Stochastic modeling

This EU will allow to:

To manipulate the main results of probability from a practical point of view. Reinforce the understanding of random phenomena with numerical illustrations. To introduce simulation methods using the Monte-Carlo method for the numerical solution of integration problems or the calculation of the probability of complex events. To complete the knowledge of the main usual laws and their properties in view of the applications to inferential statistics and statistical tests approached in Master.

This course follows the Probability theory course and will build on the results seen in this course. It will allow to complete the notions in theory and practice of randomness to approach a Master's degree in Probability and/or Statistics. It will cover the following points:

     Part I: Generating the hazard 

  - pseudo-random generators

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

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

     Part II: Monte-Carlo method 

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

     Part III: Supplements

- Gaussian vectors and link with the usual laws of inferential statistics (student, chi2) application to the construction of confidence intervals.

- simple random walks, problem of maximizing the expectation of a cost function in financial math.

 The practical work for the implementation of the numerical methods will be done using the R software.

Analysis and probability courses in L1, L2 and L3, in particular:

- HAX506X Probability Theory

Recommended prerequisites:

Basic programming in R

Hourly volumes:

            CM : 18

            TD : 15

            TP : 12

            Land: -