30561 - STOCHASTIC PROCESSES AND SIMULATION IN NATURAL SCIENCES

Programming: basic Python, basic familiarity with numpy, fundamental theoretical computer science notions Maths: linear algebra, calculus, probability

The aim of the course is to introduce the theoretical and numerical tools for the analysis and simulation of stochastic and natural processes, which are ubiquitous in many of the program's other subjects (Finance, Physics, Statistical Machine Learning etc.)

• Stochastic simulation and Monte Carlo methods
• Markov chains, Markov Chain Monte Carlo
• Poisson processes and other continuous-time stochastic models
• Basics of numerical calculus
• Numerical methods for ordinary and partial differential equations

• Characterize and describe Monte Carlo and Markov Chain Monte Carlo methods
• Formulate probabilistic models based on Markov chains, Poisson processes and other continuous time processes
• Analyze the above stochastic processes using probability theory and other mathematical tools
• List and explain fundamental methods to solve numerically differential equations
• Recognize numerical issues and identify workaround strategies
• Estimate the computational cost of implementing all of the above

• Determine whether a Monte Carlo method is appropriate for a task, and if so choose the best approach
• Develop a Markov Chain Monte Carlo algorithm for a given problem
• Translate phenomena involving randomness and uncertainty into appropriate probabilistic models
• Characterize the average and long-run behavior of a given stochastic process
• Simulate a process described by a set of differential equations

• Face-to-face lectures

The teaching method is face-to-face lectures.

The written general exam will contain theoretical questions and exercises, intended to verify that the students have acquired both the basic mathematical knowledge (about MCMC, stochastic processes, differential equations) and the analytical skills to relate the different techniques to given problem instances.

The group project will consist in implementing from scratch a simulation or a numerical method for a problem that was not discussed in class. The students can demonstrate that they have internalized the theoretical aspects, that they can design a strategy and implement it in code.

The written exam will form 80% of the final grade, and the group project the remaining 20%. Both parts will have an individual threshold (to be determined) and the final grade will be the sum of the two grades.

Textbooks for this course have not been decided upon yet. A decision will be made before January 2023.