GIMAS7AB PROBABILITY THEORY STOCHASTIC PROCESSES

GIMAS7AB

Probability theory for stochastic processes

ECTS Credits: 4

Duration: 42 hours

Semester: S7

Person(s) in charge:

Denis VILLEMONAIS, Associate Professor, denis.villemonais@mines-nancy.univ-lorraine.fr

Keywords: Probabilities

Prerequisites: Basics in functional analysis and probabilities

Objective:

present certain stochastic processes

Program and contents:

Objectives

This course presents certain stochastic processes and, more particularly, it introduces the tools which are necessary to understand mathematical models in finance. It will also be useful for better understanding the stochastic methods used for modelling all phenomena containing random variables.

Content

• Gaussian random variables: the different analyses of these random variables.
• Gaussian random vectors: analysis via a covariance matrix or density, convergence toward the normal law and application of the chi-square test.
• Condition number: definition of conditional expectation, link between conditional expectation and projection,  definition of conditional laws.
• Martingales: definition of stopping times and martingales, stopping properties of a martingale, convergence of several martingales.
• Markov chains: definition and examples, canonical chains, Markov properties and stopping times, description of the chain by using the potential, invariant measure and convergence of the chain toward this measure.

Assessment methods

Two tests, continuous monitoring, project

Abilities: 

Levels

Description and operational vocabulary

Know 

Understand 

Apply 

Analyze 

Summarize

Assess

Evaluation:

• Written test

• Continuous assessment

• Oral presentation

• Project

• Written report