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GIMAS8AA

Monte Carlo Method & Application to Random Processes

 

ECTS Credits: 2

Duration: 21 hours

 

Semester: S8

Person(s) in charge:

Rémi PEYRE, Associate Professor, remi.peyre@mines-nancy.univ-lorraine.fr

Keywords: Monte Carlo methods; Random processes; Simulation

Prerequisites: Probabiliy theory (M1 level); basics in MATLAB

Objective:

Mastering the Monte Carlo method and being able to simulate random processes

Program and contents:

The first part of this course introduces the Monte Carlo method, which consists in estimating a probabilistic expectation using simulations; this way, a deterministic quantity is computed by the means of a random device.

   In this part we will explain how to compute the desired value ; and also how to assess the confidence interval for the estimator we get—which is quite important too. Then, improving that confidence interval is worth: to do that, we will introduce the so-called variance reduction techniques. We will introduce four of these techniques: importance sampling, conditioning, control variable, and common random numbers.

   The second part of this course is an introduction to the study of random processes indexed by time: these processes are an important field in mathematical engineering; moreover it is a context where applying the Monte Carlo method is frequently needed.

   In this second part, our goal will be to understand in a concrete way what Markovian random processes are, and how one can simulate them numerically. So, we will deal with stochastic differential equations, as well as with jump processes. We will focus on the general ideas and the practical devices, rather than bothering with the underlying technicalities.

   A wide part of this course will be devoted to implementing the concepts into computers. The software used for that will be MATLAB.


 

Abilities: 

Levels

Description and operational verbs

Know 

Knowing how to compute the confidence interval for a Monte Carlo method

Knowing the main variance reduction techniques

Understand 

Understanding the principle of the Monte Carlo method

Understanding the meaning of a Brownian stochastic differential equation, possibly with jumps

Apply 

Implement numerically the Monte Carlo method on a computer

Simulating a Brownian stochastic differential equation, possibly with jumps

Analyze 

Being able to choose a relevant reduction variance technique for a given problem

Summarize

 

Assess

 

Evaluation:

  • Written test
  • Continuous assessment
  • Oral presentation
  • Project
  • Written report
  • Aucune étiquette