 Lecture, 1 Exercise
• Week 49:  2 Lectures, 1 Exercise
• Week 50:  1 Lecture
• Week 51: 1 Lecture, 1 Exercise

Course responsible

Philipp Birken

Teacher

Evelyn Buckwar

Registration

Registration closes 26 October 2022

Registration is now closed.