Stochastic modeling

This EU will enable:

Manipulate the main results of probabilities from a practical perspective. Reinforce understanding of random phenomena with numerical illustrations. Introduce simulation methods using the Monte Carlo method for the numerical resolution of integration problems or the calculation of probabilities for complex events. Supplement knowledge of the main commonly used laws and their properties with a view to applications in inferential statistics and statistical tests covered in the Master's program.

This course follows on from the Probability Theory course and will build on the results seen in that course. It will supplement the theoretical and practical concepts of randomness in preparation for a Master's degree in Probability and/or Statistics. It will cover the following topics:

     Part I: Generating Randomness 

  - pseudo-random generators

  - Random variable simulations: distribution inversion method, rejection method, other distributions (Box-Muller method for simulating a normal distribution, mixtures, simulation of a Poisson random variable from the sum of independent exponential variables)

   - digital illustrations of the main results of the probability course: law of large numbers, Moivre-Laplace theorem.

     Part II: Monte Carlo Method 

   - Monte Carlo method for approximate calculation of integrals and variance reduction: antithetical variables, control variables, preferential sampling. Application to the simulation of rare events.

     Part III: Supplements

- Gaussian vectors and their link to standard inferential statistics laws (Student's t-test, chi-square test) applied to the construction of confidence intervals.

- Simple random walks, problem of maximizing the expectation of a cost function in financial mathematics.

 Practical work for the implementation of numerical methods will be carried out using R software.

The analysis and probability courses in L1, L2, and L3, in particular:

- HAX506X Probability Theory

Recommended prerequisites:

Basics of programming in R

Hourly volumes:

            CM: 18

            TD: 15

            TP: 12

            Land: -