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Probability and interactions
Probability courses in SMA
Contact person: Professor Juhan Aru
Bachelor (3rd semester):
- Probabilités
Bachelor (5th and 6th semesters):
- Advanced probability I : discrete stochastic processes
- Advanced probability II : continuous time processes
- Measure theory (optional)
Master:
- Can take both advanced probability courses from BA5/6
- Theory of stochastic calculus
- Stochastic PDEs (every second year)
- Advanced stochasticl analysis (every second year)
- Markov processes (every second year)
- Multiscale stochastic dynamics (every second year)
- Gaussian processes (every second year)
- Topics in random geometry (every second year)
More applied courses:
- Probabilistic models in modern AI
- Martingales in Financial Mathematics
- Statistical mechanics and Gibbs measures
- Foundation of probabilistic proof
- Numerical integration of stochastic differential equations
- Probabilistic methods in combinatorics
- Functional data analysis
PhD
- Several of the Master level courses
- And courses by invited professors (Gaussian free field and isomorphism theorems, Malliavin calculus, Liouville CFT, Interacting particle systems and SPDEs...)
Description :
All students take the 3rd semester course “Probabilités” during the 3rd semester.
The courses “Advanced probability I” and “Advanced probability II” build on the 3rd semester course “Probabilités” and take the theory further. They introduce important concepts related to the evolution of random phenomena over space and time. The first course emphasizes discrete processes, including Markov chains and martingales; it could be useful for students interested in statistics and Monte-Carlo methods. The second one studies continuous time processes, including Brownian motion. Both courses prepare for more specialized courses at the Master level. Additionally measure theory is a course that helps to build very solid foundations.
Among the master level courses “Théorie du calcul stochastique” is a first rigorous introduction to stochastic calculus for mathematics students , "Markov processes" discusses the general theory of Markov processes in greather depth, "Gaussian processes" gives an overview of the magic world of Gaussians in both low and high dimensions, "Introduction to SPDEs" and "Advanced stochastic analysis" introduce the studetns to the fast-developing field of stochastic PDEs and related topics etc...These are all more pure math in spirit.
There are also some more applied courses. For example “Martingales in Financial Mathematics” applies martingale methods and stochastic calculus in the context of mathematical finance. It has “Théorie du calcul stochastique” as a prerequisite. This course aims at introducing mathematics students to the main mathematical ideas that are used in modern financial mathematics. Students who would like to develop a significant background in financial mathematics should also consider taking statistics courses such as “Time Series” or “Quantitative Risk Management”, as well as courses in Financial Engineering, such as “Fixed income analysis”, “Financial econometrics” or “Advanced derivatives”.
Some other Math courses related to probability:
All the courses in statistics are related to probability.
The course “Numerical integration of stochastic differential equations” studies numerical methods for approximating stochastic processes. It has “Probabilités” and Numerical Analysis as prerequisites.
The course “Computational finance” presents numerical methods used in financial applications. Some background in stochastic processes and stochastic calculus are prerequisites.
The course “Probabilistic method” explores the use of randomness in discrete mathematics. It has “Probabilités” as a prerequisite.
Many courses in analysis like measure theory or ergodic theorems are strongly related to probability, as are sevaral courses on PDEs.
Related Minors :