# Martingales and Monte-Carlo Methods

## Objectives

At the end of this module, the student will have understood and be able to explain (main concepts) :
- conditional expectation, information filtering, the main properties of
martingales and their applications in modelling.
- Basics of simulation of random variables, Monte-Carlo methods, Markov
Chain Monte-Carlo (MCMC) Methods, Stochastic Algorithms (Robbins-Monro,
Simulated annealing)
The student will be able to:
· Compute a conditional expectation and to use its main properties, understand
the different types of probabilistic convergences, to show that a random process is
a martingale, to use decomposition, stopping, control and convergence theorems for
martingales.
· Simulate a random variable by different methods, use probabilistic
algorithms for solving numerical problems (computing expectations, searching for
extrema), choose appropriate techniques for variance reduction and error estimation.

## Needed prerequisite

Probability and Statistics
Further lectures in probability

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...