logo Insalogo Insa

Advanced probability and Monte Carlo methods

Presentation

Programme (detailed contents):

-          Conditional expectation, filtration, martingale, submartingale and supermartingale, Doob’s theorem, optional stopping theorem, convergence theorems, law of large numbers and central limit theorem for martingales. 

-          Background on deterministic gradient descent, Introduction to Robbins-Monro algorithms and links with classical results (Law of Large Numbers), Robbins-Siegmund Lemma, Robbins-Monro Convergence Theorems, Applications (Two-Armed Bandit, quantile, quantization, Linear Regression in high dimension).

-          Generation of random numbers, simulation by inversion of the distribution function, by the reject method and by some specific methods, Monte-Carlo Methods (convergence, rate of convergence, variance reduction by using different methods).

Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

-          The notion of conditional expectation, the main properties of martingales and their classical use in modelling,

-          Stochastic algorithms of Robbins-Monro type.

-          The fundamental principles of simulating random variables and Monte-Carlo methods.

 

 

 

The student will be able to:

-          To compute a conditional expectation, to show that a random process is a martingale, to use the various theorems (Doob, optional stopping and convergences), in particular for the maximum  likelihood estimation.

 

-          Build and study the convergence of stochastic optimization algorithms, apply these methods to different problems (quantile, quantization,…)

 

-          Simulate a random variable by different methods, use probabilistic,  choose appropriate techniques for variance reduction and error estimation.

Needed prerequisite

 A basic course on probabilities.

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...