The course is based on the study of the main tools of probability theory that are used in finance and financial engineering. Although the applications are related to these domains and many examples will be studied in class, it is mainly about mathematics. The main objective of this course is to make the student comfortable with the mathematical concepts commonly used in financial engineering: sigma-field, martingale, stopping time, Brownian motion, stochastic integral, diffusion processes,and risk neutral measures. The course is divided into two main blocks: discrete and continuous models. Each of these blocks is again subdivided in two parts: a more theoretical section where the mathematical concepts are introduced and a second section in which the mathematical tools are used.

**
Chapter 1. Fundamental set, sigma-field, measurable
function, probability measure
**
Probability space

Exercises 1

**Chapter 2.
Stochastic processes, filtration, stopping time**
Stochastic
processes

Exercises 2

**Chapter 3. Conditional
expectation**
Conditionnal
expectation

Exercises 3

**Chapter 4. Discrete time
martingales **
Martingales

Exercises 4

**
Chapter 5. Introduction to risk neutral measures **
Binomial model

Exercises 5

**Chapter 6. Replication and
risk neutral measures **

Exercises 6

**Chapter. Snell Envelope **
American contingent claim

Exercises 7

**
Chapter 8. Convergence of Random Variable Series **
Convergence

Exercises 9

brownien1.xls brownien2.m brownien3.m

**Chapter 10. Ito's Integral **
Stochastic integral

Exercises 10

Int_stoch.xls

**Chapter 11. Stochastic
Differential Equations and Ito's Lemma **
SDE

Exercices 11

**Chapter 13. Applications du théorème de représentation des martingales
**
Martingale
representation theorem and hedging

Exercises 13

**Chapter 14 Autre application **
Bonds