MathMods :: Joint MSc

• This course addresses the basic methods used for simulating random variables and implementing Monte-Carlo and Quasi Monte-Carlo methods.

• Simulation of stochastic processes used in neuroscience and mathematical finance, such as Brownian motion and solutions to stochastic differential equations, will be addressed.

• The course will introduce sampling methods in finite dimension, discretization of diffusion processes, strong and weak errors. Exercices will be done on paper and on the computer (using Python language)

Geometric Statistics by Prof. Khazhgali Kozhasov

Statistical concepts like principal component analysis, (empirical) mean or covariance (matrix) are inherent to data and probability distributions living in linear spaces. Geometric statistics aims at providing tools for analysing data that populate (possibly) non-linear spaces such as manifolds. As the notion of metric is essential for this goal, Riemanniangeometry provides a solid ground for the theory. In the course we are going to introduce necessary geometric results, give essentials on probability distributions and then discuss “nonlinear” generalizations of some classical concepts from statistics. The exposition will be accompanied by numerous examples with a view towards applications. Familiarity with calculus on manifolds or basic differential geometry is recommended.

Statistical learning from and on graphs by Prof. Marco Cornelli

This course explores in detail some areas in machine/statistical learning that either use graphs in order to learn from data leaving in Euclidean spaces (e.g. Bayesian networks, spectral clustering) or directly model graph data (e.g. social network analysis, graph neural networks). In all model based approaches that we consider, maximum likelihood inference is adopted for the numerical estimation of the model parameters, with an important focus on EM and variational EM algorithms. Both R and Python will be employed to illustrate some implementations/applications of the proposed approaches and allow the student to become acquainted with the related libraries.

Probabilistic numerical methods are widely used in machine learning algorithms as well as in mathematical finance for pricing financial derivatives and computing strategies. The course will present the basic methods used for simulating random variables and implementing the Monte-Carlo methods. Simulation in Scilab of stochastic processes used in mathematical finance, such as Brownian motion and solutions to stochastic differential equations, will be discussed as well

Sampling methods in finite dimension. Discretization of diffusion processes; strong and weak errors. Monte-Carlo methods for option pricing, variance reduction, control variates method, importance sampling. Monte-Carlo methods in risk management.

The course provides the basic knowledge in stochastic control, programming principle, dynamic programming equation, Hamilton-Jacobi-Bellman equation, control for counting processes. A second part addresses the theory of mean-field models. Applications to finance are considered.

Programming principle, dynamic programming equation, Hamilton-Jacobi-Bellman equation, control for counting processes. Mean-field models as many particle limits. Applications to finance.