WiSe 23/24: Stochastics IV:
Luca Donati
Additional information / Pre-requisites
Prerequisite: Stochastics I, II, III. ???
Comments
Content
Stochastic processes are mathematical models used to describe the dynamics of random phenomena
and are widely applied in many disciplines ranging from physics, chemistry, biology, and economics.
During the course, students will learn both the theory underlying stochastic processes and advanced numerical
methods to solve problems with real applications.
Program
The first weeks of the course are devoted to review the basic concepts of stochastic analysis already studied in previous courses. The course continues with the study and application of methods for solving stochastic analysis problems. To this end, various Python notebooks will be presented. Note that Python knowledge is not required for the course.
- Introduction to Brownian motion (1 week).
- The master equation and the Fokker-Planck equation (1 week).
- Stochastic calculus and stochastic differential equations (1 week).
- Methods to solve stochastic differential equations (2 weeks).
- Methods to solve the Master equation (2 weeks):
- System size expansion
- Gillespie algorithm
- Methods to solve the Fokker-Planck equations (3 weeks):
- Markov State Models
- The Square Root Approximation (SqRA).
- Robust Perron Cluster Cluster Analysis (PCCA+) (2 weeks).
- Estimating transition rates by Neural Networks (2 weeks).
- The Kramers problem (1 week).
- The Generalized Langevin equation (1 week).
Suggested reading
• Risken, H., The Fokker-Planck Equation: Methods of Solution and Applications, 1984.
• Gardiner, C., Stochastic Methods: A Handbook for the Natural and Social Sciences, 2009.
• Oksendal, B., Stochastic Differential Equations: An Introduction with Applications, 2010.
• Lecture notes and suggested readings.
16 Class schedule
Regular appointments